Finite Group Databasebeta

← #46847899

Subgroups — #46847899

Order 34688.

510 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
1C2×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576183C211yesmaximal, characteristic, complemented
3#31774569C211yesmaximal, characteristic, complemented
4#31774540C211yesmaximal, characteristic, complemented
5#31774287C211yesmaximal, characteristic, complemented
6#31774146C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774093C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774045C211yesmaximal, characteristic, complemented
11#31774041C211yesmaximal, characteristic, complemented
12#31774032C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773967C211yesmaximal, characteristic, complemented
15#464×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×#16864655C2211yescharacteristic, complemented
19C2×D4×D271C2211yescharacteristic
20#22576187—24no—
21#22576183—24no—
22C2×#16864641C2211yescharacteristic
23C2×#16864641C2211yescharacteristic, complemented
24C2×#16864641C2211yescharacteristic, complemented
25C2×D2168C2211yescharacteristic
26C2×D2168C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864637C2211yescharacteristic
30C2×C8×D271C2211yescharacteristic, complemented
31#22576157—21no—
32#22576157C2211yescharacteristic, complemented
33#22576142—21no—
34#22576136C2211yescharacteristic
35#22576135—21no—
36#22576135—21no—
37#22576135C2211yescharacteristic
38#22576135C2211yescharacteristic, complemented
39#22576133—21no—
40#22576131C2211yescharacteristic
41#22576130—21no—
42C4×D1084—21no—
43C4×D1084C2211yescharacteristic
44C4×D1084C2211yescharacteristic
45C22⋊C4×D271—21no—
46#22576093C2211yescharacteristic, complemented
47#22576084C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576071C2211yescharacteristic, complemented
54C2×Q16×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic
61C23×D271—21no—
62#16864655—22no—
63#16864655—22no—
64#16864655—22no—
65#16864655—41no—
66#16864655—41no—
67#16864655—41no—
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084—21no—
72C2×D1084—21no—
73C2×D1084—21no—
74C2×D1084—21no—
75C2×D1084D411yescharacteristic
76C2×D1084C2311yescharacteristic
77C2×D1084C2311yescharacteristic
78C2×D1084C2311yescharacteristic
79C2×C22⋊D271—21no—
80C2×C22⋊D271—21no—
81D4×D271—22no—
82D4×D271—41no—
83C2×C4×D271—21no—
84C2×C4×D271—21no—
85C2×C4×D271—21no—
86C2×C4×D271—21no—
87C2×C4×D271C2311yescharacteristic
88C2×C4×D271C2311yescharacteristic
89#16864641—22no—
90#16864641—41no—
91#16864641D414yescomplemented
92D2168—22no—
93D2168—22no—
94#16864639—22no—
95#16864639—41no—
96#16864637—22no—
97C2×C271⋊C8C2311yescharacteristic
98C2×C271⋊C8C2311yescharacteristic
99C8×D271—22no—
100C4⋊Dic271D411yescharacteristic
101C4⋊Dic271C2311yescharacteristic
102Dic271⋊C4—21no—
103C22⋊Dic271—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4—21no—
108(C2×D271)⋊C4D411yescharacteristic
109(C2×D271)⋊C4D411yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×Q8×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113Q16×C271—22no—
114SD16×C271—22no—
115C2×C8×C271C2311yescharacteristic
116C2×C8×C271C2311yescharacteristic
117C4⋊C4×C271—21no—
118C4⋊C4×C271C2311yescharacteristic
119C22⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C42×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—81no—
129C22×D271C2×D411yescharacteristic
130C22×D271C2×D411yescharacteristic
131D1084—21no—
132D1084—21no—
133D1084—21no—
134D1084—21no—
135D1084—21no—
136D1084—41no—
137D1084—42no—
138D1084—42no—
139D1084—42no—
140D1084—42no—
141D1084—81no—
142D1084—81no—
143D1084—81no—
144D1084C2×D412yes—
145C22⋊D271—42no—
146C22⋊D271—81no—
147C2×Dic271—21no—
148C2×Dic271C2×D411yescharacteristic
149C2×Dic271C2×D411yescharacteristic
150C4×D271—21no—
151C4×D271—21no—
152C4×D271—21no—
153C4×D271—21no—
154C4×D271—42no—
155C4×D271—42no—
156C4×D271—42no—
157C4×D271—81no—
158C271⋊C8—21no—
159C271⋊C8C2×D412yes—
160C23×C271—21no—
161Q8×C271—21no—
162Q8×C271—21no—
163Q8×C271—21no—
164Q8×C271C2×D412yes—
165D4×C271—21no—
166D4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2411yescharacteristic
173C8×C271—21no—
174C8×C271—21no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271C22×D411yescharacteristic
179C2×D271—21no—
180C2×D271—21no—
181C2×D271—22no—
182C2×D271—21no—
183C2×D271—21no—
184C2×D271—21no—
185C2×D271—21no—
186C2×D271—41no—
187C2×D271—41no—
188C2×D271—41no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—84no—
195C4×C271—22no—
196C4×C271—21no—
197C4×C271—21no—
198C4×C271—41no—
199C4×C271—41no—
200C4×C271C22×D411yescharacteristic
201C4×C271H(5,𝔽2)11yescharacteristic
202Dic271—21no—
203Dic271—21no—
204Dic271—41no—
205C2×C271—81no—
206C2×C271D4211yescharacteristic
207C2×C271#57511yescharacteristic
208C2×C271#57611yescharacteristic
209D271—41no—
210D271—41no—
211D271—42no—
212D271—81no—
213D271—81no—
214C271#326811yescharacteristic, complemented, Sylow
215#3268—2711nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2711no—
217C2×#135—2711no—
218#550—2711no—
219#546—2711no—
220#496—2711no—
221#468—2711no—
222#466—2711no—
223#464D27111yescharacteristic, complemented
224#459—2711no—
225#452—2711no—
226#449—2711no—
227#449—2711no—
228#447—2711no—
229#440—2711no—
230#435—2711no—
231C2×#41—2711no—
232C2×#41—2711no—
233C2×#41—2711no—
234C22×D4—2711no—
235AGL(1,ℤ/8)—5424no—
236#135—5424no—
237C2×Q16C2×D27111yescharacteristic, complemented
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16C2×D27111yescharacteristic
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8—2711no—
246C2×M16—2711no—
247C22×C8—2711no—
248#127—2711no—
249#127—5421no—
250#124C2×D27111yescharacteristic
251#123—5421no—
252C22⋊Q8—2711no—
253#121—2711no—
254#121—2711no—
255#121—2711no—
256#121—5421no—
257#121—5421no—
258#121—5421no—
259#120—5421no—
260C4×Q8C2×D27111yescharacteristic, complemented
261C4×D4—2711no—
262C4×D4—2711no—
263C4×D4—5421no—
264C2×C22⋊C4—5421no—
265#107—2711no—
266C4⋊C8C2×D27111yescharacteristic, complemented
267#103—2711no—
268#103C2×D27111yescharacteristic, complemented
269#102—2711no—
270#102—2711no—
271#102—2711no—
272#102C2×D27111yescharacteristic
273C22⋊C8—2711no—
274C22⋊C8—2711no—
275#97—2711no—
276C24—5421no—
277#41—5422no—
278#41—5422no—
279#41—5422no—
280#41—10841no—
281#41—10841no—
282#41—10841no—
283C2×Q8C22×D27111yescharacteristic
284C2×Q8C22×D27111yescharacteristic
285C2×D4—2711no—
286C2×D4—2711no—
287C2×D4—2711no—
288C2×D4—2711no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4—5421no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5422no—
296C2×D4—5421no—
297C2×D4—5421no—
298C2×D4—5421no—
299C2×D4—10841no—
300C2×D4C22×D27111yescharacteristic
301C22×C4—2711no—
302C22×C4—2711no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4—5421no—
306C22×C4—5421no—
307Q16—22no—
308SD16—22no—
309SD16—2714no—
310SD16—5422no—
311SD16—10841no—
312D8—5422no—
313D8—5422no—
314D8—5422no—
315D8—10841no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—5422no—
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—2711no—
324C4⋊C4—2711no—
325C4⋊C4—5421no—
326C4⋊C4C22×D27111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2711no—
330C22⋊C4—2711no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C42—21no—
337C42C22×D27111yescharacteristic
338C23—21no—
339C23—2711no—
340C23—2711no—
341C23—5421no—
342C23—5421no—
343C23—5421no—
344C23—10842no—
345C23—10842no—
346C23—21681no—
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8D4×D27112yes—
351D4—21no—
352D4—21no—
353D4—2712no—
354D4—5421no—
355D4—5421no—
356D4—5421no—
357D4—5421no—
358D4—5421no—
359D4—10841no—
360D4—10842no—
361D4—10842no—
362D4—10842no—
363D4—10842no—
364D4—10842no—
365D4—21681no—
366D4—21681no—
367D4—21681no—
368D4—21681no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—2711no—
373C2×C4—2711no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—10842no—
380C2×C4—10842no—
381C2×C4—10842no—
382C2×C4—21681no—
383C2×C4D4×D27111yescharacteristic
384C2×C4D4×D27111yescharacteristic
385C2×C4C23×D27111yescharacteristic
386C8—21no—
387C8—21no—
388C8—2712no—
389C8—5421no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—5421no—
394C22—5421no—
395C22—5422no—
396C22—5421no—
397C22—5421no—
398C22—5421no—
399C22—5421no—
400C22—10841no—
401C22—10841no—
402C22—10841no—
403C22—10841no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—21684no—
409C22C2×D4×D27111yescharacteristic
410C4—22no—
411C4—21no—
412C4—21no—
413C4—41no—
414C4—41no—
415C4—5421no—
416C4—5421no—
417C4—10841no—
418C4C2×D4×D27111yescharacteristic
419C4#2257620611yescharacteristic
420C2—81no—
421C2—10841no—
422C2—10841no—
423C2—10842no—
424C2—21681no—
425C2—21681no—
426C2#3177453111yescharacteristic
427C2#3177469611yescharacteristic
428C2#3177469911yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#464×C271C2
socleC22×C271C22×D4
hypercenter#464D271
hypocenterC271#3268
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576183C2semidirect
3. #31774569C2semidirect
4. #31774540C2semidirect
5. #31774287C2semidirect
6. #31774146C2semidirect
7. #31774132C2semidirect
8. #31774093C2semidirect
9. #31774058C2semidirect
10. #31774045C2semidirect
11. #31774041C2semidirect
12. #31774032C2semidirect
13. #31773995C2semidirect
14. #31773967C2semidirect
15. #464×C271C2semidirect
18. C2×#16864655C22semidirect
23. C2×#16864641C22semidirect
24. C2×#16864641C22semidirect
30. C2×C8×D271C22semidirect
32. #22576157C22semidirect
38. #22576135C22semidirect
46. #22576093C22semidirect
47. #22576084C22semidirect
48. #22576082C22semidirect
49. #22576082C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576071C22semidirect
54. C2×Q16×C271C22semidirect
57. C4×Q8×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
91. #16864641D4semidirect
214. C271#3268semidirect
223. #464D271semidirect
237. C2×Q16C2×D271semidirect
260. C4×Q8C2×D271semidirect
266. C4⋊C8C2×D271semidirect
268. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761832
#317745692
#317745402
#317742872
#317741462
#317741322
#317740932
#317740582
#317740452
#317740412
#317740322
#317739952
#317739672
#464×C2712
#3268271