Finite Group Databasebeta

← #46848355

Subgroups — #46848355

Order 34688.

420 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#31774551C211yesmaximal, characteristic, complemented
2#31774256C211yesmaximal, characteristic, complemented
3#31774230C211yesmaximal, characteristic, complemented
4#31774208C211yesmaximal, characteristic, complemented
5#31774192C211yesmaximal, characteristic, complemented
6#31774145C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774109C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774036C211yesmaximal, characteristic, complemented
11#31773995C211yesmaximal, characteristic, complemented
12#31773969C211yesmaximal, characteristic, complemented
13#31773898C211yesmaximal, characteristic, complemented
14C2×#22576084C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864641C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×#16864638C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22#22576160C2211yescharacteristic
23#22576144—21no—
24#22576144—21no—
25#22576140—21no—
26#22576134C2211yescharacteristic
27#22576134C2211yescharacteristic
28C4×D1084C2211yescharacteristic
29C2×C4⋊Dic271C2211yescharacteristic
30C2×C4⋊Dic271C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576093C2211yescharacteristic
33#22576093C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576084C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576084C2211yescharacteristic
39#22576084C2211yescharacteristic
40#22576084C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576081C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47#22576071C2211yescharacteristic
48C2×#41×C271C2211yescharacteristic
49C22×Q8×C271C2211yescharacteristic
50#124×C271—21no—
51#124×C271C2211yescharacteristic
52C22⋊Q8×C271—21no—
53C22⋊Q8×C271C2211yescharacteristic
54#121×C271—21no—
55#121×C271C2211yescharacteristic
56C4×Q8×C271C2211yescharacteristic
57C4×D4×C271—21no—
58C4×D4×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C22×Dic271—21no—
65C22×Dic271—21no—
66C2×C4×D271—21no—
67#16864641—22no—
68#16864641—41no—
69#16864639—41no—
70#16864638—41no—
71C2×C271⋊C8—41no—
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C2×C271⋊C8C2311yescharacteristic
76C4⋊Dic271—21no—
77C4⋊Dic271—21no—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82(C2×D271)⋊C4—21no—
83(C2×D271)⋊C4—21no—
84(C2×D271)⋊C4—21no—
85#41×C271—22no—
86#41×C271—41no—
87C2×Q8×C271—21no—
88C2×Q8×C271—22no—
89C2×Q8×C271—41no—
90C2×Q8×C271C2311yescharacteristic
91C2×Q8×C271C2311yescharacteristic
92C2×D4×C271—21no—
93C2×D4×C271—21no—
94C2×D4×C271D411yescharacteristic
95C2×D4×C271C2311yescharacteristic
96C22×C4×C271—21no—
97C22×C4×C271—21no—
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C4⋊C4×C271—21no—
101C4⋊C4×C271—21no—
102C4⋊C4×C271D411yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271—21no—
108C22⋊C4×C271D411yescharacteristic
109C22⋊C4×C271D411yescharacteristic
110C42×C271—21no—
111C42×C271C2311yescharacteristic
112C22×D271—21no—
113Dic542—42no—
114D1084—21no—
115D1084—21no—
116C22⋊D271—42no—
117C22⋊D271—42no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—42no—
122C2×Dic271—42no—
123C4×D271—42no—
124C271⋊C8—21no—
125C271⋊C8—21no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C23×C271—21no—
129C23×C271C2×D411yescharacteristic
130C23×C271C2×D411yescharacteristic
131Q8×C271—21no—
132Q8×C271—21no—
133Q8×C271—21no—
134Q8×C271—42no—
135Q8×C271#4112yescomplemented
136D4×C271—22no—
137D4×C271—21no—
138D4×C271—21no—
139D4×C271—42no—
140D4×C271—42no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—42no—
149C2×C4×C271—42no—
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2×D411yescharacteristic
153C2×C4×C271C2×D411yescharacteristic
154C2×C4×C271C2411yescharacteristic
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—21no—
159C22×C271—21no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271AGL(1,ℤ/8)12yes—
164C22×C271C22×D411yescharacteristic
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—41no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—22no—
171C4×C271—21no—
172C4×C271—21no—
173C4×C271—41no—
174C4×C271—41no—
175C4×C271C2×#4111yescharacteristic
176C4×C271H(5,𝔽2)11yescharacteristic
177Dic271—41no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—22no—
181C2×C271—41no—
182C2×C271—81no—
183C2×C271#54611yescharacteristic
184C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C271#57811yescharacteristic
186D271—81no—
187C271#330811yescharacteristic, complemented, Sylow
188#3308—2711nomaximal, Sylow
189#548D27111yescharacteristic, complemented
190#547—2711no—
191#489—2711no—
192#484—2711no—
193#480—2711no—
194#477—2711no—
195#468—2711no—
196#466—2711no—
197#463—2711no—
198#449—2711no—
199#448—2711no—
200#440—2711no—
201#435—2711no—
202#417—2711no—
203C2×#103—2711no—
204C2×#41—5421no—
205C2×#41C2×D27111yescharacteristic
206C22×Q8C2×D27111yescharacteristic
207C2×SD16—2711no—
208C2×SD16—2711no—
209C2×D8—2711no—
210C2×M16—2711no—
211C22×C8—2711no—
212C4⋊Q8—2711no—
213#124—21no—
214#124C2×D27111yescharacteristic
215#123—5421no—
216#123—5421no—
217C22⋊Q8—21no—
218C22⋊Q8—5421no—
219C22⋊Q8C2×D27111yescharacteristic
220#121—21no—
221#121—2711no—
222#121—2711no—
223#121C2×D27111yescharacteristic
224C4×Q8C2×D27111yescharacteristic
225C4×D4—21no—
226C4×D4—2711no—
227C4×D4C2×D27111yescharacteristic
228C2×C4⋊C4—2711no—
229C2×C4⋊C4—2711no—
230#107—2711no—
231#107—2711no—
232#107—2711no—
233C4⋊C8—2711no—
234#103—2711no—
235#103—2711no—
236#103—2711no—
237#103—2711no—
238#103—2711no—
239#103—2711no—
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246#97—2711no—
247#41—22no—
248#41—41no—
249#41—10844no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—5421no—
254C2×Q8C22×D27111yescharacteristic
255C2×Q8C22×D27111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4C22⋊D27111yescharacteristic
262C2×D4C22×D27111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—5421no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4C22×D27111yescharacteristic
269C22×C4C22×D27111yescharacteristic
270SD16—5422no—
271SD16—10841no—
272D8—10841no—
273M16—10841no—
274C2×C8—2711no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8—2711no—
278C2×C8—10841no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22⋊D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4C22⋊D27111yescharacteristic
297C22⋊C4C22⋊D27111yescharacteristic
298C42—21no—
299C42C22×D27111yescharacteristic
300C23—21no—
301C23—5421no—
302C23C2×C22⋊D27111yescharacteristic
303C23C2×C22⋊D27111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—10842no—
309Q8#1686465312yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—5421no—
316D4—5421no—
317D4—10842no—
318D4—10842no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—10842no—
332C2×C4—10842no—
333C2×C4—10842no—
334C2×C4C2×C22⋊D27111yescharacteristic
335C2×C4C2×C22⋊D27111yescharacteristic
336C2×C4C2×C22⋊D27111yescharacteristic
337C2×C4C2×C22⋊D27111yescharacteristic
338C2×C4C23×D27111yescharacteristic
339C8—5421no—
340C8—5421no—
341C8—5421no—
342C8—5421no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—10841no—
352C22—10841no—
353C22—10841no—
354C22#2257618512yes—
355C22C22×C22⋊D27111yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—10841no—
364C4—10841no—
365C4—10841no—
366C4C2×#1686465311yescharacteristic
367C4#2257620511yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—21681no—
372C2#3177454111yescharacteristic
373C2C2×#2257618511yescharacteristic
374C2#3177471211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#548×C271C2
socleC22×C271C22×D4
hypercenter#548D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31774551C2semidirect
2. #31774256C2semidirect
3. #31774230C2semidirect
4. #31774208C2semidirect
5. #31774192C2semidirect
6. #31774145C2semidirect
7. #31774131C2semidirect
8. #31774109C2semidirect
9. #31774042C2semidirect
10. #31774036C2semidirect
11. #31773995C2semidirect
12. #31773969C2semidirect
13. #31773898C2semidirect
14. C2×#22576084C2semidirect
15. #548×C271C2semidirect
135. Q8×C271#41semidirect
187. C271#3308semidirect
189. #548D271semidirect
309. Q8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317745512
#317742562
#317742302
#317742082
#317741922
#317741452
#317741312
#317741092
#317740422
#317740362
#317739952
#317739692
#317738982
C2×#225760842
#548×C2712
#3308271