Finite Group Databasebeta

← #46848047

Subgroups — #46848047

Order 34688.

526 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576180C211yesmaximal, characteristic, complemented
2C2×D8×D271C211yesmaximal, characteristic, complemented
3#31774544C211yesmaximal, characteristic, complemented
4#31774535C211yesmaximal, characteristic, complemented
5#31774278C211yesmaximal, characteristic, complemented
6#31774137C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774115C211yesmaximal, characteristic, complemented
9#31774110C211yesmaximal, characteristic, complemented
10#31774049C211yesmaximal, characteristic, complemented
11#31774046C211yesmaximal, characteristic, complemented
12#31774044C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic, complemented
17C2×D4×D271C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic
19#22576180—24no—
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864643C2211yescharacteristic, complemented
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864641C2211yescharacteristic, complemented
24C2×D2168C2211yescharacteristic
25C2×#16864639C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27D8×D271—24no—
28C22×C271⋊C8C2211yescharacteristic, complemented
29C2×C8×D271C2211yescharacteristic
30#22576149C2211yescharacteristic, complemented
31#22576147C2211yescharacteristic, complemented
32#22576143—21no—
33#22576142—21no—
34#22576140C2211yescharacteristic
35#22576136C2211yescharacteristic
36#22576134C2211yescharacteristic, complemented
37#22576132—21no—
38#22576131—21no—
39#22576128—21no—
40#22576128—21no—
41D4×Dic271C2211yescharacteristic, complemented
42#22576122C2211yescharacteristic, complemented
43C4×C22⋊D271C2211yescharacteristic
44C2×(C2×D271)⋊C4—21no—
45C22⋊C4×D271—21no—
46#22576093C2211yescharacteristic, complemented
47Dic271⋊C8C2211yescharacteristic, complemented
48#22576085C2211yescharacteristic, complemented
49#22576082C2211yescharacteristic, complemented
50#22576081C2211yescharacteristic, complemented
51#22576080C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic
53C8×Dic271C2211yescharacteristic, complemented
54C2×#41×C271C2211yescharacteristic, complemented
55C2×SD16×C271C2211yescharacteristic, complemented
56C2×D8×C271C2211yescharacteristic
57#121×C271C2211yescharacteristic, complemented
58#103×C271C2211yescharacteristic, complemented
59#102×C271C2211yescharacteristic, complemented
60C22⋊C8×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864654—22no—
64#16864654—41no—
65C2×Dic542C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271—21no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271D411yescharacteristic
73D4×D271—22no—
74D4×D271—22no—
75D4×D271—41no—
76D4×D271—41no—
77C22×Dic271—21no—
78C2×C4×D271—21no—
79C2×C4×D271—21no—
80C2×C4×D271C2311yescharacteristic
81#16864644—22no—
82#16864643—41no—
83#16864642—22no—
84#16864641—22no—
85D2168—22no—
86#16864639—22no—
87#16864639D414yescomplemented
88C2×C271⋊C8—22no—
89C2×C271⋊C8C2311yescharacteristic
90C2×C271⋊C8C2311yescharacteristic
91C8×D271—22no—
92C4⋊Dic271—21no—
93C4⋊Dic271C2311yescharacteristic
94Dic271⋊C4—21no—
95Dic271⋊C4D411yescharacteristic
96C22⋊Dic271—21no—
97C22⋊Dic271—21no—
98C22⋊Dic271—21no—
99C22⋊Dic271D411yescharacteristic
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—21no—
104(C2×D271)⋊C4—21no—
105(C2×D271)⋊C4—21no—
106(C2×D271)⋊C4D411yescharacteristic
107C4×Dic271C2311yescharacteristic
108#41×C271—22no—
109#41×C271—41no—
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271—21no—
113C2×D4×C271C2311yescharacteristic
114C2×D4×C271C2311yescharacteristic
115C22×C4×C271—21no—
116C22×C4×C271C2311yescharacteristic
117SD16×C271—41no—
118D8×C271—22no—
119C2×C8×C271C2311yescharacteristic
120C2×C8×C271C2311yescharacteristic
121C4⋊C4×C271C2311yescharacteristic
122C22⋊C4×C271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—42no—
126C22×D271—42no—
127C22×D271—42no—
128C22×D271—42no—
129C22×D271—81no—
130C22×D271—81no—
131C22×D271C2×D411yescharacteristic
132Dic542—21no—
133Dic542—21no—
134D1084—21no—
135D1084—21no—
136D1084—21no—
137D1084C2×D412yes—
138C22⋊D271—22no—
139C22⋊D271—42no—
140C22⋊D271—42no—
141C22⋊D271—42no—
142C22⋊D271—81no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—81no—
146C2×Dic271C2×D411yescharacteristic
147C2×Dic271C2×D411yescharacteristic
148C2×Dic271C2×D411yescharacteristic
149C4×D271—21no—
150C4×D271—21no—
151C4×D271—42no—
152C4×D271—81no—
153C271⋊C8—21no—
154C271⋊C8C2×D412yes—
155C23×C271—21no—
156C23×C271—21no—
157C23×C271C2×D411yescharacteristic
158Q8×C271—21no—
159Q8×C271—21no—
160D4×C271—21no—
161D4×C271—21no—
162D4×C271—21no—
163D4×C271—42no—
164D4×C271—81no—
165D4×C271C2×D412yes—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—21no—
169C2×C4×C271—21no—
170C2×C4×C271—42no—
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2411yescharacteristic
173C8×C271—21no—
174C8×C271—21no—
175C22×C271—21no—
176C22×C271—22no—
177C22×C271—21no—
178C22×C271—21no—
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—22no—
188C2×D271—41no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—84no—
194C2×D271—84no—
195C2×D271#13512yescomplemented
196C4×C271—21no—
197C4×C271—41no—
198C4×C271—41no—
199C4×C271C22×D411yescharacteristic
200C4×C271H(5,𝔽2)11yescharacteristic
201Dic271—21no—
202Dic271—21no—
203Dic271—21no—
204Dic271—41no—
205Dic271—41no—
206C2×C271—42no—
207C2×C271—41no—
208C2×C271—81no—
209C2×C271D4211yescharacteristic
210C2×C271C2×#13511yescharacteristic
211C2×C271#57611yescharacteristic
212D271—22no—
213D271—42no—
214D271—81no—
215C271#328011yescharacteristic, complemented, Sylow
216#3280—2711nomaximal, Sylow
217C2×#135—2711no—
218C22×D8—2711no—
219#546—2711no—
220#546—2711no—
221#495—2711no—
222#467—2711no—
223#466—2711no—
224#463—2711no—
225#463—2711no—
226#450—2711no—
227#450—2711no—
228#449—2711no—
229#449D27111yescharacteristic, complemented
230C4×D8—2711no—
231C8×D4—2711no—
232C2×#41—2711no—
233C2×#41C2×D27111yescharacteristic, complemented
234C22×D4—2711no—
235C22×D4—2711no—
236#135—5424no—
237C2×Q16—2711no—
238C2×SD16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16C2×D27111yescharacteristic, complemented
242C2×D8—2711no—
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8—5424no—
246C2×D8C2×D27111yescharacteristic
247C22×C8—2711no—
248C22×C8—2711no—
249#124—2711no—
250#124—2711no—
251#123—5421no—
252#123—5421no—
253C22⋊Q8—2711no—
254C22⋊Q8—2711no—
255#121—2711no—
256#121—5421no—
257#121—5421no—
258#121C2×D27111yescharacteristic, complemented
259#120—5421no—
260#120—5421no—
261C4×D4—2711no—
262C4×D4—2711no—
263C4×D4—2711no—
264C2×C22⋊C4—5421no—
265C2×C22⋊C4—5421no—
266#107—2711no—
267C4⋊C8—2711no—
268#103—2711no—
269#103C2×D27111yescharacteristic, complemented
270#102—2711no—
271#102—2711no—
272#102—2711no—
273#102C2×D27111yescharacteristic, complemented
274C22⋊C8—2711no—
275C22⋊C8C2×D27111yescharacteristic, complemented
276C4×C8—2711no—
277C24—5421no—
278C24—5421no—
279#41—22no—
280#41—41no—
281#41—5422no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2711no—
288C2×D4—2711no—
289C2×D4—2711no—
290C2×D4—5421no—
291C2×D4—5421no—
292C2×D4—5422no—
293C2×D4—5422no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—10841no—
297C2×D4—10841no—
298C2×D4C22×D27111yescharacteristic
299C2×D4C22×D27111yescharacteristic
300C22×C4—21no—
301C22×C4—2711no—
302C22×C4—5421no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4C22×D27111yescharacteristic
306Q16—5422no—
307SD16—41no—
308SD16—5422no—
309SD16—5422no—
310SD16—10841no—
311D8—22no—
312D8—2714no—
313D8—5422no—
314D8—5422no—
315C2×C8—2711no—
316C2×C8—2711no—
317C2×C8—5422no—
318C2×C8—5422no—
319C2×C8C22×D27111yescharacteristic
320C2×C8C22×D27111yescharacteristic
321C4⋊C4—2711no—
322C4⋊C4—2711no—
323C4⋊C4—5421no—
324C4⋊C4—5421no—
325C4⋊C4C22×D27111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—2711no—
328C22⋊C4—2711no—
329C22⋊C4—5421no—
330C22⋊C4—5421no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C42—2711no—
339C23—21no—
340C23—21no—
341C23—2711no—
342C23—5421no—
343C23—5421no—
344C23—10842no—
345C23—10842no—
346C23—10842no—
347C23—10842no—
348C23—21681no—
349C23—21681no—
350C23D4×D27111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—5421no—
354Q8—5421no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—81no—
360D4—2712no—
361D4—5422no—
362D4—5421no—
363D4—5421no—
364D4—5421no—
365D4—10842no—
366D4—10842no—
367D4—10842no—
368D4—21681no—
369D4D4×D27112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—2711no—
376C2×C4—2711no—
377C2×C4—2711no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—5421no—
381C2×C4—5421no—
382C2×C4—10842no—
383C2×C4—21681no—
384C2×C4—21681no—
385C2×C4D4×D27111yescharacteristic
386C2×C4C23×D27111yescharacteristic
387C8—21no—
388C8—21no—
389C8—2712no—
390C8—5421no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—2712no—
401C22—5421no—
402C22—5421no—
403C22—5422no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—21684no—
410C22—21684no—
411C22C2×D4×D27111yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—5421no—
416C4—5421no—
417C4—5421no—
418C4—10841no—
419C4—10841no—
420C4C2×D4×D27111yescharacteristic
421C4#2257620511yescharacteristic
422C2—41no—
423C2—42no—
424C2—81no—
425C2—5422no—
426C2—10842no—
427C2—21681no—
428C2#3177452911yescharacteristic
429C2#135×D27111yescharacteristic
430C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3280
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#22576180C2semidirect
2. C2×D8×D271C2semidirect
3. #31774544C2semidirect
4. #31774535C2semidirect
5. #31774278C2semidirect
6. #31774137C2semidirect
7. #31774131C2semidirect
8. #31774115C2semidirect
9. #31774110C2semidirect
10. #31774049C2semidirect
11. #31774046C2semidirect
12. #31774044C2semidirect
13. #31773977C2semidirect
14. (C22⋊D271)⋊C8C2semidirect
15. #449×C271C2semidirect
16. C2×#16864654C22semidirect
20. C2×#16864644C22semidirect
21. C2×#16864643C22semidirect
22. C2×#16864642C22semidirect
23. C2×#16864641C22semidirect
28. C22×C271⋊C8C22semidirect
30. #22576149C22semidirect
31. #22576147C22semidirect
36. #22576134C22semidirect
41. D4×Dic271C22semidirect
42. #22576122C22semidirect
46. #22576093C22semidirect
47. Dic271⋊C8C22semidirect
48. #22576085C22semidirect
49. #22576082C22semidirect
50. #22576081C22semidirect
51. #22576080C22semidirect
53. C8×Dic271C22semidirect
54. C2×#41×C271C22semidirect
55. C2×SD16×C271C22semidirect
57. #121×C271C22semidirect
58. #103×C271C22semidirect
59. #102×C271C22semidirect
60. C22⋊C8×C271C22semidirect
87. #16864639D4semidirect
195. C2×D271#135semidirect
215. C271#3280semidirect
229. #449D271semidirect
233. C2×#41C2×D271semidirect
241. C2×SD16C2×D271semidirect
258. #121C2×D271semidirect
269. #103C2×D271semidirect
273. #102C2×D271semidirect
275. C22⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761802
C2×D8×D2712
#317745442
#317745352
#317742782
#317741372
#317741312
#317741152
#317741102
#317740492
#317740462
#317740442
#317739772
(C22⋊D271)⋊C82
#449×C2712
#3280271