Finite Group Databasebeta

← #72276673

Subgroups — #72276673

Order 2688.

1196 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2C2×#10620789C211yesmaximal, characteristic
3#13052147C211yesmaximal, characteristic, complemented
4C2×#6380⋊C3C212yesmaximal, complemented
5#89322⋊C3C212yesmaximal, complemented
6#11729297C311yesmaximal, characteristic, complemented, Hall
7C25×ΓL(1,𝔽8)C2211yescharacteristic
8C23×C4×ΓL(1,𝔽8)—21no—
9C22×(C22⋊D7)⋊C3C412yescomplemented
10C22×(C22⋊D7)⋊C3C2211yescharacteristic
11C2×C22⋊C4×ΓL(1,𝔽8)—21no—
12C2×C22⋊C4×ΓL(1,𝔽8)—24no—
13C2×C22⋊C4×ΓL(1,𝔽8)C2212yes—
14#10620789—21no—
15#10620789—22no—
16#10620789—21no—
17C2×(C22⋊Dic7)⋊C3—22no—
18C2×(C22⋊Dic7)⋊C3C412yescomplemented
19C2×(C22⋊Dic7)⋊C3C2212yes—
20C2×(C22⋊Dic7)⋊C3C2211yescharacteristic
21#6380⋊C3—24no—
22C22×C7×C22⋊C4C2×C311yescharacteristic, complemented
23C2×#6516C2×C311yescharacteristic
24#90019C2×C311yescharacteristic, complemented
25C2×#6380C2×C312yescomplemented
26#89322C2×C312yescomplemented
27C3×#1898—71nomaximal, Hall
28C24×ΓL(1,𝔽8)—22no—
29C24×ΓL(1,𝔽8)—22no—
30C24×ΓL(1,𝔽8)—44no—
31C24×ΓL(1,𝔽8)C2×C412yes—
32C24×ΓL(1,𝔽8)D411yescharacteristic
33C24×ΓL(1,𝔽8)D412yes—
34C24×ΓL(1,𝔽8)D411yescharacteristic
35C24×ΓL(1,𝔽8)C2311yescharacteristic
36C23×AGL(1,𝔽7)C2×C411yescharacteristic, complemented
37C22×C4×ΓL(1,𝔽8)—24no—
38C22×C4×ΓL(1,𝔽8)—22no—
39C22×C4×ΓL(1,𝔽8)—44no—
40C2×(C22⋊D7)⋊C3—22no—
41C2×(C22⋊D7)⋊C3—22no—
42C2×(C22⋊D7)⋊C3—22no—
43C2×(C22⋊D7)⋊C3—21no—
44C2×(C22⋊D7)⋊C3—21no—
45C2×(C22⋊D7)⋊C3—22no—
46C2×(C22⋊D7)⋊C3—44no—
47C22×ΓL(1,𝔽8)⋊C4—22no—
48C22×ΓL(1,𝔽8)⋊C4C2×C411yescharacteristic, complemented
49C22×ΓL(1,𝔽8)⋊C4C2×C412yescomplemented
50C22⋊C4×ΓL(1,𝔽8)—24no—
51C22⋊C4×ΓL(1,𝔽8)—22no—
52C22⋊C4×ΓL(1,𝔽8)D414yescomplemented
53(C22⋊Dic7)⋊C3—22no—
54(C22⋊Dic7)⋊C3—22no—
55(C22⋊Dic7)⋊C3—21no—
56(C22⋊Dic7)⋊C3—21no—
57(C22⋊Dic7)⋊C3—22no—
58(C22⋊Dic7)⋊C3—22no—
59(C22⋊Dic7)⋊C3—44no—
60C25×C7C22×C311yescharacteristic
61C23×C4×C7—21no—
62C22×C22⋊D7C3×C412yescomplemented
63C22×C22⋊D7C22×C311yescharacteristic
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4C22×C312yes—
67#6516—22no—
68#6516—21no—
69#6516—21no—
70C2×C22⋊Dic7—22no—
71C2×C22⋊Dic7C3×C412yescomplemented
72C2×C22⋊Dic7C22×C312yes—
73C2×C22⋊Dic7C22×C311yescharacteristic
74#6380—24no—
75C2×C3×#120—71no—
76C22×C3×C22⋊C4—71no—
77C2×C3×#99—72no—
78C3×#379—71no—
79C3×#342—72no—
80C23×ΓL(1,𝔽8)—24no—
81C23×ΓL(1,𝔽8)—21no—
82C23×ΓL(1,𝔽8)—21no—
83C23×ΓL(1,𝔽8)—24no—
84C23×ΓL(1,𝔽8)—24no—
85C23×ΓL(1,𝔽8)—24no—
86C23×ΓL(1,𝔽8)—21no—
87C23×ΓL(1,𝔽8)—24no—
88C23×ΓL(1,𝔽8)—21no—
89C23×ΓL(1,𝔽8)—42no—
90C23×ΓL(1,𝔽8)—42no—
91C23×ΓL(1,𝔽8)—42no—
92C23×ΓL(1,𝔽8)—42no—
93C23×ΓL(1,𝔽8)—48no—
94C23×ΓL(1,𝔽8)—48no—
95C23×ΓL(1,𝔽8)C22⋊C412yes—
96C23×ΓL(1,𝔽8)C22⋊C412yes—
97C23×ΓL(1,𝔽8)C22×C411yescharacteristic
98C23×ΓL(1,𝔽8)C2×D411yescharacteristic
99C23×ΓL(1,𝔽8)C2×D411yescharacteristic
100C23×ΓL(1,𝔽8)C2×D412yes—
101C23×ΓL(1,𝔽8)#4112yes—
102C22×AGL(1,𝔽7)—21no—
103C22×AGL(1,𝔽7)—22no—
104C22×AGL(1,𝔽7)—42no—
105C2×C4×ΓL(1,𝔽8)—24no—
106C2×C4×ΓL(1,𝔽8)—48no—
107C2×C4×ΓL(1,𝔽8)—44no—
108(C22⋊D7)⋊C3—44no—
109(C22⋊D7)⋊C3—44no—
110(C22⋊D7)⋊C3—41no—
111(C22⋊D7)⋊C3—42no—
112(C22⋊D7)⋊C3—41no—
113C2×ΓL(1,𝔽8)⋊C4—21no—
114C2×ΓL(1,𝔽8)⋊C4—22no—
115C2×ΓL(1,𝔽8)⋊C4—22no—
116C2×ΓL(1,𝔽8)⋊C4—42no—
117C2×ΓL(1,𝔽8)⋊C4—44no—
118C2×ΓL(1,𝔽8)⋊C4—42no—
119#1898—71noSylow
120C24×C7—22no—
121C24×C7—22no—
122C24×C7—44no—
123C24×C7C2×C3×C412yes—
124C24×C7C3×D411yescharacteristic
125C24×C7C3×D412yes—
126C24×C7C3×D411yescharacteristic
127C24×C7C23×C311yescharacteristic
128C23×D7C2×C3×C411yescharacteristic, complemented
129C22×C4×C7—24no—
130C22×C4×C7—22no—
131C22×C4×C7—44no—
132C2×C22⋊D7—22no—
133C2×C22⋊D7—22no—
134C2×C22⋊D7—21no—
135C2×C22⋊D7—21no—
136C2×C22⋊D7—22no—
137C2×C22⋊D7—22no—
138C2×C22⋊D7—44no—
139C22×Dic7—22no—
140C22×Dic7C2×C3×C411yescharacteristic, complemented
141C22×Dic7C2×C3×C412yescomplemented
142C7×C22⋊C4—24no—
143C7×C22⋊C4—22no—
144C7×C22⋊C4C3×D414yescomplemented
145C22⋊Dic7—22no—
146C22⋊Dic7—22no—
147C22⋊Dic7—21no—
148C22⋊Dic7—21no—
149C22⋊Dic7—22no—
150C22⋊Dic7—22no—
151C22⋊Dic7—44no—
152C25×C3—71no—
153C22×C3×D4—71no—
154C22×C3×D4—72no—
155C23×C3×C4—141no—
156C3×#120—142no—
157C3×#120—141no—
158C3×#120—141no—
159C2×C3×C22⋊C4—72no—
160C2×C3×C22⋊C4—71no—
161C2×C3×C22⋊C4—72no—
162C2×C3×C22⋊C4—72no—
163C2×C3×C22⋊C4—144no—
164C2×C3×C22⋊C4—141no—
165C2×C3×C22⋊C4—142no—
166C3×#99—144no—
167C22×ΓL(1,𝔽8)—24no—
168C22×ΓL(1,𝔽8)—24no—
169C22×ΓL(1,𝔽8)—22no—
170C22×ΓL(1,𝔽8)—22no—
171C22×ΓL(1,𝔽8)—22no—
172C22×ΓL(1,𝔽8)—22no—
173C22×ΓL(1,𝔽8)—22no—
174C22×ΓL(1,𝔽8)—22no—
175C22×ΓL(1,𝔽8)—22no—
176C22×ΓL(1,𝔽8)—22no—
177C22×ΓL(1,𝔽8)—42no—
178C22×ΓL(1,𝔽8)—42no—
179C22×ΓL(1,𝔽8)—48no—
180C22×ΓL(1,𝔽8)—48no—
181C22×ΓL(1,𝔽8)—42no—
182C22×ΓL(1,𝔽8)—42no—
183C22×ΓL(1,𝔽8)#9914yescomplemented
184C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
185C22×ΓL(1,𝔽8)C4×D412yes—
186C22×ΓL(1,𝔽8)#12011yescharacteristic
187C22×ΓL(1,𝔽8)#12112yes—
188C22×ΓL(1,𝔽8)#12311yescharacteristic
189C2×AGL(1,𝔽7)—41no—
190C2×AGL(1,𝔽7)—42no—
191C2×AGL(1,𝔽7)—82no—
192C4×ΓL(1,𝔽8)—44no—
193ΓL(1,𝔽8)⋊C4—41no—
194ΓL(1,𝔽8)⋊C4—42no—
195ΓL(1,𝔽8)⋊C4—82no—
196C2×#120—71no—
197C22×C22⋊C4AGL(1,𝔽7)11yescharacteristic, complemented
198C2×#99—72no—
199#379—71no—
200#342—72no—
201C23×C7—24no—
202C23×C7—21no—
203C23×C7—21no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—21no—
208C23×C7—24no—
209C23×C7—21no—
210C23×C7—42no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—48no—
215C23×C7—48no—
216C23×C7C3×C22⋊C412yes—
217C23×C7C3×C22⋊C412yes—
218C23×C7C22×C3×C411yescharacteristic
219C23×C7C2×C3×D411yescharacteristic
220C23×C7C2×C3×D411yescharacteristic
221C23×C7C2×C3×D412yes—
222C23×C7C3×#4112yes—
223C22×D7—21no—
224C22×D7—22no—
225C22×D7—42no—
226C2×C4×C7—24no—
227C2×C4×C7—48no—
228C2×C4×C7—44no—
229C22⋊D7—44no—
230C22⋊D7—44no—
231C22⋊D7—41no—
232C22⋊D7—42no—
233C22⋊D7—41no—
234C2×Dic7—21no—
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—42no—
238C2×Dic7—44no—
239C2×Dic7—42no—
240C24×C3—71no—
241C24×C3—72no—
242C24×C3—72no—
243C24×C3—71no—
244C24×C3—71no—
245C24×C3—71no—
246C24×C3—142no—
247C24×C3—142no—
248C24×C3—284no—
249C2×C3×D4—142no—
250C2×C3×D4—141no—
251C2×C3×D4—141no—
252C2×C3×D4—142no—
253C2×C3×D4—142no—
254C2×C3×D4—142no—
255C2×C3×D4—284no—
256C22×C3×C4—71no—
257C22×C3×C4—72no—
258C22×C3×C4—142no—
259C22×C3×C4—144no—
260C22×C3×C4—142no—
261C22×C3×C4—284no—
262C3×C22⋊C4—74no—
263C3×C22⋊C4—142no—
264C3×C22⋊C4—144no—
265C3×C22⋊C4—141no—
266C3×C22⋊C4—141no—
267C3×C22⋊C4—142no—
268C3×C22⋊C4—142no—
269C3×C22⋊C4—142no—
270C3×C22⋊C4—142no—
271C3×C22⋊C4—284no—
272C2×ΓL(1,𝔽8)—22no—
273C2×ΓL(1,𝔽8)—21no—
274C2×ΓL(1,𝔽8)—22no—
275C2×ΓL(1,𝔽8)—24no—
276C2×ΓL(1,𝔽8)—21no—
277C2×ΓL(1,𝔽8)—42no—
278C2×ΓL(1,𝔽8)#38611yescharacteristic
279C2×ΓL(1,𝔽8)C2×#9912yes—
280AGL(1,𝔽7)—81no—
281C25C2×AGL(1,𝔽7)11yescharacteristic
282C22×D4—71no—
283C22×D4—72no—
284C23×C4—21no—
285#120—142no—
286#120—141no—
287#120—141no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—72no—
291C2×C22⋊C4—71no—
292C2×C22⋊C4—72no—
293C2×C22⋊C4—142no—
294C2×C22⋊C4C2×AGL(1,𝔽7)12yes—
295#99—144no—
296C22×C7—24no—
297C22×C7—24no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7—22no—
301C22×C7—22no—
302C22×C7—22no—
303C22×C7—22no—
304C22×C7—22no—
305C22×C7—22no—
306C22×C7—42no—
307C22×C7—42no—
308C22×C7—48no—
309C22×C7—48no—
310C22×C7—42no—
311C22×C7—42no—
312C22×C7C3×#9914yescomplemented
313C22×C7C2×C3×C22⋊C411yescharacteristic
314C22×C7C3×C4×D412yes—
315C22×C7C3×#12011yescharacteristic
316C22×C7C3×#12112yes—
317C22×C7C3×#12311yescharacteristic
318C2×D7—41no—
319C2×D7—42no—
320C2×D7—82no—
321C4×C7—44no—
322Dic7—41no—
323Dic7—42no—
324Dic7—82no—
325C23×C3—71no—
326C23×C3—71no—
327C23×C3—72no—
328C23×C3—71no—
329C23×C3—72no—
330C23×C3—72no—
331C23×C3—72no—
332C23×C3—141no—
333C23×C3—144no—
334C23×C3—141no—
335C23×C3—141no—
336C23×C3—144no—
337C23×C3—144no—
338C23×C3—144no—
339C23×C3—141no—
340C23×C3—144no—
341C23×C3—141no—
342C23×C3—142no—
343C23×C3—282no—
344C23×C3—282no—
345C23×C3—282no—
346C23×C3—282no—
347C23×C3—288no—
348C23×C3—288no—
349C23×C3—282no—
350C3×D4—281no—
351C3×D4—282no—
352C3×D4—281no—
353C3×D4—284no—
354C3×D4—284no—
355C2×C3×C4—144no—
356C2×C3×C4—141no—
357C2×C3×C4—142no—
358C2×C3×C4—142no—
359C2×C3×C4—284no—
360C2×C3×C4—288no—
361C2×C3×C4—282no—
362C2×C3×C4—282no—
363C2×C3×C4—284no—
364ΓL(1,𝔽8)#189811yescharacteristic, complemented, Hall
365C24—22no—
366C24—22no—
367C24—44no—
368C24—71no—
369C24C4×AGL(1,𝔽7)12yes—
370C24C4⋊AGL(1,𝔽7)12yes—
371C24(C22⋊D7)⋊C311yescharacteristic
372C24(C22⋊D7)⋊C311yescharacteristic
373C24C22×AGL(1,𝔽7)11yescharacteristic
374C2×D4—141no—
375C2×D4—142no—
376C2×D4—142no—
377C2×D4—141no—
378C2×D4—142no—
379C2×D4—142no—
380C2×D4—284no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—44no—
384C22×C4—71no—
385C22×C4—72no—
386C22×C4—142no—
387C22⋊C4—24no—
388C22⋊C4—22no—
389C22⋊C4—141no—
390C22⋊C4—141no—
391C22⋊C4—142no—
392C22⋊C4—142no—
393C22⋊C4—142no—
394C22⋊C4—142no—
395C22⋊C4—284no—
396C22⋊C4(C22⋊D7)⋊C314yescomplemented
397C2×C7—22no—
398C2×C7—21no—
399C2×C7—22no—
400C2×C7—24no—
401C2×C7—21no—
402C2×C7—42no—
403C2×C7C3×#38611yescharacteristic
404C2×C7C2×C3×#9912yes—
405D7—81no—
406C22×C3—74no—
407C22×C3—71no—
408C22×C3—72no—
409C22×C3—71no—
410C22×C3—72no—
411C22×C3—71no—
412C22×C3—142no—
413C22×C3—142no—
414C22×C3—142no—
415C22×C3—142no—
416C22×C3—142no—
417C22×C3—142no—
418C22×C3—142no—
419C22×C3—142no—
420C22×C3—144no—
421C22×C3—144no—
422C22×C3—282no—
423C22×C3—282no—
424C22×C3—288no—
425C22×C3—288no—
426C22×C3—282no—
427C22×C3—282no—
428C22×C3—281no—
429C22×C3—282no—
430C22×C3—562no—
431C3×C4—284no—
432C3×C4—281no—
433C3×C4—282no—
434C3×C4—562no—
435C23—24no—
436C23—21no—
437C23—21no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—21no—
442C23—24no—
443C23—21no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—48no—
449C23—48no—
450C23—141no—
451C23—142no—
452C23—282no—
453C23(C2×AGL(1,𝔽7))⋊C412yes—
454C23(C2×AGL(1,𝔽7))⋊C412yes—
455C23C2×C4×AGL(1,𝔽7)11yescharacteristic
456C23C2×C4⋊AGL(1,𝔽7)11yescharacteristic
457C23#1157⋊C312yes—
458C23C2×(C22⋊D7)⋊C311yescharacteristic
459C23C2×(C22⋊D7)⋊C312yes—
460D4—284no—
461D4—284no—
462D4—281no—
463D4—282no—
464D4—281no—
465C2×C4—24no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—142no—
469C2×C4—141no—
470C2×C4—142no—
471C2×C4—282no—
472C2×C4—284no—
473C2×C4—282no—
474C7C3×#189811yescharacteristic, complemented, Sylow
475C2×C3—72no—
476C2×C3—71no—
477C2×C3—141no—
478C2×C3—142no—
479C2×C3—144no—
480C2×C3—141no—
481C2×C3—142no—
482C2×C3—282no—
483C2×C3—561no—
484C22—24no—
485C22—24no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—48no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—282no—
501C22—281no—
502C22—562no—
503C22#6380⋊C314yescomplemented
504C22C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
505C22C4×(C22⋊D7)⋊C312yes—
506C22#6492⋊C311yescharacteristic
507C22#6493⋊C312yes—
508C22#1062078911yescharacteristic
509C4—44no—
510C4—282no—
511C4—281no—
512C4—562no—
513C3—71noSylow
514C2—21no—
515C2—22no—
516C2—24no—
517C2—21no—
518C2—22no—
519C2—42no—
520C2—561no—
521C2C2×#6380⋊C312yes—
522C2#89985⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C2×AGL(1,𝔽7))⋊C4
commutatorC23×C7C22×C3×C4
FrattiniC24C22×AGL(1,𝔽7)
FittingC22×C7×C22⋊C4C2×C3
socleC22×C7C2×C3×C22⋊C4
prime radicalC2×#10620789C2
cosocleC24×C7C23×C3
hypercenterC22×C22⋊C4AGL(1,𝔽7)
hypocenterC7C3×#1898
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C4×ΓL(1,𝔽8)C2semidirect
3. #13052147C2semidirect
4. C2×#6380⋊C3C2semidirect
5. #89322⋊C3C2semidirect
6. #11729297C3semidirect
9. C22×(C22⋊D7)⋊C3C4semidirect
18. C2×(C22⋊Dic7)⋊C3C4semidirect
22. C22×C7×C22⋊C4C2×C3semidirect
24. #90019C2×C3semidirect
25. C2×#6380C2×C3semidirect
26. #89322C2×C3semidirect
36. C23×AGL(1,𝔽7)C2×C4semidirect
48. C22×ΓL(1,𝔽8)⋊C4C2×C4semidirect
49. C22×ΓL(1,𝔽8)⋊C4C2×C4semidirect
52. C22⋊C4×ΓL(1,𝔽8)D4semidirect
62. C22×C22⋊D7C3×C4semidirect
71. C2×C22⋊Dic7C3×C4semidirect
128. C23×D7C2×C3×C4semidirect
140. C22×Dic7C2×C3×C4semidirect
141. C22×Dic7C2×C3×C4semidirect
144. C7×C22⋊C4C3×D4semidirect
183. C22×ΓL(1,𝔽8)#99semidirect
197. C22×C22⋊C4AGL(1,𝔽7)semidirect
312. C22×C7C3×#99semidirect
364. ΓL(1,𝔽8)#1898semidirect
396. C22⋊C4(C22⋊D7)⋊C3semidirect
474. C7C3×#1898semidirect
503. C22#6380⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×ΓL(1,𝔽8)2
C2×#106207892
#130521472
C2×#6380⋊C32
#89322⋊C32
#117292973
C3×#18987