Finite Group Databasebeta

← #46846555

Subgroups — #46846555

Order 34688.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774521C211yesmaximal, characteristic, complemented
2#31774369C211yesmaximal, characteristic, complemented
3#31774240C211yesmaximal, characteristic, complemented
4#31774224C211yesmaximal, characteristic, complemented
5#31774184C211yesmaximal, characteristic, complemented
6#31774113C211yesmaximal, characteristic, complemented
7#31774112C211yesmaximal, characteristic, complemented
8#31774049C211yesmaximal, characteristic, complemented
9#31774032C211yesmaximal, characteristic
10#31774006C211yesmaximal, characteristic, complemented
11#31773996C211yesmaximal, characteristic, complemented
12#31773988C211yesmaximal, characteristic, complemented
13#31773979C211yesmaximal, characteristic, complemented
14#31773957C211yesmaximal, characteristic, complemented
15#474×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C2×#16864643C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×D2168C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21#22576155—21no—
22#22576149—21no—
23#22576148C2211yescharacteristic, complemented
24#22576145—21no—
25#22576140C2211yescharacteristic
26#22576136—21no—
27#22576135C2211yescharacteristic
28#22576125C2211yescharacteristic, complemented
29C4×D1084C2211yescharacteristic
30D4×Dic271C2211yescharacteristic
31#22576122C2211yescharacteristic
32#22576122C2211yescharacteristic, complemented
33C4×C22⋊D271—21no—
34(C22⋊D271)⋊C4—21no—
35#22576118C2211yescharacteristic
36#22576117C2211yescharacteristic
37C2×(C2×D271)⋊C4—21no—
38C22⋊C4×D271—21no—
39#22576091C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41C4⋊(C271⋊C8)C2211yescharacteristic, complemented
42#22576085C2211yescharacteristic, complemented
43#22576083C2211yescharacteristic, complemented
44#22576082C2211yescharacteristic, complemented
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic, complemented
48(C2×D271)⋊C8C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52C4⋊Q8×C271C2211yescharacteristic, complemented
53C4×D4×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic, complemented
55#107×C271C2211yescharacteristic, complemented
56C4⋊C8×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60C2×Dic542C2311yescharacteristic
61C2×D1084—21no—
62C2×D1084C2311yescharacteristic
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65D4×D271—22no—
66D4×D271—22no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271C2311yescharacteristic
72#16864643—22no—
73#16864642—22no—
74D2168—22no—
75#16864639—22no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80Dic271⋊C4—21no—
81Dic271⋊C4D411yescharacteristic
82Dic271⋊C4D411yescharacteristic
83C22⋊Dic271—21no—
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271—21no—
98C2×D4×C271C2311yescharacteristic
99C22×C4×C271—21no—
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271—21no—
103C4⋊C4×C271C2311yescharacteristic
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109C22×D271—24no—
110C22×D271—21no—
111C22×D271—81no—
112C22×D271—81no—
113C22×D271C2×D411yescharacteristic
114Dic542—21no—
115Dic542—21no—
116D1084—21no—
117D1084—21no—
118D1084—21no—
119D1084—81no—
120D1084#4112yes—
121C22⋊D271—24no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—81no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—81no—
131C4×D271—81no—
132C4×D271—81no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136Q8×C271—81no—
137D4×C271—21no—
138D4×C271#4112yes—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—81no—
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271—22no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C22×D411yescharacteristic
152C2×D271—21no—
153C2×D271—22no—
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—84no—
162C4×C271—21no—
163C4×C271—21no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271C2×#4111yescharacteristic
168C4×C271C2×#4111yescharacteristic
169Dic271—21no—
170Dic271—21no—
171Dic271—21no—
172Dic271—41no—
173Dic271—41no—
174C2×C271—42no—
175C2×C271#52511yescharacteristic
176C2×C271#57611yescharacteristic
177C2×C271#57711yescharacteristic
178D271—41no—
179D271—42no—
180D271—81no—
181C271#311811yescharacteristic, complemented, Sylow
182#3118—2711nomaximal, Sylow
183#542—2711no—
184#518—2711no—
185#485—2711no—
186#482—2711no—
187#476—2711no—
188#474D27111yescharacteristic, complemented
189#463—2711no—
190#463—2711no—
191#450—2711no—
192#447—2711no—
193#442—2711no—
194#440—2711no—
195C4×SD16—2711no—
196C4×D8—2711no—
197#433—2711no—
198C22×D4—2711no—
199C2×SD16—2711no—
200C2×SD16—2711no—
201C2×D8—2711no—
202C2×D8—2711no—
203C4⋊Q8C2×D27111yescharacteristic, complemented
204#126—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—2711no—
209C22⋊Q8—5421no—
210#121—2711no—
211C4×Q8—2711no—
212C4×D4—2711no—
213C4×D4—2711no—
214C4×D4—2711no—
215C4×D4—2711no—
216C4×D4—5421no—
217C4×D4—5421no—
218C4×D4C2×D27111yescharacteristic
219#117—2711no—
220#117—2711no—
221C2×C22⋊C4—5421no—
222C2×C22⋊C4—5421no—
223#107C2×D27111yescharacteristic, complemented
224#107C2×D27111yescharacteristic, complemented
225#106—2711no—
226#106—2711no—
227C4⋊C8—2711no—
228C4⋊C8C2×D27111yescharacteristic, complemented
229#103—2711no—
230#103—2711no—
231#102—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235#102C2×D27111yescharacteristic
236#102C2×D27111yescharacteristic
237C22⋊C8—2711no—
238C22⋊C8—2711no—
239#97—2711no—
240C4×C8—2711no—
241C24—5421no—
242C2×Q8—21no—
243C2×Q8—2711no—
244C2×D4—2711no—
245C2×D4—2711no—
246C2×D4—5421no—
247C2×D4—5421no—
248C2×D4—5422no—
249C2×D4—5422no—
250C2×D4C22×D27111yescharacteristic
251C22×C4—21no—
252C22×C4—2711no—
253C22×C4—5421no—
254C22×C4—5421no—
255C22×C4—5421no—
256C22×C4—5421no—
257SD16—5422no—
258SD16—5422no—
259D8—5422no—
260D8—5422no—
261C2×C8—2711no—
262C2×C8—2711no—
263C2×C8C22×D27111yescharacteristic
264C2×C8C22×D27111yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4C22×D27111yescharacteristic
272C4⋊C4C22×D27111yescharacteristic
273C4⋊C4C22×D27111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—2711no—
276C22⋊C4—2711no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—2711no—
287C42—5421no—
288C42—5421no—
289C42C22×D27111yescharacteristic
290C23—21no—
291C23—2711no—
292C23—5424no—
293C23—5421no—
294C23—5421no—
295C23—21681no—
296C23—21681no—
297Q8—81no—
298Q8—5421no—
299Q8—5421no—
300D4—21no—
301D4—2712no—
302D4—5424no—
303D4—5421no—
304D4—5421no—
305D4—5421no—
306D4—21681no—
307D4#1686465512yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—2711no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—21681no—
320C2×C4—21681no—
321C2×C4—21681no—
322C2×C4—21681no—
323C2×C4D4×D27111yescharacteristic
324C2×C4D4×D27111yescharacteristic
325C2×C4C23×D27111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5421no—
329C8—5421no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—5421no—
334C22—5422no—
335C22—5421no—
336C22—5421no—
337C22—10841no—
338C22—10841no—
339C22—10841no—
340C22—10841no—
341C22—10841no—
342C22—21684no—
343C22C2×D4×D27111yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—5421no—
350C4—5421no—
351C4—5421no—
352C4—10841no—
353C4—10841no—
354C4#41×D27111yescharacteristic
355C4C2×#1686465511yescharacteristic
356C2—42no—
357C2—10841no—
358C2—10842no—
359C2—21681no—
360C2#3177441011yescharacteristic
361C2#3177469811yescharacteristic
362C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#474×C271C2
socleC22×C271C22×D4
prime radical#31774032C2
hypercenter#474D271
hypocenterC271#3118
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #31774521C2semidirect
2. #31774369C2semidirect
3. #31774240C2semidirect
4. #31774224C2semidirect
5. #31774184C2semidirect
6. #31774113C2semidirect
7. #31774112C2semidirect
8. #31774049C2semidirect
10. #31774006C2semidirect
11. #31773996C2semidirect
12. #31773988C2semidirect
13. #31773979C2semidirect
14. #31773957C2semidirect
15. #474×C271C2semidirect
23. #22576148C22semidirect
28. #22576125C22semidirect
32. #22576122C22semidirect
39. #22576091C22semidirect
40. #22576091C22semidirect
41. C4⋊(C271⋊C8)C22semidirect
42. #22576085C22semidirect
43. #22576083C22semidirect
44. #22576082C22semidirect
47. #22576080C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
52. C4⋊Q8×C271C22semidirect
54. #107×C271C22semidirect
55. #107×C271C22semidirect
56. C4⋊C8×C271C22semidirect
181. C271#3118semidirect
188. #474D271semidirect
203. C4⋊Q8C2×D271semidirect
223. #107C2×D271semidirect
224. #107C2×D271semidirect
228. C4⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745212
#317743692
#317742402
#317742242
#317741842
#317741132
#317741122
#317740492
#317740322
#317740062
#317739962
#317739882
#317739792
#317739572
#474×C2712
#3118271