Finite Group Databasebeta

← #46848053

Subgroups — #46848053

Order 34688.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864641C211yesmaximal, characteristic, complemented
2C2×SD16×D271C211yesmaximal, characteristic, complemented
3Q8×C22⋊D271C211yesmaximal, characteristic, complemented
4#31774531C211yesmaximal, characteristic
5#31774268C211yesmaximal, characteristic, complemented
6#31774129C211yesmaximal, characteristic, complemented
7#31774124C211yesmaximal, characteristic, complemented
8#31774101C211yesmaximal, characteristic, complemented
9#31774095C211yesmaximal, characteristic, complemented
10#31774050C211yesmaximal, characteristic, complemented
11#31774046C211yesmaximal, characteristic, complemented
12#31774037C211yesmaximal, characteristic, complemented
13#31773982C211yesmaximal, characteristic, complemented
14(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
15#448×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic, complemented
17C22×D1084C2211yescharacteristic
18C2×D4×D271—21no—
19C2×D4×D271C2211yescharacteristic
20C2×#16864643C2211yescharacteristic
21C2×#16864643C2211yescharacteristic, complemented
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864641—24no—
24C2×#16864641C2211yescharacteristic
25C2×#16864641C2211yescharacteristic
26C2×#16864641C2211yescharacteristic, complemented
27C2×#16864641C2211yescharacteristic, complemented
28SD16×D271—24no—
29C22×C271⋊C8C2211yescharacteristic, complemented
30C2×C8×D271C2211yescharacteristic, complemented
31#22576158—21no—
32#22576158C2211yescharacteristic, complemented
33#22576156C2211yescharacteristic
34#22576139—21no—
35#22576139C2211yescharacteristic, complemented
36#22576136—21no—
37#22576136C2211yescharacteristic, complemented
38#22576135C2211yescharacteristic
39#22576133—21no—
40#22576130—21no—
41#22576128—21no—
42Q8×Dic271C2211yescharacteristic, complemented
43#22576122C2211yescharacteristic
44C4×C22⋊D271—21no—
45C4×C22⋊D271C2211yescharacteristic
46#22576091C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic
50#22576082C2211yescharacteristic, complemented
51#22576081C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53C8×Dic271C2211yescharacteristic, complemented
54C22×Q8×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C2×SD16×C271C2211yescharacteristic, complemented
57#121×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60C22⋊C8×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63C2×Dic542—21no—
64C2×Dic542C2311yescharacteristic
65Q8×D271—22no—
66Q8×D271—41no—
67C2×D1084—22no—
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084—41no—
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271D411yescharacteristic
77D4×D271—22no—
78D4×D271—41no—
79D4×D271—44no—
80C2×C4×D271—21no—
81C2×C4×D271—21no—
82C2×C4×D271C2311yescharacteristic
83#16864643—22no—
84#16864643—41no—
85#16864642—22no—
86#16864641—22no—
87#16864641—22no—
88#16864641—22no—
89#16864641D414yescomplemented
90C2×C271⋊C8—22no—
91C2×C271⋊C8C2311yescharacteristic
92C2×C271⋊C8C2311yescharacteristic
93C8×D271—22no—
94C4⋊Dic271—21no—
95C4⋊Dic271C2311yescharacteristic
96Dic271⋊C4—21no—
97Dic271⋊C4—21no—
98Dic271⋊C4—21no—
99Dic271⋊C4—21no—
100Dic271⋊C4D411yescharacteristic
101C22⋊Dic271—21no—
102C22⋊Dic271D411yescharacteristic
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4D411yescharacteristic
107C4×Dic271—21no—
108C4×Dic271C2311yescharacteristic
109C2×Q8×C271—21no—
110C2×Q8×C271—22no—
111C2×Q8×C271—41no—
112C2×Q8×C271C2311yescharacteristic
113C2×Q8×C271C2311yescharacteristic
114C2×D4×C271—21no—
115C2×D4×C271C2311yescharacteristic
116C22×C4×C271—21no—
117C22×C4×C271C2311yescharacteristic
118SD16×C271—22no—
119SD16×C271—41no—
120C2×C8×C271C2311yescharacteristic
121C2×C8×C271C2311yescharacteristic
122C4⋊C4×C271C2311yescharacteristic
123C22⋊C4×C271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—42no—
130C22×D271—81no—
131C22×D271—81no—
132C22×D271C2×D411yescharacteristic
133Dic542—21no—
134Dic542—21no—
135Dic542—42no—
136D1084—21no—
137D1084—21no—
138D1084—21no—
139D1084—42no—
140D1084—42no—
141D1084C2×D412yes—
142C22⋊D271—42no—
143C22⋊D271—42no—
144C22⋊D271—81no—
145C22⋊D271SD1614yescomplemented
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271C2×D411yescharacteristic
149C2×Dic271C2×D411yescharacteristic
150C2×Dic271C2×D411yescharacteristic
151C4×D271—21no—
152C4×D271—21no—
153C4×D271—42no—
154C4×D271—42no—
155C271⋊C8—21no—
156C271⋊C8C2×D412yescomplemented
157C23×C271—21no—
158C23×C271C2×D411yescharacteristic
159Q8×C271—21no—
160Q8×C271—21no—
161Q8×C271—21no—
162Q8×C271—42no—
163Q8×C271C2×D412yescomplemented
164D4×C271—21no—
165D4×C271—21no—
166D4×C271—42no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—21no—
171C2×C4×C271—21no—
172C2×C4×C271—42no—
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2411yescharacteristic
175C8×C271—21no—
176C8×C271—21no—
177C22×C271—21no—
178C22×C271—21no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271C2×SD1612yescomplemented
183C22×C271C22×D411yescharacteristic
184C2×D271—21no—
185C2×D271—21no—
186C2×D271—22no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—84no—
193C2×D271—84no—
194C2×D271C2×SD1612yescomplemented
195C4×C271—21no—
196C4×C271—22no—
197C4×C271—41no—
198C4×C271—41no—
199C4×C271C22×D411yescharacteristic
200C4×C271H(5,𝔽2)11yescharacteristic
201Dic271—21no—
202Dic271—21no—
203Dic271—41no—
204Dic271—41no—
205Dic271C2×SD1612yescomplemented
206C2×C271—22no—
207C2×C271—81no—
208C2×C271D4211yescharacteristic
209C2×C271C22×SD1611yescharacteristic
210C2×C271#57711yescharacteristic
211D271—22no—
212D271—42no—
213D271—81no—
214C271D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—2711nomaximal, Sylow
216C22×SD16—2711no—
217C22×SD16—2711no—
218D4×Q8—2711no—
219D42—2711no—
220#492—2711no—
221#465—2711no—
222#465—2711no—
223#461—2711no—
224#460—2711no—
225#450—2711no—
226#450—2711no—
227#448—2711no—
228#448D27111yescharacteristic, complemented
229C4×SD16—2711no—
230C8×D4—2711no—
231C22×Q8—2711no—
232C22×Q8C2×D27111yescharacteristic, complemented
233C22×D4—2711no—
234C22×D4—2711no—
235C22×D4—5421no—
236C2×SD16—2711no—
237C2×SD16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16—2711no—
242C2×SD16—2711no—
243C2×SD16—5424no—
244C2×SD16—5424no—
245C2×SD16C2×D27111yescharacteristic, complemented
246C2×SD16C2×D27111yescharacteristic, complemented
247C22×C8—2711no—
248C22×C8—2711no—
249C4⋊Q8—2711no—
250C4⋊Q8—5421no—
251#127—2711no—
252C22⋊Q8—2711no—
253C22⋊Q8—2711no—
254C22⋊Q8—5421no—
255C22⋊Q8—5421no—
256#121—2711no—
257#121—5421no—
258#121C2×D27111yescharacteristic
259#120—5421no—
260#120—5421no—
261C4×Q8—2711no—
262C4×D4—2711no—
263C4×D4—2711no—
264C4×D4—5421no—
265#106—2711no—
266C4⋊C8—2711no—
267#103—2711no—
268#103C2×D27111yescharacteristic, complemented
269#103C2×D27111yescharacteristic, complemented
270#102—2711no—
271#102—2711no—
272#102—2711no—
273C22⋊C8—2711no—
274C22⋊C8C2×D27111yescharacteristic, complemented
275C4×C8—2711no—
276C24—5421no—
277C24—5421no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—2711no—
282C2×Q8—5421no—
283C2×Q8—5422no—
284C2×Q8—10841no—
285C2×Q8C22×D27111yescharacteristic
286C2×Q8C22×D27111yescharacteristic
287C2×D4—21no—
288C2×D4—2711no—
289C2×D4—2711no—
290C2×D4—2711no—
291C2×D4—5422no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5422no—
297C2×D4—5421no—
298C2×D4—10841no—
299C2×D4—10841no—
300C2×D4—10844no—
301C2×D4C22×D27111yescharacteristic
302C22×C4—21no—
303C22×C4—2711no—
304C22×C4—5421no—
305C22×C4—5421no—
306C22×C4C22×D27111yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—2714no—
310SD16—5422no—
311SD16—5422no—
312SD16—5422no—
313SD16—5422no—
314SD16—5422no—
315SD16—10841no—
316C2×C8—2711no—
317C2×C8—2711no—
318C2×C8—5422no—
319C2×C8—5422no—
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—2711no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4—5421no—
326C4⋊C4—5421no—
327C4⋊C4—5421no—
328C4⋊C4—5421no—
329C4⋊C4C22×D27111yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—2711no—
332C22⋊C4—2711no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C42—2711no—
338C42—5421no—
339C23—21no—
340C23—2711no—
341C23—5421no—
342C23—5421no—
343C23—10842no—
344C23—10842no—
345C23—10842no—
346C23—10842no—
347C23—21681no—
348C23—21681no—
349C23D4×D27111yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—21no—
353Q8—42no—
354Q8—5421no—
355Q8—5421no—
356Q8—10842no—
357Q8D4×D27112yescomplemented
358D4—21no—
359D4—21no—
360D4—42no—
361D4—2714no—
362D4—2712no—
363D4—5421no—
364D4—5421no—
365D4—5421no—
366D4—10842no—
367D4—10842no—
368D4—10842no—
369D4—10842no—
370D4—21681no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—2711no—
378C2×C4—2711no—
379C2×C4—2711no—
380C2×C4—5421no—
381C2×C4—5421no—
382C2×C4—5421no—
383C2×C4—5421no—
384C2×C4—10842no—
385C2×C4—10842no—
386C2×C4D4×D27111yescharacteristic
387C2×C4C23×D27111yescharacteristic
388C8—21no—
389C8—21no—
390C8—2712no—
391C8—5421no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—2712no—
398C22—5421no—
399C22—5421no—
400C22—5422no—
401C22—10841no—
402C22—10841no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—21684no—
407C22—21684no—
408C22SD16×D27112yescomplemented
409C22C2×D4×D27111yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—2712no—
415C4—5421no—
416C4—5421no—
417C4—10841no—
418C4—10841no—
419C4C2×D4×D27111yescharacteristic
420C4#2257620511yescharacteristic
421C2—22no—
422C2—81no—
423C2—5422no—
424C2—10842no—
425C2—21681no—
426C2#3177452911yescharacteristic
427C2C2×SD16×D27111yescharacteristic
428C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#448×C271C2
socleC22×C271C22×D4
prime radical#31774531C2
hypercenter#448D271
hypocenterC271D4×SD16
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#16864641C2semidirect
2. C2×SD16×D271C2semidirect
3. Q8×C22⋊D271C2semidirect
5. #31774268C2semidirect
6. #31774129C2semidirect
7. #31774124C2semidirect
8. #31774101C2semidirect
9. #31774095C2semidirect
10. #31774050C2semidirect
11. #31774046C2semidirect
12. #31774037C2semidirect
13. #31773982C2semidirect
14. (C22⋊D271)⋊C8C2semidirect
15. #448×C271C2semidirect
16. C2×Q8×D271C22semidirect
21. C2×#16864643C22semidirect
22. C2×#16864642C22semidirect
26. C2×#16864641C22semidirect
27. C2×#16864641C22semidirect
29. C22×C271⋊C8C22semidirect
30. C2×C8×D271C22semidirect
32. #22576158C22semidirect
35. #22576139C22semidirect
37. #22576136C22semidirect
42. Q8×Dic271C22semidirect
46. #22576091C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576083C22semidirect
50. #22576082C22semidirect
51. #22576081C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. C8×Dic271C22semidirect
54. C22×Q8×C271C22semidirect
55. C2×SD16×C271C22semidirect
56. C2×SD16×C271C22semidirect
58. #103×C271C22semidirect
59. #103×C271C22semidirect
60. C22⋊C8×C271C22semidirect
89. #16864641D4semidirect
145. C22⋊D271SD16semidirect
156. C271⋊C8C2×D4semidirect
163. Q8×C271C2×D4semidirect
182. C22×C271C2×SD16semidirect
194. C2×D271C2×SD16semidirect
205. Dic271C2×SD16semidirect
214. C271D4×SD16semidirect
228. #448D271semidirect
232. C22×Q8C2×D271semidirect
245. C2×SD16C2×D271semidirect
246. C2×SD16C2×D271semidirect
268. #103C2×D271semidirect
269. #103C2×D271semidirect
274. C22⋊C8C2×D271semidirect
357. Q8D4×D271semidirect
408. C22SD16×D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646412
C2×SD16×D2712
Q8×C22⋊D2712
#317745312
#317742682
#317741292
#317741242
#317741012
#317740952
#317740502
#317740462
#317740372
#317739822
(C22⋊D271)⋊C82
#448×C2712
D4×SD16271