Finite Group Databasebeta

← #46840025

Subgroups — #46840025

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×#22576139C211yesmaximal, characteristic, complemented
2C2×#22576137C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic271C211yesmaximal, characteristic
4#31773833C211yesmaximal, characteristic, complemented
5#31773820C211yesmaximal, characteristic, complemented
6#31773820C211yesmaximal, characteristic, complemented
7#31773809C211yesmaximal, characteristic, complemented
8#31773772C211yesmaximal, characteristic, complemented
9#31773770C211yesmaximal, characteristic, complemented
10#31773770C211yesmaximal, characteristic, complemented
11#31773769C211yesmaximal, characteristic, complemented
12#31773757C211yesmaximal, characteristic, complemented
13#31773755C211yesmaximal, characteristic, complemented
14#31773753C211yesmaximal, characteristic
15C22×C22⋊C4×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic, complemented
17C23×Dic271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19#22576139—24no—
20#22576137—24no—
21C2×C4⋊Dic271C2211yescharacteristic, complemented
22C2×C4⋊Dic271C2211yescharacteristic, complemented
23C2×C4⋊Dic271C2211yescharacteristic, complemented
24C2×Dic271⋊C4C2211yescharacteristic, complemented
25C2×Dic271⋊C4C2211yescharacteristic, complemented
26C2×Dic271⋊C4C2211yescharacteristic, complemented
27C2×Dic271⋊C4C2211yescharacteristic, complemented
28C2×Dic271⋊C4C2211yescharacteristic, complemented
29C2×Dic271⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic271—24no—
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271C2211yescharacteristic
38C2×C22⋊Dic271C2211yescharacteristic
39C2×C22⋊Dic271C2211yescharacteristic
40C2×C22⋊Dic271C2211yescharacteristic
41C271⋊#95C2211yescharacteristic, complemented
42C271⋊#95C2211yescharacteristic, complemented
43C271⋊#95C2211yescharacteristic, complemented
44C271⋊#95C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C25×C271C2211yescharacteristic
48C23×C4×C271C2211yescharacteristic
49C23×C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271—24no—
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×Dic542—22no—
56C2×Dic542—22no—
57C2×Dic542—41no—
58C22×Dic271—22no—
59C22×Dic271—22no—
60C22×Dic271—22no—
61C22×Dic271—22no—
62C22×Dic271—41no—
63C22×Dic271—41no—
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C4⋊Dic271—22no—
73C4⋊Dic271—22no—
74C4⋊Dic271—41no—
75Dic271⋊C4—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—41no—
80Dic271⋊C4—41no—
81C22⋊Dic271—22no—
82C22⋊Dic271—22no—
83C22⋊Dic271—22no—
84C22⋊Dic271—22no—
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87C22⋊Dic271—22no—
88C22⋊Dic271—41no—
89C22⋊Dic271—41no—
90C22⋊Dic271D414yes—
91C24×C271—22no—
92C24×C271—22no—
93C24×C271—22no—
94C24×C271—22no—
95C24×C271—41no—
96C24×C271—41no—
97C24×C271Q814yescomplemented
98C24×C271C2311yescharacteristic
99C24×C271C2311yescharacteristic
100C24×C271C2311yescharacteristic
101C22×C4×C271—22no—
102C22×C4×C271—22no—
103C22×C4×C271—22no—
104C22×C4×C271—22no—
105C22×C4×C271—41no—
106C22×C4×C271—41no—
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C22⋊C4×C271—22no—
112C22⋊C4×C271—22no—
113C22⋊C4×C271—22no—
114C22⋊C4×C271D414yes—
115Dic542—44no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271—21no—
128C2×Dic271—21no—
129C2×Dic271—21no—
130C2×Dic271—21no—
131C2×Dic271—21no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—21no—
137C2×Dic271—21no—
138C2×Dic271—44no—
139C2×Dic271—44no—
140C2×Dic271C2×D412yes—
141C2×Dic271C2×D412yes—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271—21no—
145C23×C271—21no—
146C23×C271—21no—
147C23×C271—21no—
148C23×C271—21no—
149C23×C271—21no—
150C23×C271—28no—
151C23×C271—28no—
152C23×C271—21no—
153C23×C271—21no—
154C23×C271—21no—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271—21no—
158C23×C271—44no—
159C23×C271—44no—
160C23×C271—44no—
161C23×C271—44no—
162C23×C271—44no—
163C23×C271C2×D412yes—
164C23×C271C2×D412yes—
165C23×C271C2×Q812yes—
166C23×C271C2×Q812yes—
167C23×C271C2×Q812yes—
168C23×C271#4112yes—
169C23×C271#4112yes—
170C23×C271C2411yescharacteristic
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×C271—44no—
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×C271—22no—
192C22×C271—22no—
193C22×C271—22no—
194C22×C271—22no—
195C22×C271—22no—
196C22×C271—22no—
197C22×C271—22no—
198C22×C271—22no—
199C22×C271—22no—
200C22×C271—416no—
201C22×C271—41no—
202C22×C271—41no—
203C22×C271—41no—
204C22×C271—41no—
205C22×C271C22⋊Q814yescomplemented
206C22×C271C22⋊Q814yescomplemented
207C22×C271C22×D411yescharacteristic
208C22×C271C22×D411yescharacteristic
209C22×C271C22×Q811yescharacteristic
210C22×C271C2×#4111yescharacteristic
211C22×C271C2×#4111yescharacteristic
212C22×C271H(5,𝔽2)11yescharacteristic
213C22×C271H(5,𝔽2)11yescharacteristic
214C4×C271—22no—
215C4×C271—22no—
216C4×C271—41no—
217C4×C271—41no—
218Dic271—22no—
219Dic271—22no—
220Dic271—41no—
221Dic271—41no—
222Dic271—41no—
223Dic271—41no—
224Dic271—41no—
225Dic271—41no—
226C2×C271—24no—
227C2×C271—24no—
228C2×C271—42no—
229C2×C271C2×C22⋊Q811yescharacteristic
230C2×C271C2×C22⋊Q811yescharacteristic
231C2×C271#54311yescharacteristic
232C2×C271D4211yescharacteristic
233C2×C271#54611yescharacteristic
234C2×C271#54611yescharacteristic
235C2×C271#55111yescharacteristic
236C271#243711yescharacteristic, complemented, Sylow
237#2437—2711nomaximal, Sylow
238C2×C22⋊Q8—2711no—
239C2×C22⋊Q8—2711no—
240C22×C22⋊C4—2711no—
241C22×C22⋊C4D27111yescharacteristic
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—
253C25C2×D27111yescharacteristic
254C22×Q8—2711no—
255C23×C4—2711no—
256C23×C4—2711no—
257C23×C4C2×D27111yescharacteristic
258C23×C4C2×D27111yescharacteristic
259C22⋊Q8—5424no—
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⋊C4—2711no—
268C2×C4⋊C4—2711no—
269C2×C4⋊C4—2711no—
270C2×C22⋊C4—24no—
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—5424no—
282C2×C22⋊C4C2×D27111yescharacteristic
283C2×C22⋊C4C2×D27111yescharacteristic
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—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24Dic54214yescomplemented
299C24C22×D27111yescharacteristic
300C24C22×D27111yescharacteristic
301C24C22×D27111yescharacteristic
302C2×Q8—5422no—
303C2×Q8—5422no—
304C2×Q8—10841no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—2711no—
312C22×C4—2711no—
313C22×C4—2711no—
314C22×C4—2711no—
315C22×C4—2711no—
316C22×C4—2711no—
317C22×C4—2711no—
318C22×C4—2711no—
319C22×C4—5422no—
320C22×C4—5422no—
321C22×C4—5422no—
322C22×C4—5422no—
323C22×C4—10841no—
324C22×C4—10841no—
325C22×C4C22×D27111yescharacteristic
326C22×C4C22×D27111yescharacteristic
327C22×C4C22×D27111yescharacteristic
328C22×C4C22×D27111yescharacteristic
329C4⋊C4—5422no—
330C4⋊C4—5422no—
331C4⋊C4—5422no—
332C4⋊C4—5422no—
333C4⋊C4—5422no—
334C4⋊C4—5422no—
335C4⋊C4—10841no—
336C4⋊C4—10841no—
337C4⋊C4—10841no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—2714no—
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—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—28no—
361C23—28no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23D4×D27112yes—
374C23C2×C22⋊D27112yes—
375C23C2×Dic54212yes—
376C23C2×Dic54212yes—
377C23C2×Dic54212yes—
378C23#1686465312yes—
379C23#1686465412yes—
380C23C23×D27111yescharacteristic
381Q8—10844no—
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—44no—
393C2×C4—44no—
394C2×C4—2712no—
395C2×C4—2712no—
396C2×C4—5421no—
397C2×C4—5421no—
398C2×C4—5421no—
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—5421no—
417C2×C4—5421no—
418C2×C4—10844no—
419C2×C4—10844no—
420C2×C4C2×C22⋊D27112yes—
421C2×C4C2×C22⋊D27112yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—416no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#2257613714yescomplemented
443C22#2257613914yescomplemented
444C22C2×D4×D27111yescharacteristic
445C22C22×C22⋊D27111yescharacteristic
446C22C22×Dic54211yescharacteristic
447C22C2×#1686465311yescharacteristic
448C22C2×#1686465411yescharacteristic
449C22#2257620511yescharacteristic
450C22#2257620511yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—5422no—
456C4—5422no—
457C4—10841no—
458C4—10841no—
459C4—10841no—
460C4—10841no—
461C4—10841no—
462C4—10841no—
463C2—24no—
464C2—24no—
465C2—42no—
466C2C2×#2257613711yescharacteristic
467C2C2×#2257613911yescharacteristic
468C2#3177452311yescharacteristic
469C2D4×C22⋊D27111yescharacteristic
470C2#3177453711yescharacteristic
471C2#3177454111yescharacteristic
472C2#3177457311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC22×C22⋊C4×C271C2
prime radicalC25×C271C22
hypercenterC22×C22⋊C4D271
hypocenterC271#2437
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#22576139C2semidirect
2. C2×#22576137C2semidirect
4. #31773833C2semidirect
5. #31773820C2semidirect
6. #31773820C2semidirect
7. #31773809C2semidirect
8. #31773772C2semidirect
9. #31773770C2semidirect
10. #31773770C2semidirect
11. #31773769C2semidirect
12. #31773757C2semidirect
13. #31773755C2semidirect
16. C22×Dic542C22semidirect
21. C2×C4⋊Dic271C22semidirect
22. C2×C4⋊Dic271C22semidirect
23. C2×C4⋊Dic271C22semidirect
24. C2×Dic271⋊C4C22semidirect
25. C2×Dic271⋊C4C22semidirect
26. C2×Dic271⋊C4C22semidirect
27. C2×Dic271⋊C4C22semidirect
28. C2×Dic271⋊C4C22semidirect
29. C2×Dic271⋊C4C22semidirect
41. C271⋊#95C22semidirect
42. C271⋊#95C22semidirect
43. C271⋊#95C22semidirect
44. C271⋊#95C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
97. C24×C271Q8semidirect
205. C22×C271C22⋊Q8semidirect
206. C22×C271C22⋊Q8semidirect
236. C271#2437semidirect
298. C24Dic542semidirect
442. C22#22576137semidirect
443. C22#22576139semidirect

Maximal subgroups

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

subgroupindex
C2×#225761392
C2×#225761372
C22×C22⋊Dic2712
#317738332
#317738202
#317738092
#317737722
#317737702
#317737692
#317737572
#317737552
#317737532
C22×C22⋊C4×C2712
#2437271