Finite Group Databasebeta

← #46847971

Subgroups — #46847971

Order 34688.

520 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×#22576186C211yesmaximal, characteristic, complemented
2C2×#22576185C211yesmaximal, characteristic, complemented
3#31774543C211yesmaximal, characteristic, complemented
4#31774279C211yesmaximal, characteristic, complemented
5#31774145C211yesmaximal, characteristic, complemented
6#31774141C211yesmaximal, characteristic, complemented
7#31774112C211yesmaximal, characteristic, complemented
8#31774109C211yesmaximal, characteristic, complemented
9#31774059C211yesmaximal, characteristic, complemented
10#31774057C211yesmaximal, characteristic, complemented
11#31774031C211yesmaximal, characteristic, complemented
12#31774030C211yesmaximal, characteristic, complemented
13C4×#16864639C211yesmaximal, characteristic, complemented
14#31773975C211yesmaximal, characteristic, complemented
15#546×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×D1084C2211yescharacteristic, complemented
18#22576186—24no—
19#22576185—24no—
20C2×#16864644C2211yescharacteristic, complemented
21C2×#16864642C2211yescharacteristic, complemented
22C2×#16864642C2211yescharacteristic, complemented
23C2×#16864641C2211yescharacteristic, complemented
24C2×#16864641C2211yescharacteristic, complemented
25C2×#16864639C2211yescharacteristic
26C2×#16864639C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864638C2211yescharacteristic, complemented
30C2×#16864638C2211yescharacteristic, complemented
31#22576148C2211yescharacteristic, complemented
32#22576145—21no—
33#22576139C2211yescharacteristic, complemented
34#22576134C2211yescharacteristic
35#22576133—21no—
36#22576128—21no—
37C4×D1084C2211yescharacteristic, complemented
38C22⋊C4×D271—21no—
39#22576093C2211yescharacteristic, complemented
40C4⋊(C271⋊C8)C2211yescharacteristic, complemented
41#22576084C2211yescharacteristic, complemented
42#22576083C2211yescharacteristic, complemented
43#22576081C2211yescharacteristic, complemented
44#22576081C2211yescharacteristic, complemented
45#22576080C2211yescharacteristic, complemented
46#22576080C2211yescharacteristic, complemented
47C22⋊(C271⋊C8)C2211yescharacteristic, complemented
48C22⋊(C271⋊C8)C2211yescharacteristic, complemented
49C4×C271⋊C8C2211yescharacteristic, complemented
50C2×#41×C271C2211yescharacteristic
51C22×D4×C271C2211yescharacteristic, complemented
52#124×C271C2211yescharacteristic, complemented
53#123×C271—21no—
54C22⋊Q8×C271C2211yescharacteristic, complemented
55#121×C271—21no—
56#121×C271C2211yescharacteristic
57#120×C271—21no—
58C4×D4×C271C2211yescharacteristic
59C4×D4×C271C2211yescharacteristic, complemented
60C2×C22⋊C4×C271—21no—
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—41no—
75#16864642—22no—
76#16864642—22no—
77#16864641—22no—
78#16864641—22no—
79#16864639—22no—
80#16864639—22no—
81#16864639—41no—
82#16864639D414yescomplemented
83#16864638—22no—
84#16864638—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—
96C24×C271—21no—
97#41×C271—22no—
98#41×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×D4×C271—22no—
101C2×D4×C271—21no—
102C2×D4×C271—21no—
103C2×D4×C271—21no—
104C2×D4×C271—41no—
105C2×D4×C271D411yescharacteristic
106C2×D4×C271C2311yescharacteristic
107C2×D4×C271C2311yescharacteristic
108C22×C4×C271—21no—
109C22×C4×C271—21no—
110C22×C4×C271C2311yescharacteristic
111C22×C4×C271C2311yescharacteristic
112C4⋊C4×C271—21no—
113C4⋊C4×C271D411yescharacteristic
114C4⋊C4×C271C2311yescharacteristic
115C22⋊C4×C271—21no—
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C22⋊C4×C271—21no—
120C22⋊C4×C271D411yescharacteristic
121C22⋊C4×C271D411yescharacteristic
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×C271—21no—
146C23×C271—21no—
147C23×C271—42no—
148C23×C271—42no—
149C23×C271—81no—
150C23×C271C2×D411yescharacteristic
151C23×C271C2×D411yescharacteristic
152Q8×C271—21no—
153Q8×C271—21no—
154D4×C271—21no—
155D4×C271—21no—
156D4×C271—21no—
157D4×C271—41no—
158D4×C271—42no—
159D4×C271—42no—
160D4×C271—81no—
161D4×C271C2×D412yes—
162C2×C4×C271—21no—
163C2×C4×C271—21no—
164C2×C4×C271—21no—
165C2×C4×C271—21no—
166C2×C4×C271—21no—
167C2×C4×C271—21no—
168C2×C4×C271—42no—
169C2×C4×C271—81no—
170C2×C4×C271C2×D411yescharacteristic
171C2×C4×C271C2×D411yescharacteristic
172C2×C4×C271C2×D411yescharacteristic
173C2×C4×C271C2×D411yescharacteristic
174C2×C4×C271C2411yescharacteristic
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—22no—
178C22×C271—21no—
179C22×C271—21no—
180C22×C271—21no—
181C22×C271—21no—
182C22×C271—41no—
183C22×C271—41no—
184C22×C271—41no—
185C22×C271—41no—
186C22×C271—41no—
187C22×C271—84no—
188C22×C271C22×D411yescharacteristic
189C2×D271—22no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—41no—
195C2×D271—84no—
196C4×C271—21no—
197C4×C271—21no—
198C4×C271—21no—
199C4×C271—21no—
200C4×C271—41no—
201C4×C271—41no—
202C4×C271C22×D411yescharacteristic
203C4×C271H(5,𝔽2)11yescharacteristic
204Dic271—41no—
205Dic271—41no—
206C2×C271—41no—
207C2×C271—41no—
208C2×C271—42no—
209C2×C271—81no—
210C2×C271D4211yescharacteristic
211C2×C271#57511yescharacteristic
212C2×C271#57611yescharacteristic
213D271—42no—
214D271—81no—
215C271#327311yescharacteristic, complemented, Sylow
216#3273—2711nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2711no—
218C2×AGL(1,ℤ/8)—2711no—
219#546—2711no—
220#546D27111yescharacteristic, complemented
221#495—2711no—
222#468—2711no—
223#468—2711no—
224#463—2711no—
225#463—2711no—
226#452—2711no—
227#452—2711no—
228#447—2711no—
229#447—2711no—
230C4×D8—2711no—
231#436—2711no—
232C2×#41—2711no—
233C2×#41C2×D27111yescharacteristic
234C22×D4—2711no—
235C22×D4C2×D27111yescharacteristic, complemented
236AGL(1,ℤ/8)—5424no—
237AGL(1,ℤ/8)—5424no—
238C2×Q16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16—2711no—
242C2×SD16—2711no—
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8—2711no—
246C2×D8—2711no—
247C2×M16—2711no—
248C2×M16—2711no—
249#124—2711no—
250#124C2×D27111yescharacteristic, complemented
251#123—21no—
252#123—5421no—
253C22⋊Q8—2711no—
254C22⋊Q8C2×D27111yescharacteristic, complemented
255#121—21no—
256#121—2711no—
257#121—5421no—
258#121C2×D27111yescharacteristic
259#120—21no—
260#120—5421no—
261C4×D4—2711no—
262C4×D4C2×D27111yescharacteristic
263C4×D4C2×D27111yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—5421no—
266#107—2711no—
267C4⋊C8—2711no—
268#103—2711no—
269#103—2711no—
270#102—2711no—
271#102—2711no—
272#102—2711no—
273#102—2711no—
274C22⋊C8—2711no—
275C22⋊C8—2711no—
276C4×C8—2711no—
277C24—21no—
278C24—5421no—
279#41—22no—
280#41—41no—
281#41—5422no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2711no—
291C2×D4—2711no—
292C2×D4—5422no—
293C2×D4—5421no—
294C2×D4—5421no—
295C2×D4—5421no—
296C2×D4—10841no—
297C2×D4C22⋊D27111yescharacteristic
298C2×D4C22×D27111yescharacteristic
299C2×D4C22×D27111yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—5421no—
303C22×C4—5421no—
304C22×C4C22×D27111yescharacteristic
305C22×C4C22×D27111yescharacteristic
306Q16—10841no—
307SD16—5422no—
308SD16—5422no—
309SD16—5422no—
310SD16—5422no—
311D8—2714no—
312D8—5422no—
313D8—5422no—
314D8—10841no—
315M16—5422no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
319C2×C8—2711no—
320C2×C8—2711no—
321C4⋊C4—21no—
322C4⋊C4—2711no—
323C4⋊C4—5421no—
324C4⋊C4C22⋊D27111yescharacteristic
325C4⋊C4C22×D27111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—5421no—
332C22⋊C4—5421no—
333C22⋊C4—5421no—
334C22⋊C4—5421no—
335C22⋊C4—5421no—
336C22⋊C4C22⋊D27111yescharacteristic
337C22⋊C4C22⋊D27111yescharacteristic
338C42C22×D27111yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—5421no—
345C23—5421no—
346C23—10842no—
347C23—10842no—
348C23—21681no—
349C23C2×C22⋊D27111yescharacteristic
350C23C2×C22⋊D27111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—5421no—
354Q8—5421no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—2712no—
363D4—5421no—
364D4—5421no—
365D4—5421no—
366D4—10842no—
367D4—10842no—
368D4—21681no—
369D4D4×D27112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—10842no—
381C2×C4—21681no—
382C2×C4C2×C22⋊D27111yescharacteristic
383C2×C4C2×C22⋊D27111yescharacteristic
384C2×C4C2×C22⋊D27111yescharacteristic
385C2×C4C2×C22⋊D27111yescharacteristic
386C2×C4C23×D27111yescharacteristic
387C8—2712no—
388C8—5421no—
389C8—5421no—
390C8—5421no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—5422no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—10841no—
410C22—21684no—
411C22C22×C22⋊D27111yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—10841no—
419C4—10841no—
420C4C2×D4×D27111yescharacteristic
421C4#2257620511yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—10842no—
427C2—21681no—
428C2D4×C22⋊D27111yescharacteristic
429C2#3177469511yescharacteristic
430C2#3177470011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#546×C271C2
socleC22×C271C22×D4
hypercenter#546D271
hypocenterC271#3273
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. C2×#22576185C2semidirect
3. #31774543C2semidirect
4. #31774279C2semidirect
5. #31774145C2semidirect
6. #31774141C2semidirect
7. #31774112C2semidirect
8. #31774109C2semidirect
9. #31774059C2semidirect
10. #31774057C2semidirect
11. #31774031C2semidirect
12. #31774030C2semidirect
13. C4×#16864639C2semidirect
14. #31773975C2semidirect
15. #546×C271C2semidirect
17. C22×D1084C22semidirect
20. C2×#16864644C22semidirect
21. C2×#16864642C22semidirect
22. C2×#16864642C22semidirect
23. C2×#16864641C22semidirect
24. C2×#16864641C22semidirect
29. C2×#16864638C22semidirect
30. C2×#16864638C22semidirect
31. #22576148C22semidirect
33. #22576139C22semidirect
37. C4×D1084C22semidirect
39. #22576093C22semidirect
40. C4⋊(C271⋊C8)C22semidirect
41. #22576084C22semidirect
42. #22576083C22semidirect
43. #22576081C22semidirect
44. #22576081C22semidirect
45. #22576080C22semidirect
46. #22576080C22semidirect
47. C22⋊(C271⋊C8)C22semidirect
48. C22⋊(C271⋊C8)C22semidirect
49. C4×C271⋊C8C22semidirect
51. C22×D4×C271C22semidirect
52. #124×C271C22semidirect
54. C22⋊Q8×C271C22semidirect
59. C4×D4×C271C22semidirect
82. #16864639D4semidirect
215. C271#3273semidirect
220. #546D271semidirect
235. C22×D4C2×D271semidirect
250. #124C2×D271semidirect
254. C22⋊Q8C2×D271semidirect
263. C4×D4C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
C2×#225761852
#317745432
#317742792
#317741452
#317741412
#317741122
#317741092
#317740592
#317740572
#317740312
#317740302
C4×#168646392
#317739752
#546×C2712
#3273271