Finite Group Databasebeta

← #46848270

Subgroups — #46848270

Order 34688.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774599C211yesmaximal, characteristic, complemented
2#31774558C211yesmaximal, characteristic, complemented
3#31774253C211yesmaximal, characteristic, complemented
4#31774236C211yesmaximal, characteristic, complemented
5#31774216C211yesmaximal, characteristic, complemented
6#31774202C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774114C211yesmaximal, characteristic, complemented
10#31774041C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14C2×#22576084C211yesmaximal, characteristic, complemented
15#448×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576153—21no—
21#22576151C2211yescharacteristic
22#22576149—21no—
23#22576149C2211yescharacteristic
24#22576144—21no—
25#22576136—21no—
26#22576136C2211yescharacteristic
27#22576134—21no—
28#22576134C2211yescharacteristic
29#22576132—21no—
30#22576131C2211yescharacteristic
31Q8×Dic271C2211yescharacteristic
32D4×Dic271—21no—
33D4×Dic271C2211yescharacteristic
34C4×C22⋊D271—21no—
35C4×C22⋊D271C2211yescharacteristic
36#22576118C2211yescharacteristic
37C2×C4⋊Dic271C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576092C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C22×Q8×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—41no—
61C2×D1084—21no—
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C22⋊D271—21no—
65C2×C22⋊D271D411yescharacteristic
66C22×Dic271—21no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69C2×C4×D271C2311yescharacteristic
70#16864639—41no—
71C2×C271⋊C8—41no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C8×D271—41no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77C4⋊Dic271—21no—
78C4⋊Dic271C2311yescharacteristic
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271D411yescharacteristic
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
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96C2×Q8×C271—21no—
97C2×Q8×C271—22no—
98C2×Q8×C271—41no—
99C2×Q8×C271C2311yescharacteristic
100C2×Q8×C271C2311yescharacteristic
101C2×D4×C271—21no—
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105SD16×C271—22no—
106SD16×C271—41no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113D1084—21no—
114D1084—21no—
115D1084—42no—
116C22⋊D271—22no—
117C22⋊D271—42no—
118C22⋊D271—81no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—81no—
123C2×Dic271—81no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—21no—
128C4×D271—21no—
129C4×D271—42no—
130C271⋊C8—21no—
131C271⋊C8—21no—
132C23×C271—21no—
133C23×C271C2×D411yescharacteristic
134Q8×C271—21no—
135Q8×C271—21no—
136Q8×C271—21no—
137Q8×C271—42no—
138Q8×C271#4112yes—
139D4×C271—21no—
140D4×C271—21no—
141D4×C271—42no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—42no—
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271—41no—
157C22×C271#13512yes—
158C22×C271C22×D411yescharacteristic
159C2×D271—21no—
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C4×C271—21no—
166C4×C271—22no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271C2×#4111yescharacteristic
170C4×C271H(5,𝔽2)11yescharacteristic
171Dic271—21no—
172Dic271—21no—
173Dic271—21no—
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—22no—
178C2×C271—81no—
179C2×C271#54611yescharacteristic
180C2×C271C2×#13511yescharacteristic
181C2×C271#57711yescharacteristic
182D271—41no—
183D271—81no—
184C271#330111yescharacteristic, complemented, Sylow
185#3301—2711nomaximal, Sylow
186#553—2711no—
187#548—2711no—
188#488—2711no—
189#485—2711no—
190#481—2711no—
191#479—2711no—
192#466—2711no—
193#465—2711no—
194#463—2711no—
195#449—2711no—
196#448D27111yescharacteristic, complemented
197C4×SD16—2711no—
198C8×D4—2711no—
199#416—2711no—
200C2×#103—2711no—
201C2×#41—2711no—
202C22×Q8C2×D27111yescharacteristic
203C2×SD16C2×D27111yescharacteristic
204C2×SD16C2×D27111yescharacteristic
205C2×D8—2711no—
206C22×C8—2711no—
207C22×C8—2711no—
208#126—5421no—
209#125—2711no—
210#124—2711no—
211#124—5421no—
212#123—5421no—
213C22⋊Q8—2711no—
214C22⋊Q8—5421no—
215#121—2711no—
216#121—2711no—
217#121—5421no—
218#121—5421no—
219#121C2×D27111yescharacteristic
220C4×Q8—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223C4×D4—5421no—
224C4×D4—5421no—
225#117—2711no—
226C2×C4⋊C4—2711no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103C2×D27111yescharacteristic
236#103C2×D27111yescharacteristic
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241C22⋊C8—2711no—
242C22⋊C8C2×D27111yescharacteristic
243C4×C8—2711no—
244#41—5422no—
245#41—10841no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D27111yescharacteristic
250C2×Q8C22×D27111yescharacteristic
251C2×D4—21no—
252C2×D4—2711no—
253C2×D4—2711no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4C22×D27111yescharacteristic
264SD16—22no—
265SD16—41no—
266D8—10841no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—10841no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2711no—
285C22⋊C4—2711no—
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—
295C42—5421no—
296C23—21no—
297C23—2711no—
298C23—5421no—
299C23D4×D27111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686465312yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—5422no—
309D4—5421no—
310D4—5421no—
311D4—10842no—
312D4—10842no—
313D4—21681no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—2711no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—10842no—
329C2×C4—21681no—
330C2×C4—21681no—
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—5421no—
343C22—5421no—
344C22—5421no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22#2257618112yes—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—5421no—
355C4—5421no—
356C4—5421no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4C2×#1686465311yescharacteristic
361C4#2257620511yescharacteristic
362C2—22no—
363C2—81no—
364C2—10841no—
365C2—21681no—
366C2#3177453411yescharacteristic
367C2C2×#2257618111yescharacteristic
368C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#448×C271C2
socleC22×C271C22×D4
hypercenter#448D271
hypocenterC271#3301
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #31774599C2semidirect
2. #31774558C2semidirect
3. #31774253C2semidirect
4. #31774236C2semidirect
5. #31774216C2semidirect
6. #31774202C2semidirect
7. #31774131C2semidirect
8. #31774123C2semidirect
9. #31774114C2semidirect
10. #31774041C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773892C2semidirect
14. C2×#22576084C2semidirect
15. #448×C271C2semidirect
184. C271#3301semidirect
196. #448D271semidirect

Maximal subgroups

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

subgroupindex
#317745992
#317745582
#317742532
#317742362
#317742162
#317742022
#317741312
#317741232
#317741142
#317740412
SD16×Dic2712
(C22⋊D271)⋊C82
#317738922
C2×#225760842
#448×C2712
#3301271