Finite Group Databasebeta

← #46847910

Subgroups — #46847910

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×#22576184C211yesmaximal, characteristic, complemented
2C2×#22576179C211yesmaximal, characteristic, complemented
3#31774567C211yesmaximal, characteristic, complemented
4#31774542C211yesmaximal, characteristic, complemented
5#31774284C211yesmaximal, characteristic, complemented
6#31774142C211yesmaximal, characteristic, complemented
7#31774132C211yesmaximal, characteristic, complemented
8#31774120C211yesmaximal, characteristic, complemented
9#31774060C211yesmaximal, characteristic, complemented
10#31774043C211yesmaximal, characteristic, complemented
11#31774039C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13#31773997C211yesmaximal, characteristic, complemented
14#31773967C211yesmaximal, characteristic, complemented
15#459×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×#16864653C2211yescharacteristic, complemented
19C2×D4×D271C2211yescharacteristic
20#22576184—24no—
21#22576179—24no—
22C2×Dic1084C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic, complemented
26C2×#16864642C2211yescharacteristic, complemented
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864637C2211yescharacteristic
30C2×C8×D271C2211yescharacteristic, complemented
31#22576156—21no—
32#22576148C2211yescharacteristic
33#22576144—21no—
34#22576140C2211yescharacteristic
35#22576135C2211yescharacteristic
36#22576134—21no—
37#22576132—21no—
38#22576132—21no—
39#22576131C2211yescharacteristic
40#22576131C2211yescharacteristic, complemented
41#22576130—21no—
42C4×Dic542C2211yescharacteristic, complemented
43C4×D1084C2211yescharacteristic
44D4×Dic271—21no—
45C2×(C2×D271)⋊C4—21no—
46#22576093C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576083C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic
50#22576081C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53#22576071C2211yescharacteristic, complemented
54C2×D8×C271C2211yescharacteristic
55C2×D8×C271C2211yescharacteristic
56#127×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62#16864653—22no—
63#16864653—22no—
64#16864653—22no—
65#16864653—41no—
66#16864653—41no—
67#16864653—41no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2311yescharacteristic
70C2×D1084—21no—
71C2×D1084D411yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77C2×C22⋊D271—21no—
78C2×C22⋊D271—21no—
79D4×D271—22no—
80D4×D271—41no—
81C22×Dic271—21no—
82C22×Dic271—21no—
83C22×Dic271—21no—
84C2×C4×D271C2311yescharacteristic
85C2×C4×D271C2311yescharacteristic
86Dic1084—22no—
87#16864643—22no—
88#16864642—22no—
89#16864642—41no—
90#16864642D414yescomplemented
91#16864639—22no—
92#16864639—41no—
93#16864637—22no—
94C2×C271⋊C8C2311yescharacteristic
95C2×C271⋊C8C2311yescharacteristic
96C8×D271—22no—
97C4⋊Dic271—21no—
98C4⋊Dic271D411yescharacteristic
99C4⋊Dic271C2311yescharacteristic
100Dic271⋊C4—21no—
101C22⋊Dic271—21no—
102C22⋊Dic271—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4—21no—
107(C2×D271)⋊C4D411yescharacteristic
108(C2×D271)⋊C4D411yescharacteristic
109C4×Dic271—21no—
110C2×D4×C271—21no—
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271—21no—
116D8×C271—22no—
117D8×C271—22no—
118C2×C8×C271C2311yescharacteristic
119C2×C8×C271C2311yescharacteristic
120C4⋊C4×C271C2311yescharacteristic
121C22⋊C4×C271—21no—
122C42×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271—42no—
125C22×D271—42no—
126C22×D271—81no—
127C22×D271C2×D411yescharacteristic
128C22×D271C2×D411yescharacteristic
129Dic542—21no—
130Dic542—21no—
131Dic542—21no—
132Dic542C2×D412yes—
133D1084—21no—
134D1084—21no—
135D1084—41no—
136D1084—81no—
137C22⋊D271—42no—
138C22⋊D271—42no—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C22⋊D271—81no—
142C22⋊D271—81no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—42no—
147C2×Dic271—42no—
148C2×Dic271—42no—
149C2×Dic271C2×D411yescharacteristic
150C2×Dic271C2×D411yescharacteristic
151C4×D271—21no—
152C4×D271—21no—
153C4×D271—21no—
154C4×D271—21no—
155C271⋊C8—21no—
156C271⋊C8C2×D412yes—
157C23×C271—21no—
158C23×C271—21no—
159C23×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—21no—
163D4×C271—21no—
164D4×C271—21no—
165D4×C271—42no—
166D4×C271—81no—
167D4×C271C2×D412yes—
168C2×C4×C271—21no—
169C2×C4×C271—81no—
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2411yescharacteristic
173C8×C271—21no—
174C8×C271—21no—
175C22×C271—22no—
176C22×C271—41no—
177C22×C271—41no—
178C22×C271—41no—
179C22×C271—41no—
180C22×C271—41no—
181C22×C271—41no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271C22×D411yescharacteristic
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—21no—
188C2×D271—21no—
189C2×D271—21no—
190C2×D271—21no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—84no—
195C4×C271—21no—
196C4×C271—21no—
197C4×C271—41no—
198C4×C271C22×D411yescharacteristic
199C4×C271H(5,𝔽2)11yescharacteristic
200Dic271—21no—
201Dic271—21no—
202Dic271—22no—
203Dic271—41no—
204Dic271—41no—
205C2×C271—42no—
206C2×C271—81no—
207C2×C271—81no—
208C2×C271D4211yescharacteristic
209C2×C271#57511yescharacteristic
210C2×C271#57611yescharacteristic
211D271—41no—
212D271—41no—
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#464—2711no—
224#459D27111yescharacteristic, complemented
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×Q16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16—2711no—
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8C2×D27111yescharacteristic
245C2×D8C2×D27111yescharacteristic
246C2×M16—2711no—
247C22×C8—2711no—
248#127—5421no—
249#127C2×D27111yescharacteristic, complemented
250#124—2711no—
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×Q8—2711no—
261C4×D4—2711no—
262C4×D4—5421no—
263C4×D4C2×D27111yescharacteristic
264C2×C22⋊C4—5421no—
265#107—2711no—
266C4⋊C8C2×D27111yescharacteristic, complemented
267#103—2711no—
268#103—2711no—
269#102—2711no—
270#102—2711no—
271#102C2×D27111yescharacteristic, complemented
272#102C2×D27111yescharacteristic, complemented
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×Q8—2711no—
284C2×Q8—2711no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2711no—
288C2×D4—2711no—
289C2×D4—5421no—
290C2×D4—5421no—
291C2×D4—5422no—
292C2×D4—5421no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—5421no—
297C2×D4—10841no—
298C2×D4C22×D27111yescharacteristic
299C2×D4C22×D27111yescharacteristic
300C2×D4C22×D27111yescharacteristic
301C22×C4—21no—
302C22×C4—2711no—
303C22×C4—2711no—
304C22×C4—5421no—
305C22×C4—5421no—
306C22×C4—5421no—
307Q16—5422no—
308SD16—2714no—
309SD16—5422no—
310SD16—5422no—
311SD16—10841no—
312D8—22no—
313D8—22no—
314D8—5422no—
315D8—10841no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—5422no—
320C2×C8C22×D27111yescharacteristic
321C2×C8C22×D27111yescharacteristic
322C4⋊C4—2711no—
323C4⋊C4—2711no—
324C4⋊C4—5421no—
325C4⋊C4—5421no—
326C4⋊C4C22×D27111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—2711no—
329C22⋊C4—2711no—
330C22⋊C4—5421no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C42—5421no—
337C42C22×D27111yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2711no—
342C23—2711no—
343C23—5421no—
344C23—10842no—
345C23—10842no—
346C23—21681no—
347Q8—2712no—
348Q8—5421no—
349Q8—5421no—
350Q8—5421no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—5421no—
359D4—5421no—
360D4—10841no—
361D4—10842no—
362D4—10842no—
363D4—10842no—
364D4—10842no—
365D4—21681no—
366D4—21681no—
367D4—21681no—
368D4D4×D27112yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—2711no—
372C2×C4—2711no—
373C2×C4—5421no—
374C2×C4—5421no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—10842no—
381C2×C4—10842no—
382C2×C4—10842no—
383C2×C4D4×D27111yescharacteristic
384C2×C4D4×D27111yescharacteristic
385C2×C4C23×D27111yescharacteristic
386C8—21no—
387C8—21no—
388C8—2712no—
389C8—5421no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5421no—
400C22—5421no—
401C22—5421no—
402C22—5421no—
403C22—5421no—
404C22—5421no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—21684no—
409C22C2×D4×D27111yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5421no—
414C4—5421no—
415C4—5422no—
416C4—10841no—
417C4—10841no—
418C4C2×D4×D27111yescharacteristic
419C4#2257620611yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—10841no—
424C2—10841no—
425C2—21681no—
426C2#3177453111yescharacteristic
427C2#3177469311yescharacteristic
428C2#3177469811yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#459×C271C2
socleC22×C271C22×D4
hypercenter#459D271
hypocenterC271#3268
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#22576184C2semidirect
2. C2×#22576179C2semidirect
3. #31774567C2semidirect
4. #31774542C2semidirect
5. #31774284C2semidirect
6. #31774142C2semidirect
7. #31774132C2semidirect
8. #31774120C2semidirect
9. #31774060C2semidirect
10. #31774043C2semidirect
11. #31774039C2semidirect
12. #31774029C2semidirect
13. #31773997C2semidirect
14. #31773967C2semidirect
15. #459×C271C2semidirect
18. C2×#16864653C22semidirect
22. C2×Dic1084C22semidirect
25. C2×#16864642C22semidirect
26. C2×#16864642C22semidirect
30. C2×C8×D271C22semidirect
40. #22576131C22semidirect
42. C4×Dic542C22semidirect
46. #22576093C22semidirect
47. #22576085C22semidirect
48. #22576083C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. #22576071C22semidirect
56. #127×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #102×C271C22semidirect
60. #102×C271C22semidirect
90. #16864642D4semidirect
214. C271#3268semidirect
224. #459D271semidirect
249. #127C2×D271semidirect
266. C4⋊C8C2×D271semidirect
271. #102C2×D271semidirect
272. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761842
C2×#225761792
#317745672
#317745422
#317742842
#317741422
#317741322
#317741202
#317740602
#317740432
#317740392
#317740292
#317739972
#317739672
#459×C2712
#3268271