Finite Group Databasebeta

← #72277355

Subgroups — #72277355

Order 2688.

684 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90482⋊C3C211yesmaximal, characteristic
2#90436⋊C3C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#407C211yesmaximal, characteristic, complemented
4#89979⋊C3C211yesmaximal, characteristic, complemented
5#89904⋊C3C211yesmaximal, characteristic, complemented
6#89354⋊C3C211yesmaximal, characteristic, complemented
7#89351⋊C3C211yesmaximal, characteristic, complemented
8#11727746C311yesmaximal, characteristic, complemented, Hall
9C23×C4×ΓL(1,𝔽8)C2211yescharacteristic
10C2×#1157⋊C3C2211yescharacteristic
11C2×M16×ΓL(1,𝔽8)C2211yescharacteristic
12C2×M16×ΓL(1,𝔽8)C2211yescharacteristic
13#10620789C412yes—
14#6493⋊C3—21no—
15#6492⋊C3C412yes—
16C4×(C22⋊D7)⋊C3—22no—
17#6489⋊C3C2211yescharacteristic
18#6487⋊C3—21no—
19C2×#1136⋊C3C2211yescharacteristic
20C2×#1136⋊C3C2211yescharacteristic
21#6470⋊C3—22no—
22M16×AGL(1,𝔽7)—22no—
23#6464⋊C3—22no—
24ΓL(1,𝔽8)×C22⋊C8—22no—
25(C22⋊(C7⋊C8))⋊C3—22no—
26#6398⋊C3—21no—
27#6397⋊C3—21no—
28#90482C2×C311yescharacteristic
29#90436C2×C311yescharacteristic, complemented
30C7×#407C2×C311yescharacteristic, complemented
31#89979C2×C311yescharacteristic, complemented
32#89904C2×C311yescharacteristic, complemented
33#89354C2×C311yescharacteristic, complemented
34#89351C2×C311yescharacteristic, complemented
35C3×#1891—71nomaximal, Hall
36C24×ΓL(1,𝔽8)C2×C411yescharacteristic
37C22×C4×ΓL(1,𝔽8)—21no—
38C22×C4×ΓL(1,𝔽8)—21no—
39C22×C4×ΓL(1,𝔽8)—21no—
40C22×C4×ΓL(1,𝔽8)—21no—
41C22×C4×ΓL(1,𝔽8)—21no—
42C22×C4×ΓL(1,𝔽8)—21no—
43C22×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
44C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
45C2×(C22⋊D7)⋊C3—22no—
46C2×(C22⋊D7)⋊C3C2×C412yes—
47#1157⋊C3—21no—
48#1157⋊C3—21no—
49#1157⋊C3—21no—
50#1157⋊C3—21no—
51C2×C4⋊AGL(1,𝔽7)D411yescharacteristic
52C2×C4×AGL(1,𝔽7)D412yes—
53C2×ΓL(1,𝔽8)⋊Q8D411yescharacteristic
54M16×ΓL(1,𝔽8)—21no—
55M16×ΓL(1,𝔽8)—21no—
56M16×ΓL(1,𝔽8)—21no—
57M16×ΓL(1,𝔽8)—21no—
58C2×C8×ΓL(1,𝔽8)—21no—
59C2×C8×ΓL(1,𝔽8)D412yescomplemented
60(C22⋊Dic7)⋊C3—22no—
61(C22⋊Dic7)⋊C3C2×C412yes—
62(C2×AGL(1,𝔽7))⋊C4—22no—
63C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
64(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
65C4×ΓL(1,𝔽8)⋊C4—21no—
66#1136⋊C3—21no—
67#1136⋊C3—21no—
68#1136⋊C3—21no—
69#1136⋊C3—21no—
70C2×ΓL(1,𝔽8)⋊C8—21no—
71C2×ΓL(1,𝔽8)⋊C8D412yescomplemented
72#1131⋊C3—24no—
73C8×AGL(1,𝔽7)—24no—
74C23×C4×C7C22×C311yescharacteristic
75C2×#1157C22×C311yescharacteristic
76C2×C7×M16C22×C311yescharacteristic
77C2×C7×M16C22×C311yescharacteristic
78#6516C3×C412yes—
79#6493—21no—
80#6492C3×C412yes—
81C4×C22⋊D7—22no—
82#6489C22×C311yescharacteristic
83#6487—21no—
84C2×#1136C22×C311yescharacteristic
85C2×#1136C22×C311yescharacteristic
86#6470—22no—
87D7×M16—22no—
88#6464—22no—
89C7×C22⋊C8—22no—
90C22⋊(C7⋊C8)—22no—
91#6398—21no—
92#6397—21no—
93C3×#568—71no—
94C3×#525—71no—
95C3×#413—71no—
96C3×#407—71no—
97C3×#407—71no—
98C3×#344—71no—
99C3×#344—71no—
100C23×ΓL(1,𝔽8)—21no—
101C23×ΓL(1,𝔽8)—21no—
102C23×ΓL(1,𝔽8)—21no—
103C23×ΓL(1,𝔽8)—42no—
104C23×ΓL(1,𝔽8)C22×C411yescharacteristic
105C22×AGL(1,𝔽7)C22⋊C412yes—
106C2×C4×ΓL(1,𝔽8)—21no—
107C2×C4×ΓL(1,𝔽8)—21no—
108C2×C4×ΓL(1,𝔽8)—21no—
109C2×C4×ΓL(1,𝔽8)—41no—
110C2×C4×ΓL(1,𝔽8)—41no—
111C2×C4×ΓL(1,𝔽8)—41no—
112C2×C4×ΓL(1,𝔽8)—41no—
113C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
114C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
115C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
116C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
117C2×C4×ΓL(1,𝔽8)#4111yescharacteristic
118C2×C4×ΓL(1,𝔽8)#4111yescharacteristic
119(C22⋊D7)⋊C3—22no—
120(C22⋊D7)⋊C3—22no—
121(C22⋊D7)⋊C3—44no—
122C2×ΓL(1,𝔽8)⋊C4—22no—
123C2×ΓL(1,𝔽8)⋊C4C22⋊C412yes—
124C4⋊AGL(1,𝔽7)—21no—
125C4⋊AGL(1,𝔽7)—21no—
126C4×AGL(1,𝔽7)—22no—
127C4×AGL(1,𝔽7)—22no—
128ΓL(1,𝔽8)⋊Q8—21no—
129ΓL(1,𝔽8)⋊Q8—21no—
130C8×ΓL(1,𝔽8)—22no—
131C8×ΓL(1,𝔽8)—41no—
132ΓL(1,𝔽8)⋊C8—22no—
133ΓL(1,𝔽8)⋊C8—41no—
134#1891—71noSylow
135C24×C7C2×C3×C411yescharacteristic
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7C2×C3×C411yescharacteristic
143C22×C4×C7C23×C311yescharacteristic
144C2×C22⋊D7—22no—
145C2×C22⋊D7C2×C3×C412yes—
146#1157—21no—
147#1157—21no—
148#1157—21no—
149#1157—21no—
150C2×D28C3×D411yescharacteristic
151C2×C4×D7C3×D412yes—
152C2×Dic14C3×D411yescharacteristic
153C7×M16—21no—
154C7×M16—21no—
155C7×M16—21no—
156C7×M16—21no—
157C2×C7×C8—21no—
158C2×C7×C8C3×D412yescomplemented
159C22⋊Dic7—22no—
160C22⋊Dic7C2×C3×C412yes—
161(C2×D7)⋊C4—22no—
162C4⋊Dic7—21no—
163Dic7⋊C4—22no—
164C4×Dic7—21no—
165#1136—21no—
166#1136—21no—
167#1136—21no—
168#1136—21no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8C3×D412yescomplemented
171#1131—24no—
172C8×D7—24no—
173C2×C3×#41—71no—
174C23×C3×C4—71no—
175C3×#131—142no—
176C3×#131—142no—
177C2×C3×M16—71no—
178C2×C3×M16—71no—
179C2×C3×M16—71no—
180C2×C3×M16—71no—
181C2×C3×M16—142no—
182C3×#123—72no—
183C3×C22⋊Q8—141no—
184C3×#121—141no—
185C3×#120—72no—
186C3×C4×D4—142no—
187C3×#117—71no—
188C3×#101—141no—
189C3×#100—141no—
190C3×C22⋊C8—142no—
191C3×C22⋊C8—142no—
192C22×ΓL(1,𝔽8)—22no—
193C22×ΓL(1,𝔽8)—22no—
194C22×ΓL(1,𝔽8)—22no—
195C22×ΓL(1,𝔽8)—41no—
196C22×ΓL(1,𝔽8)—41no—
197C22×ΓL(1,𝔽8)—41no—
198C22×ΓL(1,𝔽8)—41no—
199C22×ΓL(1,𝔽8)—41no—
200C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
201C22×ΓL(1,𝔽8)C4×D411yescharacteristic
202C22×ΓL(1,𝔽8)C4×D411yescharacteristic
203C2×AGL(1,𝔽7)—22no—
204C2×AGL(1,𝔽7)—42no—
205C4×ΓL(1,𝔽8)—41no—
206C4×ΓL(1,𝔽8)#12011yescharacteristic
207C4×ΓL(1,𝔽8)#12111yescharacteristic
208C4×ΓL(1,𝔽8)#12111yescharacteristic
209C4×ΓL(1,𝔽8)#12311yescharacteristic
210ΓL(1,𝔽8)⋊C4—22no—
211ΓL(1,𝔽8)⋊C4—42no—
212#568—71no—
213#525—71no—
214#413—71no—
215#407—71no—
216#407AGL(1,𝔽7)11yescharacteristic, complemented
217#344—71no—
218#344—71no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—42no—
223C23×C7C22×C3×C411yescharacteristic
224C22×D7C3×C22⋊C412yes—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—41no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7C2×C3×D411yescharacteristic
233C2×C4×C7C2×C3×D411yescharacteristic
234C2×C4×C7C2×C3×D411yescharacteristic
235C2×C4×C7C2×C3×D411yescharacteristic
236C2×C4×C7C3×#4111yescharacteristic
237C2×C4×C7C3×#4111yescharacteristic
238C22⋊D7—22no—
239C22⋊D7—22no—
240C22⋊D7—44no—
241C2×Dic7—22no—
242C2×Dic7C3×C22⋊C412yes—
243D28—21no—
244D28—21no—
245C4×D7—22no—
246C4×D7—22no—
247Dic14—21no—
248Dic14—21no—
249C7×C8—22no—
250C7×C8—41no—
251C7⋊C8—22no—
252C7⋊C8—41no—
253C24×C3—71no—
254C3×#41—141no—
255C3×#41—141no—
256C3×#41—141no—
257C3×#41—141no—
258C2×C3×Q8—71no—
259C2×C3×D4—71no—
260C2×C3×D4—72no—
261C2×C3×D4—142no—
262C22×C3×C4—72no—
263C22×C3×C4—71no—
264C22×C3×C4—71no—
265C22×C3×C4—141no—
266C22×C3×C4—141no—
267C22×C3×C4—141no—
268C22×C3×C4—141no—
269C22×C3×C4—141no—
270C22×C3×C4—141no—
271C3×M16—141no—
272C3×M16—141no—
273C3×M16—141no—
274C3×M16—141no—
275C3×M16—141no—
276C3×M16—141no—
277C3×M16—144no—
278C3×M16—141no—
279C3×M16—141no—
280C2×C3×C8—72no—
281C2×C3×C8—72no—
282C2×C3×C8—141no—
283C2×C3×C8—144no—
284C2×C3×C8—141no—
285C3×C4⋊C4—141no—
286C3×C4⋊C4—142no—
287C3×C22⋊C4—72no—
288C3×C22⋊C4—142no—
289C3×C22⋊C4—142no—
290C3×C42—141no—
291C2×ΓL(1,𝔽8)—21no—
292C2×ΓL(1,𝔽8)—21no—
293C2×ΓL(1,𝔽8)—21no—
294C2×ΓL(1,𝔽8)—42no—
295C2×ΓL(1,𝔽8)#38611yescharacteristic
296AGL(1,𝔽7)—42no—
297C2×#41—71no—
298C23×C4C2×AGL(1,𝔽7)11yescharacteristic
299#131—142no—
300#131—142no—
301C2×M16—71no—
302C2×M16—71no—
303C2×M16—142no—
304C2×M16C2×AGL(1,𝔽7)11yescharacteristic
305C2×M16C2×AGL(1,𝔽7)11yescharacteristic
306#123—72no—
307C22⋊Q8—141no—
308#121—141no—
309#120—72no—
310C4×D4—142no—
311#117—71no—
312#101—141no—
313#100—141no—
314C22⋊C8—22no—
315C22⋊C8—142no—
316C22×C7—22no—
317C22×C7—22no—
318C22×C7—22no—
319C22×C7—41no—
320C22×C7—41no—
321C22×C7—41no—
322C22×C7—41no—
323C22×C7—41no—
324C22×C7C2×C3×C22⋊C411yescharacteristic
325C22×C7C3×C4×D411yescharacteristic
326C22×C7C3×C4×D411yescharacteristic
327C2×D7—22no—
328C2×D7—42no—
329C4×C7—41no—
330C4×C7C3×#12011yescharacteristic
331C4×C7C3×#12111yescharacteristic
332C4×C7C3×#12111yescharacteristic
333C4×C7C3×#12311yescharacteristic
334Dic7—22no—
335Dic7—42no—
336C23×C3—72no—
337C23×C3—71no—
338C23×C3—141no—
339C23×C3—141no—
340C23×C3—141no—
341C23×C3—282no—
342C3×Q8—141no—
343C3×Q8—141no—
344C3×D4—141no—
345C3×D4—141no—
346C3×D4—142no—
347C3×D4—142no—
348C3×D4—284no—
349C2×C3×C4—72no—
350C2×C3×C4—71no—
351C2×C3×C4—71no—
352C2×C3×C4—71no—
353C2×C3×C4—71no—
354C2×C3×C4—71no—
355C2×C3×C4—71no—
356C2×C3×C4—141no—
357C2×C3×C4—141no—
358C2×C3×C4—141no—
359C2×C3×C4—142no—
360C2×C3×C4—142no—
361C2×C3×C4—142no—
362C2×C3×C4—281no—
363C2×C3×C4—281no—
364C2×C3×C4—281no—
365C2×C3×C4—281no—
366C3×C8—142no—
367C3×C8—142no—
368C3×C8—281no—
369C3×C8—281no—
370ΓL(1,𝔽8)#189111yescharacteristic, complemented, Hall
371C24C4×AGL(1,𝔽7)11yescharacteristic
372#41—141no—
373#41—141no—
374#41—141no—
375#41—141no—
376C2×Q8—71no—
377C2×D4—72no—
378C2×D4—71no—
379C2×D4—142no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—72no—
387C22×C4C4×AGL(1,𝔽7)11yescharacteristic
388C22×C4C22×AGL(1,𝔽7)11yescharacteristic
389M16—21no—
390M16—21no—
391M16—21no—
392M16—21no—
393M16—144no—
394M16—141no—
395M16—141no—
396M16—141no—
397M16—141no—
398C2×C8—21no—
399C2×C8—72no—
400C2×C8—144no—
401C2×C8—141no—
402C2×C8(C22⋊D7)⋊C312yescomplemented
403C4⋊C4—142no—
404C4⋊C4—141no—
405C22⋊C4—72no—
406C22⋊C4—142no—
407C22⋊C4—142no—
408C42—141no—
409C2×C7—21no—
410C2×C7—21no—
411C2×C7—21no—
412C2×C7—42no—
413C2×C7C3×#38611yescharacteristic
414D7—42no—
415C22×C3—71no—
416C22×C3—71no—
417C22×C3—71no—
418C22×C3—142no—
419C22×C3—142no—
420C22×C3—142no—
421C22×C3—142no—
422C22×C3—281no—
423C22×C3—281no—
424C22×C3—281no—
425C22×C3—281no—
426C22×C3—281no—
427C22×C3—282no—
428C3×C4—71no—
429C3×C4—71no—
430C3×C4—71no—
431C3×C4—71no—
432C3×C4—142no—
433C3×C4—281no—
434C3×C4—282no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—42no—
439C23—72no—
440C23C2×C4×AGL(1,𝔽7)11yescharacteristic
441Q8—141no—
442Q8—141no—
443D4—142no—
444D4—142no—
445D4—141no—
446D4—141no—
447D4—284no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—72no—
456C2×C4—142no—
457C2×C4—142no—
458C2×C4—142no—
459C2×C4#1157⋊C311yescharacteristic
460C2×C4D4×AGL(1,𝔽7)11yescharacteristic
461C2×C4D4×AGL(1,𝔽7)11yescharacteristic
462C2×C4D4×AGL(1,𝔽7)11yescharacteristic
463C2×C4#1159⋊C311yescharacteristic
464C2×C4C2×(C22⋊D7)⋊C311yescharacteristic
465C8—22no—
466C8—41no—
467C8—142no—
468C8—281no—
469C7C3×#189111yescharacteristic, complemented, Sylow
470C2×C3—71no—
471C2×C3—141no—
472C2×C3—141no—
473C2×C3—141no—
474C2×C3—282no—
475C2×C3—282no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—142no—
485C22—282no—
486C22C22⋊C4×AGL(1,𝔽7)11yescharacteristic
487C22((C22⋊D7)⋊C3)⋊C411yescharacteristic
488C22C4×(C22⋊D7)⋊C311yescharacteristic
489C4—41no—
490C4—142no—
491C4—282no—
492C4#6446⋊C311yescharacteristic
493C4#6447⋊C311yescharacteristic
494C4#6500⋊C311yescharacteristic
495C4#6501⋊C311yescharacteristic
496C3—71noSylow
497C2—21no—
498C2—21no—
499C2—21no—
500C2—42no—
501C2—282no—
502C2#89728⋊C311yescharacteristic
subgroupG/H
centerC4#6500⋊C3
commutatorC23×C7C22×C3×C4
FrattiniC22×C4C22×AGL(1,𝔽7)
FittingC7×#407C2×C3
socleC2×C7C3×#386
prime radical#90482⋊C3C2
cosocleC22×C4×C7C23×C3
hypercenter#407AGL(1,𝔽7)
hypocenterC7C3×#1891
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #90436⋊C3C2semidirect
3. ΓL(1,𝔽8)×#407C2semidirect
4. #89979⋊C3C2semidirect
5. #89904⋊C3C2semidirect
6. #89354⋊C3C2semidirect
7. #89351⋊C3C2semidirect
8. #11727746C3semidirect
29. #90436C2×C3semidirect
30. C7×#407C2×C3semidirect
31. #89979C2×C3semidirect
32. #89904C2×C3semidirect
33. #89354C2×C3semidirect
34. #89351C2×C3semidirect
59. C2×C8×ΓL(1,𝔽8)D4semidirect
71. C2×ΓL(1,𝔽8)⋊C8D4semidirect
158. C2×C7×C8C3×D4semidirect
170. C2×C7⋊C8C3×D4semidirect
216. #407AGL(1,𝔽7)semidirect
370. ΓL(1,𝔽8)#1891semidirect
402. C2×C8(C22⋊D7)⋊C3semidirect
469. C7C3×#1891semidirect

Maximal subgroups

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

subgroupindex
#90482⋊C32
#90436⋊C32
ΓL(1,𝔽8)×#4072
#89979⋊C32
#89904⋊C32
#89354⋊C32
#89351⋊C32
#117277463
C3×#18917