Finite Group Databasebeta

← #72277326

Subgroups — #72277326

Order 2688.

1240 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10621518C211yesmaximal, characteristic
2ΓL(1,𝔽8)×#417C211yesmaximal, characteristic, complemented
3C2×#6411⋊C3C211yesmaximal, characteristic, complemented
4C2×#6399⋊C3C211yesmaximal, characteristic, complemented
5C2×#6397⋊C3C211yesmaximal, characteristic, complemented
6#89760⋊C3C211yesmaximal, characteristic, complemented
7#89738⋊C3C211yesmaximal, characteristic, complemented
8#11728059C311yesmaximal, characteristic, complemented, Hall
9C2×#41×ΓL(1,𝔽8)C2211yescharacteristic
10#10621518—22no—
11#10621518—22no—
12#10621518—22no—
13#10621518C414yescomplemented
14C2×#66659—21no—
15C2×D4×AGL(1,𝔽7)—22no—
16C2×D4×AGL(1,𝔽7)—21no—
17C2×#1157⋊C3—21no—
18C2×#1157⋊C3C2211yescharacteristic
19C22×C4⋊AGL(1,𝔽7)—21no—
20C22×C4⋊AGL(1,𝔽7)C2211yescharacteristic
21C2×M16×ΓL(1,𝔽8)C2211yescharacteristic, complemented
22C2×C4⋊C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
23C2×(C2×AGL(1,𝔽7))⋊C4—22no—
24C2×C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic, complemented
25C2×#1136⋊C3C2211yescharacteristic, complemented
26ΓL(1,𝔽8)×#103—21no—
27ΓL(1,𝔽8)×#102—21no—
28#6411⋊C3—22no—
29#6399⋊C3—22no—
30#6397⋊C3—22no—
31#6384⋊C3—21no—
32#6383⋊C3—21no—
33C2×#6553C2×C311yescharacteristic
34C7×#417C2×C311yescharacteristic, complemented
35C2×#6411C2×C311yescharacteristic, complemented
36C2×#6399C2×C311yescharacteristic, complemented
37C2×#6397C2×C311yescharacteristic, complemented
38#89760C2×C311yescharacteristic, complemented
39#89738C2×C311yescharacteristic, complemented
40C3×#1792—71nomaximal, Hall
41C23×AGL(1,𝔽7)—22no—
42C23×AGL(1,𝔽7)D412yes—
43#41×ΓL(1,𝔽8)—21no—
44#41×ΓL(1,𝔽8)—21no—
45#41×ΓL(1,𝔽8)C2×C412yes—
46#41×ΓL(1,𝔽8)D412yes—
47C2×Q8×ΓL(1,𝔽8)D411yescharacteristic
48C2×D4×ΓL(1,𝔽8)—21no—
49C2×D4×ΓL(1,𝔽8)D411yescharacteristic
50C22×C4×ΓL(1,𝔽8)—21no—
51C22×C4×ΓL(1,𝔽8)—21no—
52C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
53C2×(C22⋊D7)⋊C3—22no—
54C2×(C22⋊D7)⋊C3—21no—
55C22×ΓL(1,𝔽8)⋊C4—21no—
56#66659—24no—
57#66659—24no—
58D4×AGL(1,𝔽7)—28no—
59D4×AGL(1,𝔽7)—24no—
60D4×AGL(1,𝔽7)—28no—
61D4×AGL(1,𝔽7)—24no—
62#1157⋊C3—24no—
63#1157⋊C3—21no—
64#1157⋊C3—21no—
65#1157⋊C3—24no—
66#1157⋊C3C2×C412yes—
67#1157⋊C3D412yes—
68C2×C4⋊AGL(1,𝔽7)—22no—
69C2×C4⋊AGL(1,𝔽7)—24no—
70C2×C4⋊AGL(1,𝔽7)—21no—
71C2×C4⋊AGL(1,𝔽7)—21no—
72C2×C4⋊AGL(1,𝔽7)—21no—
73C2×C4⋊AGL(1,𝔽7)—21no—
74C2×C4⋊AGL(1,𝔽7)—24no—
75C2×C4⋊AGL(1,𝔽7)C2×C412yes—
76C2×C4⋊AGL(1,𝔽7)D412yes—
77C2×C4⋊AGL(1,𝔽7)D411yescharacteristic
78C2×C4×AGL(1,𝔽7)—22no—
79C2×C4×AGL(1,𝔽7)—21no—
80C2×ΓL(1,𝔽8)⋊Q8D411yescharacteristic
81M16×ΓL(1,𝔽8)—21no—
82M16×ΓL(1,𝔽8)—21no—
83C2×C8×ΓL(1,𝔽8)—21no—
84C4⋊C4×ΓL(1,𝔽8)—21no—
85C4⋊C4×ΓL(1,𝔽8)—21no—
86(C2×AGL(1,𝔽7))⋊C4—44no—
87C4×ΓL(1,𝔽8)⋊C4—21no—
88C4×ΓL(1,𝔽8)⋊C4—21no—
89#1136⋊C3—21no—
90#1136⋊C3—21no—
91C2×ΓL(1,𝔽8)⋊C8—21no—
92C2×C7×#41C22×C311yescharacteristic
93#6553—22no—
94#6553—22no—
95#6553—22no—
96#6553C3×C414yescomplemented
97C2×#1161—21no—
98C2×D4×D7—22no—
99C2×D4×D7—21no—
100C2×#1157—21no—
101C2×#1157C22×C311yescharacteristic
102C22×D28—21no—
103C22×D28C22×C311yescharacteristic
104C2×C7×M16C22×C311yescharacteristic, complemented
105C2×C7×C4⋊C4C22×C311yescharacteristic, complemented
106C2×(C2×D7)⋊C4—22no—
107C2×C4×Dic7C22×C311yescharacteristic, complemented
108C2×#1136C22×C311yescharacteristic, complemented
109C7×#103—21no—
110C7×#102—21no—
111#6411—22no—
112#6399—22no—
113#6397—22no—
114#6384—21no—
115#6383—21no—
116C2×C3×H(5,𝔽2)—71no—
117C2×C3×#104—71no—
118C2×C3×#104—71no—
119C3×#417—71no—
120C3×#417—71no—
121C2×C3×#100—71no—
122C3×#390—71no—
123C23×ΓL(1,𝔽8)—21no—
124C23×ΓL(1,𝔽8)C2×D411yescharacteristic
125C22×AGL(1,𝔽7)—22no—
126C22×AGL(1,𝔽7)—21no—
127C22×AGL(1,𝔽7)—28no—
128C22×AGL(1,𝔽7)—22no—
129C22×AGL(1,𝔽7)—42no—
130C22×AGL(1,𝔽7)—42no—
131C22×AGL(1,𝔽7)—82no—
132C22×AGL(1,𝔽7)—82no—
133C22×AGL(1,𝔽7)C22⋊C414yes—
134Q8×ΓL(1,𝔽8)—21no—
135Q8×ΓL(1,𝔽8)C22⋊C412yescomplemented
136D4×ΓL(1,𝔽8)—22no—
137D4×ΓL(1,𝔽8)—22no—
138D4×ΓL(1,𝔽8)—21no—
139D4×ΓL(1,𝔽8)C22⋊C412yescomplemented
140C2×C4×ΓL(1,𝔽8)—21no—
141C2×C4×ΓL(1,𝔽8)—22no—
142C2×C4×ΓL(1,𝔽8)—22no—
143C2×C4×ΓL(1,𝔽8)—42no—
144C2×C4×ΓL(1,𝔽8)—41no—
145C2×C4×ΓL(1,𝔽8)C22×C411yescharacteristic
146C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
147C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
148C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
149C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
150C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
151(C22⋊D7)⋊C3—28no—
152(C22⋊D7)⋊C3—22no—
153(C22⋊D7)⋊C3—22no—
154C2×ΓL(1,𝔽8)⋊C4—21no—
155C2×ΓL(1,𝔽8)⋊C4—22no—
156C2×ΓL(1,𝔽8)⋊C4—41no—
157C2×ΓL(1,𝔽8)⋊C4—41no—
158C4⋊AGL(1,𝔽7)—28no—
159C4⋊AGL(1,𝔽7)—22no—
160C4⋊AGL(1,𝔽7)—22no—
161C4⋊AGL(1,𝔽7)—22no—
162C4⋊AGL(1,𝔽7)—21no—
163C4⋊AGL(1,𝔽7)—22no—
164C4⋊AGL(1,𝔽7)C22⋊C412yescomplemented
165C4×AGL(1,𝔽7)—28no—
166C4×AGL(1,𝔽7)—22no—
167C4×AGL(1,𝔽7)—22no—
168ΓL(1,𝔽8)⋊Q8—21no—
169ΓL(1,𝔽8)⋊Q8C22⋊C412yescomplemented
170C8×ΓL(1,𝔽8)—41no—
171ΓL(1,𝔽8)⋊C8—41no—
172#1792—71noSylow
173C23×D7—22no—
174C23×D7C3×D412yes—
175C7×#41—21no—
176C7×#41—21no—
177C7×#41C2×C3×C412yes—
178C7×#41C3×D412yes—
179C2×C7×Q8C3×D411yescharacteristic
180C2×C7×D4—21no—
181C2×C7×D4C3×D411yescharacteristic
182C22×C4×C7—21no—
183C22×C4×C7—21no—
184C22×C4×C7C23×C311yescharacteristic
185C2×C22⋊D7—22no—
186C2×C22⋊D7—21no—
187C22×Dic7—21no—
188#1161—24no—
189#1161—24no—
190D4×D7—28no—
191D4×D7—24no—
192D4×D7—28no—
193D4×D7—24no—
194#1157—24no—
195#1157—21no—
196#1157—21no—
197#1157—24no—
198#1157C2×C3×C412yes—
199#1157C3×D412yes—
200C2×D28—22no—
201C2×D28—24no—
202C2×D28—21no—
203C2×D28—21no—
204C2×D28—21no—
205C2×D28—21no—
206C2×D28—24no—
207C2×D28C2×C3×C412yes—
208C2×D28C3×D412yes—
209C2×D28C3×D411yescharacteristic
210C2×C4×D7—22no—
211C2×C4×D7—21no—
212C2×Dic14C3×D411yescharacteristic
213C7×M16—21no—
214C7×M16—21no—
215C2×C7×C8—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—21no—
218(C2×D7)⋊C4—44no—
219C4×Dic7—21no—
220C4×Dic7—21no—
221#1136—21no—
222#1136—21no—
223C2×C7⋊C8—21no—
224C3×H(5,𝔽2)—74no—
225C3×H(5,𝔽2)—142no—
226C3×H(5,𝔽2)—142no—
227C3×H(5,𝔽2)—142no—
228C2×C3×#41—71no—
229C2×C3×#41—71no—
230C2×C3×#41—141no—
231C2×C3×#41—141no—
232C22×C3×D4—71no—
233C22×C3×D4—141no—
234C22×C3×D4—141no—
235C22×C3×D4—142no—
236C2×C3×M16—71no—
237C2×C3×M16—71no—
238C2×C3×C4⋊C4—71no—
239C2×C3×C22⋊C4—142no—
240C2×C3×C42—71no—
241C3×#104—142no—
242C3×#104—142no—
243C3×#103—141no—
244C3×#103—141no—
245C3×#102—141no—
246C3×#102—141no—
247C3×#100—142no—
248C22×ΓL(1,𝔽8)—21no—
249C22×ΓL(1,𝔽8)—21no—
250C22×ΓL(1,𝔽8)—22no—
251C22×ΓL(1,𝔽8)—41no—
252C22×ΓL(1,𝔽8)—41no—
253C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
254C22×ΓL(1,𝔽8)#12011yescharacteristic
255C22×ΓL(1,𝔽8)#12011yescharacteristic
256C2×AGL(1,𝔽7)—24no—
257C2×AGL(1,𝔽7)—22no—
258C2×AGL(1,𝔽7)—42no—
259C2×AGL(1,𝔽7)—41no—
260C2×AGL(1,𝔽7)—42no—
261C2×AGL(1,𝔽7)—41no—
262C2×AGL(1,𝔽7)—44no—
263C2×AGL(1,𝔽7)—82no—
264C2×AGL(1,𝔽7)—88no—
265C4×ΓL(1,𝔽8)—22no—
266C4×ΓL(1,𝔽8)—81no—
267C4×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
268C4×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
269C4×ΓL(1,𝔽8)#12011yescharacteristic
270C4×ΓL(1,𝔽8)#12011yescharacteristic
271ΓL(1,𝔽8)⋊C4—22no—
272ΓL(1,𝔽8)⋊C4—42no—
273C2×H(5,𝔽2)—71no—
274C2×#104—71no—
275C2×#104—71no—
276#417—71no—
277#417AGL(1,𝔽7)11yescharacteristic, complemented
278C2×#100—71no—
279#390—71no—
280C23×C7—21no—
281C23×C7C2×C3×D411yescharacteristic
282C22×D7—22no—
283C22×D7—21no—
284C22×D7—28no—
285C22×D7—22no—
286C22×D7—42no—
287C22×D7—42no—
288C22×D7—82no—
289C22×D7—82no—
290C22×D7C3×C22⋊C414yes—
291C7×Q8—21no—
292C7×Q8C3×C22⋊C412yescomplemented
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—21no—
296C7×D4C3×C22⋊C412yescomplemented
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—42no—
301C2×C4×C7—41no—
302C2×C4×C7C22×C3×C411yescharacteristic
303C2×C4×C7C2×C3×D411yescharacteristic
304C2×C4×C7C2×C3×D411yescharacteristic
305C2×C4×C7C2×C3×D411yescharacteristic
306C2×C4×C7C2×C3×D411yescharacteristic
307C2×C4×C7C2×C3×D411yescharacteristic
308C22⋊D7—28no—
309C22⋊D7—22no—
310C22⋊D7—22no—
311C2×Dic7—21no—
312C2×Dic7—22no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315D28—28no—
316D28—22no—
317D28—22no—
318D28—22no—
319D28—21no—
320D28—22no—
321D28C3×C22⋊C412yescomplemented
322C4×D7—28no—
323C4×D7—22no—
324C4×D7—22no—
325Dic14—21no—
326Dic14C3×C22⋊C412yescomplemented
327C7×C8—41no—
328C7⋊C8—41no—
329C24×C3—72no—
330C24×C3—142no—
331C3×#41—72no—
332C3×#41—72no—
333C3×#41—72no—
334C3×#41—72no—
335C3×#41—141no—
336C3×#41—141no—
337C3×#41—141no—
338C3×#41—141no—
339C3×#41—144no—
340C3×#41—144no—
341C3×#41—144no—
342C3×#41—144no—
343C2×C3×Q8—71no—
344C2×C3×Q8—71no—
345C2×C3×D4—71no—
346C2×C3×D4—72no—
347C2×C3×D4—72no—
348C2×C3×D4—71no—
349C2×C3×D4—141no—
350C2×C3×D4—141no—
351C2×C3×D4—141no—
352C2×C3×D4—141no—
353C2×C3×D4—141no—
354C2×C3×D4—144no—
355C2×C3×D4—144no—
356C2×C3×D4—148no—
357C2×C3×D4—141no—
358C2×C3×D4—142no—
359C2×C3×D4—144no—
360C2×C3×D4—144no—
361C2×C3×D4—148no—
362C2×C3×D4—142no—
363C22×C3×C4—71no—
364C22×C3×C4—141no—
365C22×C3×C4—141no—
366C22×C3×C4—141no—
367C22×C3×C4—142no—
368C22×C3×C4—141no—
369C3×M16—141no—
370C3×M16—141no—
371C3×M16—141no—
372C3×M16—141no—
373C2×C3×C8—141no—
374C2×C3×C8—141no—
375C3×C4⋊C4—141no—
376C3×C4⋊C4—141no—
377C3×C22⋊C4—284no—
378C3×C42—141no—
379C3×C42—141no—
380C2×ΓL(1,𝔽8)—21no—
381C2×ΓL(1,𝔽8)—21no—
382C2×ΓL(1,𝔽8)—42no—
383C2×ΓL(1,𝔽8)#37911yescharacteristic
384C2×ΓL(1,𝔽8)#45311yescharacteristic
385C2×ΓL(1,𝔽8)#45511yescharacteristic
386AGL(1,𝔽7)—44no—
387AGL(1,𝔽7)—42no—
388H(5,𝔽2)—74no—
389H(5,𝔽2)—142no—
390H(5,𝔽2)—142no—
391H(5,𝔽2)—142no—
392C2×#41—71no—
393C2×#41—141no—
394C2×#41—141no—
395C2×#41C2×AGL(1,𝔽7)11yescharacteristic
396C22×D4—71no—
397C22×D4—141no—
398C22×D4—142no—
399C22×D4—141no—
400C2×M16—71no—
401C2×M16C2×AGL(1,𝔽7)11yescharacteristic, complemented
402C2×C4⋊C4C2×AGL(1,𝔽7)11yescharacteristic, complemented
403C2×C22⋊C4—142no—
404C2×C42—71no—
405#104—142no—
406#104—142no—
407#103—21no—
408#103—141no—
409#102—21no—
410#102—141no—
411#100—142no—
412C22×C7—21no—
413C22×C7—21no—
414C22×C7—22no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7C2×C3×C22⋊C411yescharacteristic
418C22×C7C3×#12011yescharacteristic
419C22×C7C3×#12011yescharacteristic
420C2×D7—24no—
421C2×D7—22no—
422C2×D7—42no—
423C2×D7—41no—
424C2×D7—42no—
425C2×D7—41no—
426C2×D7—44no—
427C2×D7—82no—
428C2×D7—88no—
429C4×C7—22no—
430C4×C7—81no—
431C4×C7C2×C3×C22⋊C411yescharacteristic
432C4×C7C2×C3×C22⋊C411yescharacteristic
433C4×C7C3×#12011yescharacteristic
434C4×C7C3×#12011yescharacteristic
435Dic7—22no—
436Dic7—42no—
437C23×C3—74no—
438C23×C3—71no—
439C23×C3—141no—
440C23×C3—141no—
441C23×C3—142no—
442C23×C3—142no—
443C23×C3—148no—
444C23×C3—282no—
445C23×C3—282no—
446C23×C3—562no—
447C23×C3—562no—
448C3×Q8—72no—
449C3×Q8—72no—
450C3×Q8—141no—
451C3×Q8—141no—
452C3×D4—72no—
453C3×D4—72no—
454C3×D4—141no—
455C3×D4—142no—
456C3×D4—142no—
457C3×D4—142no—
458C3×D4—141no—
459C3×D4—142no—
460C3×D4—142no—
461C3×D4—142no—
462C3×D4—142no—
463C3×D4—142no—
464C3×D4—148no—
465C3×D4—148no—
466C2×C3×C4—71no—
467C2×C3×C4—71no—
468C2×C3×C4—71no—
469C2×C3×C4—71no—
470C2×C3×C4—71no—
471C2×C3×C4—71no—
472C2×C3×C4—142no—
473C2×C3×C4—142no—
474C2×C3×C4—141no—
475C2×C3×C4—142no—
476C2×C3×C4—142no—
477C2×C3×C4—141no—
478C2×C3×C4—148no—
479C2×C3×C4—142no—
480C2×C3×C4—281no—
481C2×C3×C4—282no—
482C2×C3×C4—281no—
483C2×C3×C4—281no—
484C3×C8—281no—
485C3×C8—281no—
486ΓL(1,𝔽8)#179211yescharacteristic, complemented, Hall
487C24—72no—
488C24—142no—
489#41—21no—
490#41—21no—
491#41—72no—
492#41—72no—
493#41—144no—
494#41—144no—
495#41—144no—
496#41—144no—
497#41—141no—
498#41—141no—
499#41C4×AGL(1,𝔽7)12yes—
500#41C4⋊AGL(1,𝔽7)12yes—
501C2×Q8—71no—
502C2×Q8(C22⋊D7)⋊C311yescharacteristic
503C2×D4—21no—
504C2×D4—72no—
505C2×D4—71no—
506C2×D4—72no—
507C2×D4—144no—
508C2×D4—142no—
509C2×D4—144no—
510C2×D4—148no—
511C2×D4—148no—
512C2×D4—144no—
513C2×D4—142no—
514C2×D4—141no—
515C2×D4—144no—
516C2×D4—141no—
517C2×D4—141no—
518C2×D4—141no—
519C2×D4—141no—
520C2×D4(C22⋊D7)⋊C311yescharacteristic
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—142no—
524C22×C4—141no—
525C22×C4—141no—
526C22×C4C22×AGL(1,𝔽7)11yescharacteristic
527M16—21no—
528M16—21no—
529M16—141no—
530M16—141no—
531C2×C8—21no—
532C2×C8—141no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C22⋊C4—284no—
536C42—141no—
537C42—141no—
538C2×C7—21no—
539C2×C7—21no—
540C2×C7—42no—
541C2×C7C3×#37911yescharacteristic
542C2×C7C3×#45311yescharacteristic
543C2×C7C3×#45511yescharacteristic
544D7—44no—
545D7—42no—
546C22×C3—71no—
547C22×C3—71no—
548C22×C3—71no—
549C22×C3—141no—
550C22×C3—141no—
551C22×C3—142no—
552C22×C3—142no—
553C22×C3—144no—
554C22×C3—281no—
555C22×C3—281no—
556C22×C3—282no—
557C22×C3—282no—
558C22×C3—281no—
559C22×C3—281no—
560C22×C3—284no—
561C22×C3—562no—
562C22×C3—568no—
563C3×C4—71no—
564C3×C4—71no—
565C3×C4—71no—
566C3×C4—71no—
567C3×C4—142no—
568C3×C4—142no—
569C3×C4—282no—
570C3×C4—561no—
571C23—21no—
572C23—74no—
573C23—148no—
574C23—141no—
575C23—142no—
576C23—142no—
577C23—282no—
578C23—282no—
579C23—562no—
580C23—562no—
581C23D4×AGL(1,𝔽7)11yescharacteristic
582Q8—21no—
583Q8—72no—
584Q8—141no—
585Q8(C2×AGL(1,𝔽7))⋊C412yescomplemented
586D4—22no—
587D4—22no—
588D4—21no—
589D4—72no—
590D4—148no—
591D4—148no—
592D4—142no—
593D4—142no—
594D4—142no—
595D4—141no—
596D4—142no—
597D4—142no—
598D4—142no—
599D4(C2×AGL(1,𝔽7))⋊C412yescomplemented
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—148no—
606C2×C4—141no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4C2×C4×AGL(1,𝔽7)11yescharacteristic
613C2×C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
614C2×C4D4×AGL(1,𝔽7)11yescharacteristic
615C2×C4D4×AGL(1,𝔽7)11yescharacteristic
616C2×C4D4×AGL(1,𝔽7)11yescharacteristic
617C2×C4C2×(C22⋊D7)⋊C311yescharacteristic
618C8—41no—
619C8—281no—
620C7C3×#179211yescharacteristic, complemented, Sylow
621C2×C3—71no—
622C2×C3—71no—
623C2×C3—71no—
624C2×C3—141no—
625C2×C3—141no—
626C2×C3—282no—
627C2×C3—284no—
628C2×C3—282no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—142no—
635C22—144no—
636C22—284no—
637C22—282no—
638C22—281no—
639C22—282no—
640C22—281no—
641C22—562no—
642C22—568no—
643C22C22⋊C4×AGL(1,𝔽7)11yescharacteristic
644C22#6445⋊C311yescharacteristic
645C22#6500⋊C311yescharacteristic
646C4—22no—
647C4—81no—
648C4—142no—
649C4—282no—
650C4C22⋊C4×AGL(1,𝔽7)11yescharacteristic
651C4#6445⋊C311yescharacteristic
652C4C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
653C4#6500⋊C311yescharacteristic
654C3—71noSylow
655C2—21no—
656C2—21no—
657C2—42no—
658C2—284no—
659C2—282no—
660C2#89727⋊C311yescharacteristic
661C2#89970⋊C311yescharacteristic
662C2#89974⋊C311yescharacteristic
subgroupG/H
centerC22#6500⋊C3
commutatorC2×C4×C7C22×C3×C4
FrattiniC22×C4C22×AGL(1,𝔽7)
FittingC7×#417C2×C3
socleC22×C7C3×#120
prime radicalC2×#10621518C2
cosocleC22×C4×C7C23×C3
hypercenter#417AGL(1,𝔽7)
hypocenterC7C3×#1792
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×#417C2semidirect
3. C2×#6411⋊C3C2semidirect
4. C2×#6399⋊C3C2semidirect
5. C2×#6397⋊C3C2semidirect
6. #89760⋊C3C2semidirect
7. #89738⋊C3C2semidirect
8. #11728059C3semidirect
13. #10621518C4semidirect
21. C2×M16×ΓL(1,𝔽8)C22semidirect
22. C2×C4⋊C4×ΓL(1,𝔽8)C22semidirect
24. C2×C4×ΓL(1,𝔽8)⋊C4C22semidirect
25. C2×#1136⋊C3C22semidirect
34. C7×#417C2×C3semidirect
35. C2×#6411C2×C3semidirect
36. C2×#6399C2×C3semidirect
37. C2×#6397C2×C3semidirect
38. #89760C2×C3semidirect
39. #89738C2×C3semidirect
96. #6553C3×C4semidirect
104. C2×C7×M16C22×C3semidirect
105. C2×C7×C4⋊C4C22×C3semidirect
107. C2×C4×Dic7C22×C3semidirect
108. C2×#1136C22×C3semidirect
135. Q8×ΓL(1,𝔽8)C22⋊C4semidirect
139. D4×ΓL(1,𝔽8)C22⋊C4semidirect
164. C4⋊AGL(1,𝔽7)C22⋊C4semidirect
169. ΓL(1,𝔽8)⋊Q8C22⋊C4semidirect
277. #417AGL(1,𝔽7)semidirect
292. C7×Q8C3×C22⋊C4semidirect
296. C7×D4C3×C22⋊C4semidirect
321. D28C3×C22⋊C4semidirect
326. Dic14C3×C22⋊C4semidirect
401. C2×M16C2×AGL(1,𝔽7)semidirect
402. C2×C4⋊C4C2×AGL(1,𝔽7)semidirect
486. ΓL(1,𝔽8)#1792semidirect
585. Q8(C2×AGL(1,𝔽7))⋊C4semidirect
599. D4(C2×AGL(1,𝔽7))⋊C4semidirect
620. C7C3×#1792semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#106215182
ΓL(1,𝔽8)×#4172
C2×#6411⋊C32
C2×#6399⋊C32
C2×#6397⋊C32
#89760⋊C32
#89738⋊C32
#117280593
C3×#17927