Finite Group Databasebeta

← #46847875

Subgroups — #46847875

Order 34688.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576191C211yesmaximal, characteristic, complemented
2C22×#16864641C211yesmaximal, characteristic, complemented
3#31774543C211yesmaximal, characteristic
4#31774289C211yesmaximal, characteristic, complemented
5#31774145C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774119C211yesmaximal, characteristic, complemented
8#31774099C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774053C211yesmaximal, characteristic, complemented
11#31774047C211yesmaximal, characteristic, complemented
12#31774036C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773969C211yesmaximal, characteristic, complemented
15D4×Q8×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18#22576191—24no—
19C2×#16864644C2211yescharacteristic, complemented
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864644C2211yescharacteristic, complemented
22C2×#16864641—24no—
23C2×#16864641C2211yescharacteristic
24C2×#16864641C2211yescharacteristic
25C2×#16864641C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
27C2×#16864641C2211yescharacteristic
28C2×#16864641C2211yescharacteristic
29C2×#16864638C2211yescharacteristic, complemented
30C22×C271⋊C8C2211yescharacteristic
31#22576148C2211yescharacteristic
32#22576145—21no—
33#22576139C2211yescharacteristic
34#22576134C2211yescharacteristic
35#22576133—21no—
36#22576128—21no—
37C4×D1084C2211yescharacteristic
38C22⋊C4×D271—21no—
39#22576093C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576084C2211yescharacteristic, complemented
43#22576084C2211yescharacteristic, complemented
44#22576083C2211yescharacteristic, complemented
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic
47C22⋊(C271⋊C8)C2211yescharacteristic
48C22⋊(C271⋊C8)C2211yescharacteristic, complemented
49#22576071C2211yescharacteristic, complemented
50C22×Q8×C271C2211yescharacteristic
51C22×Q8×C271C2211yescharacteristic, complemented
52C4⋊Q8×C271—21no—
53C4⋊Q8×C271C2211yescharacteristic, complemented
54C22⋊Q8×C271—21no—
55C22⋊Q8×C271—21no—
56C22⋊Q8×C271C2211yescharacteristic
57C22⋊Q8×C271C2211yescharacteristic, complemented
58C4×Q8×C271C2211yescharacteristic, complemented
59C4×D4×C271—21no—
60C4×D4×C271C2211yescharacteristic
61C23×D271—21no—
62#16864654—22no—
63#16864654—41no—
64C2×Dic542C2311yescharacteristic
65C2×D1084—22no—
66C2×D1084—21no—
67C2×D1084—41no—
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74#16864644—22no—
75#16864644—22no—
76#16864644—41no—
77#16864641—22no—
78#16864641—22no—
79#16864641—22no—
80#16864641—22no—
81#16864641—41no—
82#16864641D414yescomplemented
83#16864638—22no—
84C2×C271⋊C8—22no—
85C2×C271⋊C8C2311yescharacteristic
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C2×C271⋊C8C2311yescharacteristic
89C4⋊Dic271C2311yescharacteristic
90Dic271⋊C4—21no—
91C22⋊Dic271—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96C2×Q8×C271—21no—
97C2×Q8×C271—21no—
98C2×Q8×C271—22no—
99C2×Q8×C271—22no—
100C2×Q8×C271—41no—
101C2×Q8×C271—41no—
102C2×Q8×C271C2311yescharacteristic
103C2×Q8×C271C2311yescharacteristic
104C2×Q8×C271C2311yescharacteristic
105C2×D4×C271D411yescharacteristic
106C22×C4×C271—21no—
107C22×C4×C271—21no—
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C4⋊C4×C271—21no—
111C4⋊C4×C271—21no—
112C4⋊C4×C271—21no—
113C4⋊C4×C271—21no—
114C4⋊C4×C271—21no—
115C4⋊C4×C271D411yescharacteristic
116C4⋊C4×C271C2311yescharacteristic
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271D411yescharacteristic
120C22⋊C4×C271D411yescharacteristic
121C42×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—81no—
128Dic542—21no—
129Dic542—21no—
130D1084—21no—
131D1084—21no—
132D1084—21no—
133D1084—42no—
134D1084C2×D412yes—
135C22⋊D271—42no—
136C22⋊D271—81no—
137C2×Dic271—21no—
138C2×Dic271—21no—
139C4×D271—42no—
140C4×D271—81no—
141C271⋊C8—21no—
142C271⋊C8—21no—
143C271⋊C8—21no—
144C271⋊C8C2×D412yes—
145C23×C271C2×D411yescharacteristic
146C23×C271C2×D411yescharacteristic
147Q8×C271—21no—
148Q8×C271—21no—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—42no—
153Q8×C271—42no—
154Q8×C271C2×D412yes—
155D4×C271—22no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—42no—
165C2×C4×C271—42no—
166C2×C4×C271C2×D411yescharacteristic
167C2×C4×C271C2×D411yescharacteristic
168C2×C4×C271C2×D411yescharacteristic
169C2×C4×C271C2×D411yescharacteristic
170C2×C4×C271C2411yescharacteristic
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271#13712yescomplemented
177C22×C271C22×D411yescharacteristic
178C2×D271—22no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C2×D271—41no—
183C2×D271—41no—
184C2×D271—84no—
185C4×C271—21no—
186C4×C271—21no—
187C4×C271—22no—
188C4×C271—21no—
189C4×C271—21no—
190C4×C271—41no—
191C4×C271—41no—
192C4×C271—41no—
193C4×C271C22×D411yescharacteristic
194C4×C271H(5,𝔽2)11yescharacteristic
195Dic271—41no—
196Dic271—41no—
197C2×C271—22no—
198C2×C271—41no—
199C2×C271D4211yescharacteristic
200C2×C271C2×#13711yescharacteristic
201C2×C271#57711yescharacteristic
202D271—42no—
203D271—81no—
204C271#326611yescharacteristic, complemented, Sylow
205#3266—2711nomaximal, Sylow
206C2×#137—2711no—
207C22×SD16—2711no—
208D4×Q8D27111yescharacteristic, complemented
209#546—2711no—
210#497—2711no—
211#468—2711no—
212#467—2711no—
213#464—2711no—
214#461—2711no—
215#452—2711no—
216#451—2711no—
217#450—2711no—
218#448—2711no—
219#440—2711no—
220#435—2711no—
221C2×#41—2711no—
222C22×Q8C2×D27111yescharacteristic
223C22×Q8C2×D27111yescharacteristic, complemented
224C22×D4—2711no—
225#137—5424no—
226C2×Q16—2711no—
227C2×Q16—2711no—
228C2×Q16—2711no—
229C2×SD16—2711no—
230C2×SD16—2711no—
231C2×SD16—2711no—
232C2×SD16—2711no—
233C2×SD16—2711no—
234C2×SD16—2711no—
235C2×SD16—5424no—
236C2×M16—2711no—
237C22×C8—2711no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D27111yescharacteristic, complemented
240#124—2711no—
241#123—5421no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—2711no—
245C22⋊Q8C2×D27111yescharacteristic
246C22⋊Q8C2×D27111yescharacteristic, complemented
247#121—2711no—
248#121—5421no—
249#120—5421no—
250C4×Q8C2×D27111yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—2711no—
253C4×D4C2×D27111yescharacteristic
254C2×C22⋊C4—5421no—
255#107—2711no—
256C4⋊C8—2711no—
257#103—2711no—
258#103—2711no—
259#103—2711no—
260#103—2711no—
261#102—2711no—
262#102—2711no—
263C22⋊C8—2711no—
264C22⋊C8—2711no—
265#97—2711no—
266C24—5421no—
267#41—5422no—
268#41—10841no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—2711no—
276C2×Q8C22×D27111yescharacteristic
277C2×Q8C22×D27111yescharacteristic
278C2×Q8C22×D27111yescharacteristic
279C2×D4—2711no—
280C2×D4—2711no—
281C2×D4—5422no—
282C2×D4—5421no—
283C2×D4—5421no—
284C2×D4—5421no—
285C2×D4—10841no—
286C2×D4C22⋊D27111yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—5421no—
290C22×C4—5421no—
291C22×C4C22×D27111yescharacteristic
292C22×C4C22×D27111yescharacteristic
293Q16—5422no—
294Q16—5422no—
295Q16—10841no—
296SD16—2714no—
297SD16—5422no—
298SD16—5422no—
299SD16—5422no—
300SD16—5422no—
301SD16—10841no—
302M16—5422no—
303C2×C8—2711no—
304C2×C8—2711no—
305C2×C8—2711no—
306C2×C8—2711no—
307C2×C8—5422no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—2711no—
314C4⋊C4—5421no—
315C4⋊C4C22⋊D27111yescharacteristic
316C4⋊C4C22×D27111yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C22⋊C4—5421no—
324C22⋊C4C22⋊D27111yescharacteristic
325C22⋊C4C22⋊D27111yescharacteristic
326C42—21no—
327C42C22×D27111yescharacteristic
328C23—5421no—
329C23—5421no—
330C23—10842no—
331C23—10842no—
332C23—21681no—
333C23C2×C22⋊D27111yescharacteristic
334C23C2×C22⋊D27111yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—5421no—
343Q8—5421no—
344Q8D4×D27112yes—
345D4—22no—
346D4—2712no—
347D4—5421no—
348D4—5421no—
349D4—5421no—
350D4—10842no—
351D4—10842no—
352D4—21681no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—10842no—
366C2×C4—21681no—
367C2×C4C2×C22⋊D27111yescharacteristic
368C2×C4C2×C22⋊D27111yescharacteristic
369C2×C4C2×C22⋊D27111yescharacteristic
370C2×C4C2×C22⋊D27111yescharacteristic
371C2×C4C23×D27111yescharacteristic
372C8—2712no—
373C8—5421no—
374C8—5421no—
375C8—5421no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—5422no—
382C22—10841no—
383C22—10841no—
384C22—10841no—
385C22—10841no—
386C22—10841no—
387C22—21684no—
388C22#2257619112yescomplemented
389C22C22×C22⋊D27111yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—10841no—
399C4—10841no—
400C4C2×D4×D27111yescharacteristic
401C4#2257620511yescharacteristic
402C2—22no—
403C2—41no—
404C2—10842no—
405C2—21681no—
406C2D4×C22⋊D27111yescharacteristic
407C2C2×#2257619111yescharacteristic
408C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingD4×Q8×C271C2
socleC22×C271C22×D4
prime radical#31774543C2
hypercenterD4×Q8D271
hypocenterC271#3266
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22576191C2semidirect
2. C22×#16864641C2semidirect
4. #31774289C2semidirect
5. #31774145C2semidirect
6. #31774136C2semidirect
7. #31774119C2semidirect
8. #31774099C2semidirect
9. #31774057C2semidirect
10. #31774053C2semidirect
11. #31774047C2semidirect
12. #31774036C2semidirect
13. #31773995C2semidirect
14. #31773969C2semidirect
15. D4×Q8×C271C2semidirect
19. C2×#16864644C22semidirect
20. C2×#16864644C22semidirect
21. C2×#16864644C22semidirect
29. C2×#16864638C22semidirect
39. #22576093C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576084C22semidirect
42. #22576084C22semidirect
43. #22576084C22semidirect
44. #22576083C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. #22576071C22semidirect
51. C22×Q8×C271C22semidirect
53. C4⋊Q8×C271C22semidirect
57. C22⋊Q8×C271C22semidirect
58. C4×Q8×C271C22semidirect
82. #16864641D4semidirect
176. C22×C271#137semidirect
204. C271#3266semidirect
208. D4×Q8D271semidirect
223. C22×Q8C2×D271semidirect
239. C4⋊Q8C2×D271semidirect
246. C22⋊Q8C2×D271semidirect
250. C4×Q8C2×D271semidirect
388. C22#22576191semidirect

Maximal subgroups

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

subgroupindex
C2×#225761912
C22×#168646412
#317745432
#317742892
#317741452
#317741362
#317741192
#317740992
#317740572
#317740532
#317740472
#317740362
#317739952
#317739692
D4×Q8×C2712
#3266271