Finite Group Databasebeta

← #46847970

Subgroups — #46847970

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×#22576187C211yesmaximal, characteristic, complemented
2C2×#22576184C211yesmaximal, characteristic, complemented
3#31774542C211yesmaximal, characteristic, complemented
4#31774540C211yesmaximal, characteristic, complemented
5#31774279C211yesmaximal, characteristic, complemented
6#31774146C211yesmaximal, characteristic, complemented
7#31774142C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774062C211yesmaximal, characteristic, complemented
10#31774056C211yesmaximal, characteristic, complemented
11#31774032C211yesmaximal, characteristic, complemented
12#31774029C211yesmaximal, characteristic, complemented
13C4×#16864639C211yesmaximal, characteristic, complemented
14#31773974C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×D4×D271C2211yescharacteristic, complemented
19C2×D4×D271C2211yescharacteristic, complemented
20#22576187—24no—
21#22576184—24no—
22C2×#16864644C2211yescharacteristic, complemented
23C2×#16864643C2211yescharacteristic, complemented
24C2×#16864642C2211yescharacteristic, complemented
25C2×#16864641C2211yescharacteristic, complemented
26C2×D2168C2211yescharacteristic
27C2×#16864639C2211yescharacteristic
28C2×#16864639C2211yescharacteristic
29C2×#16864637C2211yescharacteristic, complemented
30C2×#16864637C2211yescharacteristic, complemented
31#22576148C2211yescharacteristic, complemented
32#22576144—21no—
33#22576142—21no—
34#22576140C2211yescharacteristic, complemented
35#22576136C2211yescharacteristic, complemented
36#22576135C2211yescharacteristic
37#22576134—21no—
38#22576133—21no—
39#22576131C2211yescharacteristic
40#22576130—21no—
41#22576130—21no—
42C4×D1084C2211yescharacteristic
43C4×D1084C2211yescharacteristic, complemented
44C2×(C2×D271)⋊C4—21no—
45C22⋊C4×D271—21no—
46#22576093C2211yescharacteristic, complemented
47#22576085C2211yescharacteristic, complemented
48#22576082C2211yescharacteristic, complemented
49#22576081C2211yescharacteristic, complemented
50#22576080C2211yescharacteristic, complemented
51(C2×D271)⋊C8C2211yescharacteristic, complemented
52(C2×D271)⋊C8C2211yescharacteristic, complemented
53C4×C271⋊C8C2211yescharacteristic, complemented
54C2×SD16×C271C2211yescharacteristic, complemented
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic, complemented
57C4×D4×C271C2211yescharacteristic, complemented
58C4⋊C8×C271C2211yescharacteristic, complemented
59#103×C271C2211yescharacteristic, complemented
60#102×C271C2211yescharacteristic, complemented
61C23×D271—21no—
62C23×D271—21no—
63#16864655—22no—
64#16864655—41no—
65#16864653—22no—
66#16864653—41no—
67C2×Dic542C2311yescharacteristic
68C2×D1084—21no—
69C2×D1084—21no—
70C2×D1084D411yescharacteristic
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C22⋊D271—21no—
74C2×C22⋊D271—21no—
75C2×C22⋊D271—21no—
76C2×C22⋊D271—21no—
77D4×D271—22no—
78D4×D271—22no—
79D4×D271—41no—
80D4×D271—41no—
81C22×Dic271—21no—
82C2×C4×D271—21no—
83C2×C4×D271—21no—
84C2×C4×D271C2311yescharacteristic
85C2×C4×D271C2311yescharacteristic
86#16864644—41no—
87#16864643—22no—
88#16864642—22no—
89#16864641—22no—
90D2168—22no—
91#16864639—41no—
92#16864639D414yescomplemented
93#16864637—22no—
94#16864637—22no—
95C2×C271⋊C8C2311yescharacteristic
96C2×C271⋊C8C2311yescharacteristic
97C4⋊Dic271—21no—
98C4⋊Dic271D411yescharacteristic
99C4⋊Dic271C2311yescharacteristic
100Dic271⋊C4—21no—
101C22⋊Dic271—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4—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×D4×C271C2311yescharacteristic
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114SD16×C271—22no—
115D8×C271—22no—
116C2×C8×C271C2311yescharacteristic
117C2×C8×C271C2311yescharacteristic
118C4⋊C4×C271C2311yescharacteristic
119C22⋊C4×C271—21no—
120C22⋊C4×C271—21no—
121C22⋊C4×C271—21no—
122C42×C271C2311yescharacteristic
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
132C22×D271C2×D411yescharacteristic
133Dic542—21no—
134Dic542—21no—
135D1084—21no—
136D1084—21no—
137D1084—21no—
138D1084—41no—
139D1084—42no—
140D1084—81no—
141D1084C2×D412yes—
142C22⋊D271—42no—
143C22⋊D271—42no—
144C22⋊D271—42no—
145C22⋊D271—81no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—42no—
149C2×Dic271C2×D411yescharacteristic
150C2×Dic271C2×D411yescharacteristic
151C4×D271—21no—
152C4×D271—21no—
153C4×D271—21no—
154C4×D271—21no—
155C4×D271—42no—
156C4×D271—81no—
157C271⋊C8—21no—
158C271⋊C8C2×D412yes—
159C23×C271—21no—
160C23×C271—21no—
161Q8×C271—21no—
162Q8×C271—21no—
163D4×C271—21no—
164D4×C271—21no—
165D4×C271—21no—
166D4×C271C2×D412yes—
167C2×C4×C271—21no—
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×C271C22×D411yescharacteristic
182C2×D271—21no—
183C2×D271—21no—
184C2×D271—22no—
185C2×D271—21no—
186C2×D271—21no—
187C2×D271—21no—
188C2×D271—21no—
189C2×D271—41no—
190C2×D271—41no—
191C2×D271—41no—
192C2×D271—41no—
193C2×D271—41no—
194C2×D271—84no—
195C2×D271—84no—
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—
205Dic271—41no—
206C2×C271—42no—
207C2×C271—81no—
208C2×C271D4211yescharacteristic
209C2×C271#57511yescharacteristic
210C2×C271#57611yescharacteristic
211D271—41no—
212D271—41no—
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#546—2711no—
221#495—2711no—
222#468—2711no—
223#468—2711no—
224#463—2711no—
225#463D27111yescharacteristic, complemented
226#452—2711no—
227#452—2711no—
228#447—2711no—
229#447—2711no—
230C4×D8—2711no—
231#436—2711no—
232C2×#41—2711no—
233C2×#41—2711no—
234C22×D4—2711no—
235C22×D4—2711no—
236AGL(1,ℤ/8)—5424no—
237AGL(1,ℤ/8)—5424no—
238C2×Q16—2711no—
239C2×SD16—2711no—
240C2×SD16—2711no—
241C2×SD16—2711no—
242C2×SD16C2×D27111yescharacteristic, complemented
243C2×D8—2711no—
244C2×D8—2711no—
245C2×D8—2711no—
246C2×D8C2×D27111yescharacteristic
247C2×M16—2711no—
248C2×M16—2711no—
249#124—2711no—
250#124C2×D27111yescharacteristic, complemented
251#123—5421no—
252#123—5421no—
253C22⋊Q8—2711no—
254C22⋊Q8—2711no—
255#121—2711no—
256#121—2711no—
257#121—5421no—
258#121—5421no—
259#120—5421no—
260#120—5421no—
261C4×D4—2711no—
262C4×D4—2711no—
263C4×D4C2×D27111yescharacteristic, complemented
264C2×C22⋊C4—5421no—
265C2×C22⋊C4—5421no—
266#107—2711no—
267C4⋊C8C2×D27111yescharacteristic, complemented
268#103—2711no—
269#103C2×D27111yescharacteristic, complemented
270#102—2711no—
271#102—2711no—
272#102—2711no—
273#102C2×D27111yescharacteristic, complemented
274C22⋊C8—2711no—
275C22⋊C8—2711no—
276C4×C8—2711no—
277C24—5421no—
278C24—5421no—
279#41—5422no—
280#41—5422no—
281#41—10841no—
282#41—10841no—
283C2×Q8—2711no—
284C2×Q8C22×D27111yescharacteristic
285C2×D4—2711no—
286C2×D4—2711no—
287C2×D4—2711no—
288C2×D4—5421no—
289C2×D4—5421no—
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—2711no—
303C22×C4—5421no—
304C22×C4—5421no—
305C22×C4—5421no—
306Q16—10841no—
307SD16—22no—
308SD16—5422no—
309SD16—5422no—
310SD16—5422no—
311D8—22no—
312D8—2714no—
313D8—5422no—
314D8—10841no—
315M16—5422no—
316M16—5422no—
317C2×C8—2711no—
318C2×C8—2711no—
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—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—
336C22⋊C4—5421no—
337C22⋊C4—5421no—
338C42C22×D27111yescharacteristic
339C23—21no—
340C23—21no—
341C23—2711no—
342C23—2711no—
343C23—5421no—
344C23—5421no—
345C23—10842no—
346C23—10842no—
347C23—10842no—
348C23—10842no—
349C23—21681no—
350C23—21681no—
351Q8—21no—
352Q8—21no—
353Q8—5421no—
354Q8—5421no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—2712no—
359D4—5421no—
360D4—5421no—
361D4—5421no—
362D4—10841no—
363D4—10842no—
364D4—10842no—
365D4—10842no—
366D4—10842no—
367D4—21681no—
368D4—21681no—
369D4D4×D27112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—2711no—
374C2×C4—2711no—
375C2×C4—5421no—
376C2×C4—5421no—
377C2×C4—5421no—
378C2×C4—5421no—
379C2×C4—5421no—
380C2×C4—5421no—
381C2×C4—10842no—
382C2×C4—10842no—
383C2×C4—21681no—
384C2×C4D4×D27111yescharacteristic
385C2×C4D4×D27111yescharacteristic
386C2×C4C23×D27111yescharacteristic
387C8—21no—
388C8—21no—
389C8—2712no—
390C8—5421no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—5421no—
398C22—5421no—
399C22—5422no—
400C22—5421no—
401C22—5421no—
402C22—5421no—
403C22—5421no—
404C22—10841no—
405C22—10841no—
406C22—10841no—
407C22—10841no—
408C22—10841no—
409C22—21684no—
410C22—21684no—
411C22C2×D4×D27111yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—5421no—
417C4—5421no—
418C4—10841no—
419C4—10841no—
420C4C2×D4×D27111yescharacteristic
421C4#2257620611yescharacteristic
422C2—42no—
423C2—81no—
424C2—10841no—
425C2—10841no—
426C2—10842no—
427C2—21681no—
428C2#3177453111yescharacteristic
429C2#3177469411yescharacteristic
430C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
hypercenter#463D271
hypocenterC271#3273
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22576187C2semidirect
2. C2×#22576184C2semidirect
3. #31774542C2semidirect
4. #31774540C2semidirect
5. #31774279C2semidirect
6. #31774146C2semidirect
7. #31774142C2semidirect
8. #31774113C2semidirect
9. #31774062C2semidirect
10. #31774056C2semidirect
11. #31774032C2semidirect
12. #31774029C2semidirect
13. C4×#16864639C2semidirect
14. #31773974C2semidirect
15. #463×C271C2semidirect
18. C2×D4×D271C22semidirect
19. C2×D4×D271C22semidirect
22. C2×#16864644C22semidirect
23. C2×#16864643C22semidirect
24. C2×#16864642C22semidirect
25. C2×#16864641C22semidirect
29. C2×#16864637C22semidirect
30. C2×#16864637C22semidirect
31. #22576148C22semidirect
34. #22576140C22semidirect
35. #22576136C22semidirect
43. C4×D1084C22semidirect
46. #22576093C22semidirect
47. #22576085C22semidirect
48. #22576082C22semidirect
49. #22576081C22semidirect
50. #22576080C22semidirect
51. (C2×D271)⋊C8C22semidirect
52. (C2×D271)⋊C8C22semidirect
53. C4×C271⋊C8C22semidirect
54. C2×SD16×C271C22semidirect
56. #124×C271C22semidirect
57. C4×D4×C271C22semidirect
58. C4⋊C8×C271C22semidirect
59. #103×C271C22semidirect
60. #102×C271C22semidirect
92. #16864639D4semidirect
215. C271#3273semidirect
225. #463D271semidirect
242. C2×SD16C2×D271semidirect
250. #124C2×D271semidirect
263. C4×D4C2×D271semidirect
267. C4⋊C8C2×D271semidirect
269. #103C2×D271semidirect
273. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761872
C2×#225761842
#317745422
#317745402
#317742792
#317741462
#317741422
#317741132
#317740622
#317740562
#317740322
#317740292
C4×#168646392
#317739742
#463×C2712
#3273271