Finite Group Databasebeta

← #46848356

Subgroups — #46848356

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#31774554C211yesmaximal, characteristic, complemented
2#31774548C211yesmaximal, characteristic, complemented
3#31774255C211yesmaximal, characteristic, complemented
4#31774232C211yesmaximal, characteristic, complemented
5#31774209C211yesmaximal, characteristic, complemented
6#31774190C211yesmaximal, characteristic, complemented
7#31774143C211yesmaximal, characteristic, complemented
8#31774132C211yesmaximal, characteristic, complemented
9#31774111C211yesmaximal, characteristic, complemented
10#31774042C211yesmaximal, characteristic, complemented
11#31774037C211yesmaximal, characteristic, complemented
12#31774000C211yesmaximal, characteristic, complemented
13#31773966C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#417×C271C211yesmaximal, characteristic, complemented
16C2×#16864654—21no—
17C2×Q8×D271C2211yescharacteristic
18C2×#16864643C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C2×C8×D271C2211yescharacteristic
22#22576159C2211yescharacteristic
23#22576147—21no—
24#22576147C2211yescharacteristic
25#22576145—21no—
26#22576143—21no—
27#22576139—21no—
28#22576139C2211yescharacteristic
29#22576138—21no—
30#22576135C2211yescharacteristic
31#22576134C2211yescharacteristic
32#22576131—21no—
33#22576131C2211yescharacteristic
34Q8×Dic271C2211yescharacteristic
35#22576122C2211yescharacteristic
36C4×C22⋊D271—21no—
37C4×C22⋊D271C2211yescharacteristic
38C4⋊C4×D271C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576093C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×M16×C271C2211yescharacteristic
54C2×C4⋊C4×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864654—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—41no—
65C2×D1084—21no—
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271C2311yescharacteristic
74#16864643—41no—
75#16864641—22no—
76#16864639—41no—
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C8×D271—41no—
80C4⋊Dic271—21no—
81C4⋊Dic271C2311yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271—21no—
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271D411yescharacteristic
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4D411yescharacteristic
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98#41×C271—22no—
99#41×C271—41no—
100C2×Q8×C271C2311yescharacteristic
101C2×D4×C271—21no—
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106M16×C271—41no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271—21no—
110C4⋊C4×C271C2311yescharacteristic
111C4⋊C4×C271C2311yescharacteristic
112C22×D271—21no—
113C22×D271C2×D411yescharacteristic
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117Dic542—42no—
118D1084—21no—
119D1084—21no—
120D1084—42no—
121C22⋊D271—22no—
122C22⋊D271—42no—
123C22⋊D271—42no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—21no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C4×D271—42no—
134C4×D271—42no—
135C271⋊C8—21no—
136C271⋊C8—21no—
137C23×C271—21no—
138C23×C271C2×D411yescharacteristic
139Q8×C271—21no—
140Q8×C271#4112yescomplemented
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—42no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—42no—
150C2×C4×C271—42no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271C22×D411yescharacteristic
162C2×D271—21no—
163C2×D271—21no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271AGL(1,ℤ/8)12yes—
168C4×C271—21no—
169C4×C271—22no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271C2×#4111yescharacteristic
173C4×C271H(5,𝔽2)11yescharacteristic
174Dic271—21no—
175Dic271—21no—
176Dic271—21no—
177Dic271—41no—
178Dic271—41no—
179Dic271—41no—
180C2×C271—41no—
181C2×C271—81no—
182C2×C271#54611yescharacteristic
183C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C271#57811yescharacteristic
185D271—22no—
186D271—81no—
187C271#330811yescharacteristic, complemented, Sylow
188#3308—2711nomaximal, Sylow
189#548—2711no—
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#417D27111yescharacteristic, complemented
203C2×#103—2711no—
204C2×#41—5421no—
205C2×#41C2×D27111yescharacteristic
206C22×Q8—2711no—
207C2×SD16—2711no—
208C2×SD16—2711no—
209C2×D8—2711no—
210C2×M16C2×D27111yescharacteristic
211C22×C8—2711no—
212C4⋊Q8—2711no—
213#124—2711no—
214#124—5421no—
215#123—5421no—
216#123—5421no—
217C22⋊Q8—2711no—
218C22⋊Q8—5421no—
219C22⋊Q8—5421no—
220#121—2711no—
221#121—2711no—
222#121—2711no—
223#121—5421no—
224C4×Q8—2711no—
225C4×D4—2711no—
226C4×D4—2711no—
227C4×D4—5421no—
228C2×C4⋊C4—2711no—
229C2×C4⋊C4C2×D27111yescharacteristic
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#103C2×D27111yescharacteristic
239#103C2×D27111yescharacteristic
240#102—2711no—
241#102—2711no—
242#102C2×D27111yescharacteristic
243#102C2×D27111yescharacteristic
244C22⋊C8—2711no—
245C22⋊C8—2711no—
246#97—2711no—
247#41—22no—
248#41—41no—
249#41—10844no—
250C2×Q8—2711no—
251C2×Q8—5421no—
252C2×Q8—5421no—
253C2×Q8—5422no—
254C2×Q8—10841no—
255C2×Q8C22×D27111yescharacteristic
256C2×D4—21no—
257C2×D4—2711no—
258C2×D4—2711no—
259C2×D4—5421no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4C22×D27111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—2711no—
266C22×C4—5421no—
267C22×C4—5421no—
268C22×C4—5421no—
269C22×C4C22×D27111yescharacteristic
270SD16—5422no—
271SD16—10841no—
272D8—10841no—
273M16—41no—
274C2×C8—2711no—
275C2×C8—2711no—
276C2×C8—10841no—
277C2×C8C22×D27111yescharacteristic
278C2×C8C22×D27111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—2711no—
281C4⋊C4—2711no—
282C4⋊C4—2711no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4—5421no—
287C4⋊C4C22×D27111yescharacteristic
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—2711no—
290C22⋊C4—2711no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C42—2711no—
299C42—5421no—
300C23—21no—
301C23—2711no—
302C23—5421no—
303C23D4×D27111yescharacteristic
304Q8—21no—
305Q8—5421no—
306Q8—5421no—
307Q8—10842no—
308Q8—10842no—
309Q8#1686465412yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—5422no—
314D4—5421no—
315D4—5421no—
316D4—10842no—
317D4—10842no—
318D4—10842no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—2711no—
327C2×C4—2711no—
328C2×C4—2711no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—5421no—
333C2×C4—5421no—
334C2×C4—10842no—
335C2×C4—10842no—
336C2×C4—10842no—
337C2×C4D4×D27111yescharacteristic
338C2×C4C23×D27111yescharacteristic
339C8—21no—
340C8—21no—
341C8—5421no—
342C8—5421no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—2712no—
350C22—5421no—
351C22—5421no—
352C22—10841no—
353C22—10841no—
354C22—10841no—
355C22C2×D4×D27111yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—5421no—
361C4—5421no—
362C4—5421no—
363C4—10841no—
364C4—10841no—
365C4—10841no—
366C4C2×#1686465411yescharacteristic
367C4#2257620511yescharacteristic
368C2—41no—
369C2—81no—
370C2—5422no—
371C2—21681no—
372C2#3177453711yescharacteristic
373C2AGL(1,ℤ/8)×D27111yescharacteristic
374C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#417×C271C2
socleC22×C271C22×D4
hypercenter#417D271
hypocenterC271#3308
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #31774554C2semidirect
2. #31774548C2semidirect
3. #31774255C2semidirect
4. #31774232C2semidirect
5. #31774209C2semidirect
6. #31774190C2semidirect
7. #31774143C2semidirect
8. #31774132C2semidirect
9. #31774111C2semidirect
10. #31774042C2semidirect
11. #31774037C2semidirect
12. #31774000C2semidirect
13. #31773966C2semidirect
14. #103×D271C2semidirect
15. #417×C271C2semidirect
140. Q8×C271#41semidirect
187. C271#3308semidirect
202. #417D271semidirect
309. Q8#16864654semidirect

Maximal subgroups

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

subgroupindex
#317745542
#317745482
#317742552
#317742322
#317742092
#317741902
#317741432
#317741322
#317741112
#317740422
#317740372
#317740002
#317739662
#103×D2712
#417×C2712
#3308271