Finite Group Databasebeta

← #46840027

Subgroups — #46840027

Order 34688.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576136C211yesmaximal, characteristic, complemented
2C22×(C2×D271)⋊C4C211yesmaximal, characteristic
3C2×C22⋊C4×D271C211yesmaximal, characteristic
4#31773832C211yesmaximal, characteristic, complemented
5#31773819C211yesmaximal, characteristic, complemented
6#31773819C211yesmaximal, characteristic, complemented
7#31773808C211yesmaximal, characteristic, complemented
8#31773772C211yesmaximal, characteristic, complemented
9#31773770C211yesmaximal, characteristic, complemented
10#31773768C211yesmaximal, characteristic, complemented
11#31773768C211yesmaximal, characteristic, complemented
12#31773755C211yesmaximal, characteristic, complemented
13#31773754C211yesmaximal, characteristic, complemented
14#31773752C211yesmaximal, characteristic
15C2×C22⋊Q8×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20#22576136—24no—
21C2×C4⋊Dic271C2211yescharacteristic, complemented
22C2×C4⋊Dic271C2211yescharacteristic, complemented
23C2×Dic271⋊C4C2211yescharacteristic, complemented
24C2×Dic271⋊C4C2211yescharacteristic, complemented
25C2×Dic271⋊C4C2211yescharacteristic, complemented
26C2×Dic271⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×(C2×D271)⋊C4—24no—
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C22⋊C4×D271—24no—
41C271⋊#95C2211yescharacteristic, complemented
42C271⋊#95C2211yescharacteristic, complemented
43C271⋊#95C2211yescharacteristic, complemented
44C271⋊#95C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C22×Q8×C271C2211yescharacteristic, complemented
48C23×C4×C271C2211yescharacteristic
49C22⋊Q8×C271—24no—
50C2×C4⋊C4×C271C2211yescharacteristic, complemented
51C2×C4⋊C4×C271C2211yescharacteristic, complemented
52C2×C4⋊C4×C271C2211yescharacteristic, complemented
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C23×D271—22no—
56C23×D271—22no—
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—41no—
60C23×D271—41no—
61C23×D271Q814yescomplemented
62C23×D271C2311yescharacteristic
63C23×D271C2311yescharacteristic
64C22×Dic271—22no—
65C22×Dic271—22no—
66C22×Dic271—41no—
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C22×Dic271C2311yescharacteristic
73C2×C4×D271—22no—
74C2×C4×D271—22no—
75C2×C4×D271—22no—
76C2×C4×D271—22no—
77C2×C4×D271—41no—
78C2×C4×D271—41no—
79C4⋊Dic271—22no—
80C4⋊Dic271—41no—
81Dic271⋊C4—22no—
82Dic271⋊C4—22no—
83Dic271⋊C4—41no—
84Dic271⋊C4—41no—
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87(C2×D271)⋊C4—22no—
88(C2×D271)⋊C4—22no—
89(C2×D271)⋊C4—22no—
90(C2×D271)⋊C4—22no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94(C2×D271)⋊C4—41no—
95(C2×D271)⋊C4—41no—
96(C2×D271)⋊C4D414yes—
97C24×C271C2311yescharacteristic
98C2×Q8×C271—22no—
99C2×Q8×C271—22no—
100C2×Q8×C271—41no—
101C22×C4×C271—22no—
102C22×C4×C271—22no—
103C22×C4×C271—41no—
104C22×C4×C271C2311yescharacteristic
105C22×C4×C271C2311yescharacteristic
106C22×C4×C271C2311yescharacteristic
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C4⋊C4×C271—22no—
111C4⋊C4×C271—22no—
112C4⋊C4×C271—22no—
113C22⋊C4×C271—22no—
114C22⋊C4×C271D414yes—
115C22×D271—21no—
116C22×D271—21no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—28no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—28no—
127C22×D271—44no—
128C22×D271—44no—
129C22×D271—44no—
130C22×D271—44no—
131C22×D271—44no—
132C22×D271C2×D412yes—
133C22×D271C2×Q812yes—
134C22×D271C2×Q812yes—
135C22×D271#4112yes—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C2×Dic271—21no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—44no—
154C2×Dic271C2×D412yes—
155C4×D271—44no—
156C4×D271—44no—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271—21no—
160C23×C271—21no—
161C23×C271C2×D412yes—
162C23×C271C2×Q812yes—
163C23×C271#4112yes—
164C23×C271C2411yescharacteristic
165Q8×C271—44no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—21no—
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—21no—
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271—44no—
182C2×C4×C271C2×D412yes—
183C2×C4×C271C2×D412yes—
184C2×C4×C271C2×D412yes—
185C22×C271—22no—
186C22×C271—22no—
187C22×C271—22no—
188C22×C271—22no—
189C22×C271—22no—
190C22×C271—22no—
191C22×C271C22⋊Q814yescomplemented
192C22×C271C22×D411yescharacteristic
193C22×C271C22×D411yescharacteristic
194C22×C271C22×Q811yescharacteristic
195C22×C271C2×#4111yescharacteristic
196C22×C271C2×#4111yescharacteristic
197C22×C271H(5,𝔽2)11yescharacteristic
198C22×C271H(5,𝔽2)11yescharacteristic
199C2×D271—22no—
200C2×D271—22no—
201C2×D271—22no—
202C2×D271—22no—
203C2×D271—22no—
204C2×D271—22no—
205C2×D271—22no—
206C2×D271—22no—
207C2×D271—22no—
208C2×D271—416no—
209C2×D271—41no—
210C2×D271—41no—
211C2×D271—41no—
212C2×D271—41no—
213C2×D271C22⋊Q814yescomplemented
214C4×C271—22no—
215C4×C271—22no—
216C4×C271—22no—
217C4×C271—41no—
218C4×C271—41no—
219C4×C271—41no—
220Dic271—22no—
221Dic271—41no—
222Dic271—41no—
223Dic271—41no—
224Dic271—41no—
225Dic271—41no—
226C2×C271—24no—
227C2×C271C2×C22⋊Q811yescharacteristic
228C2×C271C2×C22⋊Q811yescharacteristic
229C2×C271#54311yescharacteristic
230C2×C271D4211yescharacteristic
231C2×C271#54611yescharacteristic
232C2×C271#54611yescharacteristic
233C2×C271#55111yescharacteristic
234D271—24no—
235D271—42no—
236C271#243711yescharacteristic, complemented, Sylow
237#2437—2711nomaximal, Sylow
238C2×C22⋊Q8—2711no—
239C2×C22⋊Q8D27111yescharacteristic, complemented
240C22×C22⋊C4—2711no—
241C22×C22⋊C4—2711no—
242#399—2711no—
243#396—2711no—
244#396—2711no—
245#393—2711no—
246#385—2711no—
247#385—2711no—
248#385—2711no—
249#385—2711no—
250#380—2711no—
251#380—2711no—
252#379—2711no—
253C25—2711no—
254C22×Q8C2×D27111yescharacteristic, complemented
255C23×C4—2711no—
256C23×C4—2711no—
257C23×C4—2711no—
258C23×C4C2×D27111yescharacteristic
259C22⋊Q8—24no—
260C22⋊Q8—5424no—
261C2×C4⋊C4—2711no—
262C2×C4⋊C4—2711no—
263C2×C4⋊C4—2711no—
264C2×C4⋊C4—2711no—
265C2×C4⋊C4—2711no—
266C2×C4⋊C4—2711no—
267C2×C4⋊C4C2×D27111yescharacteristic, complemented
268C2×C4⋊C4C2×D27111yescharacteristic, complemented
269C2×C4⋊C4C2×D27111yescharacteristic, complemented
270C2×C22⋊C4—2711no—
271C2×C22⋊C4—2711no—
272C2×C22⋊C4—2711no—
273C2×C22⋊C4—2711no—
274C2×C22⋊C4—2711no—
275C2×C22⋊C4—2711no—
276C2×C22⋊C4—2711no—
277C2×C22⋊C4—2711no—
278C2×C22⋊C4—2711no—
279C2×C22⋊C4—2711no—
280C2×C22⋊C4—2711no—
281C2×C22⋊C4—2711no—
282C2×C22⋊C4—5424no—
283C2×C22⋊C4—5424no—
284C2×C22⋊C4C2×D27111yescharacteristic
285C2×C22⋊C4C2×D27111yescharacteristic
286#95—2711no—
287#95—2711no—
288#95—2711no—
289#95—2711no—
290#95—2711no—
291#95—2711no—
292C24—2711no—
293C24—2711no—
294C24—2714no—
295C24—5422no—
296C24—5422no—
297C24—5422no—
298C24—5422no—
299C24—10841no—
300C24—10841no—
301C24C22×D27111yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—41no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—2711no—
309C22×C4—2711no—
310C22×C4—2711no—
311C22×C4—2711no—
312C22×C4—2711no—
313C22×C4—2711no—
314C22×C4—5422no—
315C22×C4—5422no—
316C22×C4—5422no—
317C22×C4—5422no—
318C22×C4—5422no—
319C22×C4—5422no—
320C22×C4—10841no—
321C22×C4—10841no—
322C22×C4—10841no—
323C22×C4C22×D27111yescharacteristic
324C22×C4C22×D27111yescharacteristic
325C22×C4C22×D27111yescharacteristic
326C22×C4C22×D27111yescharacteristic
327C22×C4C22×D27111yescharacteristic
328C22×C4C22×D27111yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—5422no—
333C4⋊C4—5422no—
334C4⋊C4—5422no—
335C4⋊C4—10841no—
336C4⋊C4—10841no—
337C4⋊C4—10841no—
338C22⋊C4—22no—
339C22⋊C4—2714no—
340C22⋊C4—5422no—
341C22⋊C4—5422no—
342C22⋊C4—5422no—
343C22⋊C4—5422no—
344C22⋊C4—5422no—
345C22⋊C4—5422no—
346C22⋊C4—5422no—
347C22⋊C4—5422no—
348C22⋊C4—5422no—
349C22⋊C4—10841no—
350C22⋊C4—10841no—
351C22⋊C4C22⋊D27114yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—2712no—
357C23—2712no—
358C23—2712no—
359C23—2712no—
360C23—5428no—
361C23—5421no—
362C23—5421no—
363C23—5421no—
364C23—5421no—
365C23—5428no—
366C23—5421no—
367C23—5421no—
368C23—5421no—
369C23—5421no—
370C23—5421no—
371C23—5421no—
372C23—10844no—
373C23—10844no—
374C23—10844no—
375C23—10844no—
376C23—10844no—
377C23C2×C22⋊D27112yes—
378C23Q8×D27112yes—
379C23#1686465512yes—
380C23C23×D27111yescharacteristic
381Q8—44no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—2712no—
399C2×C4—5421no—
400C2×C4—5421no—
401C2×C4—5421no—
402C2×C4—5421no—
403C2×C4—5421no—
404C2×C4—5421no—
405C2×C4—5421no—
406C2×C4—5421no—
407C2×C4—5421no—
408C2×C4—5421no—
409C2×C4—5421no—
410C2×C4—5421no—
411C2×C4—5421no—
412C2×C4—5421no—
413C2×C4—5421no—
414C2×C4—5421no—
415C2×C4—5421no—
416C2×C4—10844no—
417C2×C4—10844no—
418C2×C4—10844no—
419C2×C4D4×D27112yes—
420C2×C4C2×C22⋊D27112yes—
421C2×C4C2×C22⋊D27112yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—2714no—
429C22—5422no—
430C22—5422no—
431C22—5422no—
432C22—5422no—
433C22—5422no—
434C22—5422no—
435C22—5422no—
436C22—5422no—
437C22—5422no—
438C22—108416no—
439C22—10841no—
440C22—10841no—
441C22—10841no—
442C22—10841no—
443C22#2257613614yescomplemented
444C22C2×D4×D27111yescharacteristic
445C22C22×C22⋊D27111yescharacteristic
446C22C2×Q8×D27111yescharacteristic
447C22#41×D27111yescharacteristic
448C22C2×#1686465511yescharacteristic
449C22#2257620511yescharacteristic
450C22#2257620611yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—5422no—
458C4—10841no—
459C4—10841no—
460C4—10841no—
461C4—10841no—
462C4—10841no—
463C2—24no—
464C2—5424no—
465C2—10842no—
466C2C22⋊Q8×D27111yescharacteristic
467C2C2×#2257613611yescharacteristic
468C2#3177452211yescharacteristic
469C2D4×C22⋊D27111yescharacteristic
470C2#3177453511yescharacteristic
471C2#3177453811yescharacteristic
472C2#3177457411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×C22⋊Q8×C271C2
prime radicalC24×D271C22
hypercenterC2×C22⋊Q8D271
hypocenterC271#2437
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22576136C2semidirect
4. #31773832C2semidirect
5. #31773819C2semidirect
6. #31773819C2semidirect
7. #31773808C2semidirect
8. #31773772C2semidirect
9. #31773770C2semidirect
10. #31773768C2semidirect
11. #31773768C2semidirect
12. #31773755C2semidirect
13. #31773754C2semidirect
15. C2×C22⋊Q8×C271C2semidirect
21. C2×C4⋊Dic271C22semidirect
22. C2×C4⋊Dic271C22semidirect
23. C2×Dic271⋊C4C22semidirect
24. C2×Dic271⋊C4C22semidirect
25. C2×Dic271⋊C4C22semidirect
26. C2×Dic271⋊C4C22semidirect
41. C271⋊#95C22semidirect
42. C271⋊#95C22semidirect
43. C271⋊#95C22semidirect
44. C271⋊#95C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C22×Q8×C271C22semidirect
50. C2×C4⋊C4×C271C22semidirect
51. C2×C4⋊C4×C271C22semidirect
52. C2×C4⋊C4×C271C22semidirect
61. C23×D271Q8semidirect
191. C22×C271C22⋊Q8semidirect
213. C2×D271C22⋊Q8semidirect
236. C271#2437semidirect
239. C2×C22⋊Q8D271semidirect
254. C22×Q8C2×D271semidirect
267. C2×C4⋊C4C2×D271semidirect
268. C2×C4⋊C4C2×D271semidirect
269. C2×C4⋊C4C2×D271semidirect
443. C22#22576136semidirect

Maximal subgroups

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

subgroupindex
C2×#225761362
C22×(C2×D271)⋊C42
C2×C22⋊C4×D2712
#317738322
#317738192
#317738082
#317737722
#317737702
#317737682
#317737552
#317737542
#317737522
C2×C22⋊Q8×C2712
#2437271