Finite Group Databasebeta

← #46846609

Subgroups — #46846609

Order 34688.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774503C211yesmaximal, characteristic, complemented
2#31774369C211yesmaximal, characteristic, complemented
3#31774224C211yesmaximal, characteristic, complemented
4#31774223C211yesmaximal, characteristic, complemented
5#31774194C211yesmaximal, characteristic, complemented
6#31774093C211yesmaximal, characteristic, complemented
7#31774092C211yesmaximal, characteristic, complemented
8#31774032C211yesmaximal, characteristic, complemented
9#31774032C211yesmaximal, characteristic, complemented
10#31774008C211yesmaximal, characteristic, complemented
11#31774006C211yesmaximal, characteristic, complemented
12#31773979C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14#31773957C211yesmaximal, characteristic, complemented
15#478×C271C211yesmaximal, characteristic, complemented
16C2×D4×D271C2211yescharacteristic
17C2×D2168C2211yescharacteristic
18C2×D2168C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21#22576157C2211yescharacteristic, complemented
22#22576145—21no—
23#22576145—21no—
24#22576135—21no—
25#22576135—21no—
26#22576135C2211yescharacteristic
27#22576135C2211yescharacteristic
28C4×D1084C2211yescharacteristic
29D4×Dic271C2211yescharacteristic
30#22576122C2211yescharacteristic
31#22576122C2211yescharacteristic, complemented
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#22576093C2211yescharacteristic, complemented
40#22576091C2211yescharacteristic, complemented
41C4⋊(C271⋊C8)C2211yescharacteristic, complemented
42#22576082C2211yescharacteristic, complemented
43#22576082C2211yescharacteristic, complemented
44#22576080C2211yescharacteristic
45#22576080C2211yescharacteristic
46#22576080C2211yescharacteristic, complemented
47#22576080C2211yescharacteristic, complemented
48(C2×D271)⋊C8C2211yescharacteristic
49(C2×D271)⋊C8C2211yescharacteristic
50#22576070C2211yescharacteristic, complemented
51C8×Dic271C2211yescharacteristic, complemented
52#125×C271C2211yescharacteristic, complemented
53C4×D4×C271C2211yescharacteristic
54#107×C271C2211yescharacteristic, complemented
55#106×C271C2211yescharacteristic, complemented
56C4⋊C8×C271C2211yescharacteristic, complemented
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C23×D271—21no—
60C2×D1084—21no—
61C2×D1084—21no—
62C2×D1084—21no—
63C2×D1084—21no—
64C2×D1084C2311yescharacteristic
65C2×D1084C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68D4×D271—22no—
69D4×D271—22no—
70C22×Dic271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76D2168—22no—
77D2168—22no—
78#16864639—22no—
79#16864639—22no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4D411yescharacteristic
84Dic271⋊C4D411yescharacteristic
85C22⋊Dic271—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)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97C2×D4×C271C2311yescharacteristic
98C22×C4×C271—21no—
99C2×C8×C271C2311yescharacteristic
100C2×C8×C271C2311yescharacteristic
101C4⋊C4×C271—21no—
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—21no—
112C22×D271—81no—
113C22×D271—81no—
114C22×D271C2×D411yescharacteristic
115D1084—21no—
116D1084—21no—
117D1084—21no—
118D1084—21no—
119D1084—21no—
120D1084—81no—
121D1084—81no—
122D1084—81no—
123D1084—81no—
124D1084#4112yes—
125C22⋊D271—24no—
126C2×Dic271—21no—
127C2×Dic271—81no—
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C2×Dic271C2×D411yescharacteristic
131C4×D271—21no—
132C4×D271—21no—
133C4×D271—81no—
134C4×D271—81no—
135C4×D271—81no—
136C4×D271—81no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139C23×C271—21no—
140D4×C271—21no—
141D4×C271#4112yes—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—81no—
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C8×C271—21no—
150C8×C271—21no—
151C22×C271—22no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271C22×D411yescharacteristic
155C2×D271—21no—
156C2×D271—22no—
157C2×D271—21no—
158C2×D271—21no—
159C2×D271—41no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—84no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271—41no—
173C4×C271C2×#4111yescharacteristic
174C4×C271C2×#4111yescharacteristic
175Dic271—21no—
176Dic271—21no—
177Dic271—21no—
178Dic271—41no—
179C2×C271—42no—
180C2×C271#52511yescharacteristic
181C2×C271#57611yescharacteristic
182C2×C271#57611yescharacteristic
183D271—41no—
184D271—42no—
185D271—81no—
186D271—81no—
187C271#312211yescharacteristic, complemented, Sylow
188#3122—2711nomaximal, Sylow
189#540—2711no—
190#518—2711no—
191#482—2711no—
192#482—2711no—
193#478—2711no—
194#478D27111yescharacteristic, complemented
195#459—2711no—
196#459—2711no—
197#447—2711no—
198#447—2711no—
199#442—2711no—
200#442—2711no—
201C4×D8—2711no—
202C4×D8—2711no—
203#433—2711no—
204C22×D4—2711no—
205C2×D8—2711no—
206C2×D8—2711no—
207C2×D8—2711no—
208C2×D8—2711no—
209#127—2711no—
210#125C2×D27111yescharacteristic, complemented
211#123—5421no—
212#123—5421no—
213#121—2711no—
214#121—2711no—
215#121—5421no—
216#121—5421no—
217C4×D4—2711no—
218C4×D4—2711no—
219C4×D4—2711no—
220C4×D4—2711no—
221C4×D4—2711no—
222C4×D4—5421no—
223C4×D4—5421no—
224C4×D4C2×D27111yescharacteristic
225#117—2711no—
226#117—2711no—
227C2×C22⋊C4—5421no—
228C2×C22⋊C4—5421no—
229#107—2711no—
230#107C2×D27111yescharacteristic, complemented
231#106—2711no—
232#106C2×D27111yescharacteristic, complemented
233C4⋊C8—2711no—
234C4⋊C8C2×D27111yescharacteristic, complemented
235#102—2711no—
236#102—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241#102C2×D27111yescharacteristic
242#102C2×D27111yescharacteristic
243C22⋊C8—2711no—
244C22⋊C8—2711no—
245#97—2711no—
246C4×C8—2711no—
247C24—5421no—
248C2×D4—2711no—
249C2×D4—2711no—
250C2×D4—2711no—
251C2×D4—5421no—
252C2×D4—5421no—
253C2×D4—5421no—
254C2×D4—5422no—
255C2×D4—5422no—
256C2×D4—5421no—
257C2×D4—5421no—
258C2×D4C22×D27111yescharacteristic
259C22×C4—21no—
260C22×C4—2711no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264C22×C4—5421no—
265C22×C4—5421no—
266D8—5422no—
267D8—5422no—
268D8—5422no—
269D8—5422no—
270C2×C8—2711no—
271C2×C8—2711no—
272C2×C8C22×D27111yescharacteristic
273C2×C8C22×D27111yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2711no—
277C4⋊C4—2711no—
278C4⋊C4—2711no—
279C4⋊C4C22×D27111yescharacteristic
280C4⋊C4C22×D27111yescharacteristic
281C4⋊C4C22×D27111yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—2711no—
284C22⋊C4—2711no—
285C22⋊C4—5421no—
286C22⋊C4—5421no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C42—2711no—
294C42—5421no—
295C42C22×D27111yescharacteristic
296C23—21no—
297C23—2711no—
298C23—5424no—
299C23—5421no—
300C23—5421no—
301C23—5421no—
302C23—21681no—
303C23—21681no—
304D4—21no—
305D4—2712no—
306D4—5424no—
307D4—5421no—
308D4—5421no—
309D4—5421no—
310D4—5421no—
311D4—5421no—
312D4—21681no—
313D4—21681no—
314D4—21681no—
315D4—21681no—
316D4#1686465512yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—2711no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—21681no—
328C2×C4—21681no—
329C2×C4—21681no—
330C2×C4—21681no—
331C2×C4—21681no—
332C2×C4D4×D27111yescharacteristic
333C2×C4D4×D27111yescharacteristic
334C2×C4C23×D27111yescharacteristic
335C8—21no—
336C8—21no—
337C8—5421no—
338C8—5421no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—5421no—
343C22—5422no—
344C22—5421no—
345C22—5421no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22—10841no—
350C22—10841no—
351C22—10841no—
352C22—10841no—
353C22—10841no—
354C22—21684no—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—5421no—
362C4—5421no—
363C4—5421no—
364C4—10841no—
365C4#41×D27111yescharacteristic
366C4C2×#1686465511yescharacteristic
367C2—42no—
368C2—10841no—
369C2—10842no—
370C2—21681no—
371C2—21681no—
372C2#3177441011yescharacteristic
373C2#3177469911yescharacteristic
374C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#478×C271C2
socleC22×C271C22×D4
hypercenter#478D271
hypocenterC271#3122
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #31774503C2semidirect
2. #31774369C2semidirect
3. #31774224C2semidirect
4. #31774223C2semidirect
5. #31774194C2semidirect
6. #31774093C2semidirect
7. #31774092C2semidirect
8. #31774032C2semidirect
9. #31774032C2semidirect
10. #31774008C2semidirect
11. #31774006C2semidirect
12. #31773979C2semidirect
13. #31773977C2semidirect
14. #31773957C2semidirect
15. #478×C271C2semidirect
21. #22576157C22semidirect
31. #22576122C22semidirect
32. #22576122C22semidirect
39. #22576093C22semidirect
40. #22576091C22semidirect
41. C4⋊(C271⋊C8)C22semidirect
42. #22576082C22semidirect
43. #22576082C22semidirect
46. #22576080C22semidirect
47. #22576080C22semidirect
50. #22576070C22semidirect
51. C8×Dic271C22semidirect
52. #125×C271C22semidirect
54. #107×C271C22semidirect
55. #106×C271C22semidirect
56. C4⋊C8×C271C22semidirect
187. C271#3122semidirect
194. #478D271semidirect
210. #125C2×D271semidirect
230. #107C2×D271semidirect
232. #106C2×D271semidirect
234. C4⋊C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745032
#317743692
#317742242
#317742232
#317741942
#317740932
#317740922
#317740322
#317740082
#317740062
#317739792
#317739772
#317739572
#478×C2712
#3122271