Finite Group Databasebeta

← #46848222

Subgroups — #46848222

Order 34688.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D271C211yesmaximal, characteristic, complemented
2#31774547C211yesmaximal, characteristic
3#31774246C211yesmaximal, characteristic, complemented
4#31774228C211yesmaximal, characteristic, complemented
5#31774226C211yesmaximal, characteristic, complemented
6#31774180C211yesmaximal, characteristic, complemented
7#31774124C211yesmaximal, characteristic, complemented
8#31774123C211yesmaximal, characteristic, complemented
9#31774100C211yesmaximal, characteristic, complemented
10#31774034C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13C2×#22576084C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#448×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×Q8×D271C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C22×C271⋊C8C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576159C2211yescharacteristic
22#22576158—21no—
23#22576158C2211yescharacteristic
24#22576142—21no—
25#22576141—21no—
26#22576139—21no—
27#22576139C2211yescharacteristic
28#22576137—21no—
29#22576136—21no—
30#22576136C2211yescharacteristic
31#22576131C2211yescharacteristic
32Q8×Dic271C2211yescharacteristic
33D4×Dic271C2211yescharacteristic
34C4×C22⋊D271—21no—
35C4×C22⋊D271C2211yescharacteristic
36C2×C4⋊Dic271C2211yescharacteristic
37C4⋊C4×D271C2211yescharacteristic
38#22576092C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576091C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576083C2211yescharacteristic
49#22576081C2211yescharacteristic
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#16864653—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—41no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271D411yescharacteristic
67C22×Dic271—21no—
68C22×Dic271—21no—
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271C2311yescharacteristic
72#16864642—41no—
73C2×C271⋊C8—41no—
74C2×C271⋊C8C2311yescharacteristic
75C2×C271⋊C8C2311yescharacteristic
76C8×D271—41no—
77C4⋊Dic271—21no—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4D411yescharacteristic
89C22⋊Dic271—21no—
90C22⋊Dic271—21no—
91C22⋊Dic271—21no—
92C22⋊Dic271D411yescharacteristic
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4D411yescharacteristic
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271—21no—
98C2×Q8×C271—22no—
99C2×Q8×C271—41no—
100C2×Q8×C271C2311yescharacteristic
101C2×Q8×C271C2311yescharacteristic
102C2×D4×C271—21no—
103C2×D4×C271C2311yescharacteristic
104C22×C4×C271—21no—
105C22×C4×C271C2311yescharacteristic
106SD16×C271—22no—
107SD16×C271—41no—
108C2×C8×C271C2311yescharacteristic
109C2×C8×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C22⋊C4×C271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—21no—
115Dic542—42no—
116Dic542—42no—
117C22⋊D271—42no—
118C22⋊D271SD1614yescomplemented
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—42no—
124C2×Dic271—42no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—42no—
131C4×D271—42no—
132C271⋊C8—21no—
133C271⋊C8—21no—
134C23×C271—21no—
135C23×C271C2×D411yescharacteristic
136Q8×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139Q8×C271—42no—
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×C271C2×D411yescharacteristic
151C2×C4×C271C2411yescharacteristic
152C8×C271—21no—
153C8×C271—21no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—41no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271C2×SD1612yes—
160C22×C271C22×D411yescharacteristic
161C2×D271—21no—
162C2×D271—21no—
163C2×D271C2×SD1612yes—
164C4×C271—21no—
165C4×C271—22no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271C2×#4111yescharacteristic
169C4×C271H(5,𝔽2)11yescharacteristic
170Dic271—21no—
171Dic271—21no—
172Dic271—41no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271C2×SD1612yes—
177C2×C271—22no—
178C2×C271—81no—
179C2×C271#54611yescharacteristic
180C2×C271C22×SD1611yescharacteristic
181C2×C271#57811yescharacteristic
182D271—22no—
183C271#329811yescharacteristic, complemented, Sylow
184#3298—2711nomaximal, Sylow
185D4×Q8—2711no—
186#547—2711no—
187#487—2711no—
188#483—2711no—
189#483—2711no—
190#475—2711no—
191#465—2711no—
192#465—2711no—
193#461—2711no—
194#448—2711no—
195#448D27111yescharacteristic, complemented
196C4×SD16—2711no—
197C8×D4—2711no—
198C2×#103—2711no—
199C2×#103—2711no—
200C2×#41—5421no—
201C22×Q8—2711no—
202C22×Q8C2×D27111yescharacteristic
203C2×SD16—2711no—
204C2×SD16C2×D27111yescharacteristic
205C2×SD16C2×D27111yescharacteristic
206C22×C8—2711no—
207C22×C8—2711no—
208C4⋊Q8—2711no—
209C4⋊Q8—2711no—
210C4⋊Q8—5421no—
211#123—5421no—
212#123—5421no—
213C22⋊Q8—2711no—
214C22⋊Q8—2711no—
215C22⋊Q8—5421no—
216C22⋊Q8—5421no—
217C22⋊Q8—5421no—
218#121—2711no—
219#121C2×D27111yescharacteristic
220C4×Q8—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223C4×D4—5421no—
224C2×C4⋊C4—2711no—
225C2×C4⋊C4—2711no—
226#106—2711no—
227#106—2711no—
228#106—2711no—
229C4⋊C8—2711no—
230#103—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#103—2711no—
237#103C2×D27111yescharacteristic
238#103C2×D27111yescharacteristic
239#102—2711no—
240C22⋊C8—2711no—
241C22⋊C8C2×D27111yescharacteristic
242C4×C8—2711no—
243#41—10844no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—2711no—
248C2×Q8—5421no—
249C2×Q8—5421no—
250C2×Q8—5422no—
251C2×Q8—10841no—
252C2×Q8C22×D27111yescharacteristic
253C2×Q8C22×D27111yescharacteristic
254C2×D4—21no—
255C2×D4—2711no—
256C2×D4—5421no—
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—2711no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264C22×C4C22×D27111yescharacteristic
265SD16—22no—
266SD16—41no—
267SD16—10841no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8—10841no—
272C2×C8C22×D27111yescharacteristic
273C2×C8C22×D27111yescharacteristic
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—2711no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4—5421no—
283C4⋊C4—5421no—
284C4⋊C4—5421no—
285C4⋊C4—5421no—
286C4⋊C4C22×D27111yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—2711no—
289C22⋊C4—2711no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C42—2711no—
295C42—5421no—
296C23—21no—
297C23—2711no—
298C23D4×D27111yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—5421no—
304Q8—5421no—
305Q8—10842no—
306Q8—10842no—
307Q8#1686465312yescomplemented
308D4—21no—
309D4—21no—
310D4—42no—
311D4—2714no—
312D4—10842no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—2711no—
320C2×C4—2711no—
321C2×C4—2711no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—10842no—
329C2×C4—10842no—
330C2×C4—10842no—
331C2×C4—10842no—
332C2×C4D4×D27111yescharacteristic
333C2×C4C23×D27111yescharacteristic
334C8—21no—
335C8—21no—
336C8—5421no—
337C8—5421no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—2712no—
344C22—5421no—
345C22—5421no—
346C22SD16×D27112yes—
347C22C2×D4×D27111yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—2712no—
353C4—5421no—
354C4—5421no—
355C4—10841no—
356C4—10841no—
357C4—10841no—
358C4—10841no—
359C4C2×#1686465311yescharacteristic
360C4#2257620511yescharacteristic
361C2—22no—
362C2—81no—
363C2—5422no—
364C2#3177453411yescharacteristic
365C2C2×SD16×D27111yescharacteristic
366C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#448×C271C2
socleC22×C271C22×D4
prime radical#31774547C2
hypercenter#448D271
hypocenterC271#3298
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. Q8×C22⋊D271C2semidirect
3. #31774246C2semidirect
4. #31774228C2semidirect
5. #31774226C2semidirect
6. #31774180C2semidirect
7. #31774124C2semidirect
8. #31774123C2semidirect
9. #31774100C2semidirect
10. #31774034C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. C2×#22576084C2semidirect
14. #103×D271C2semidirect
15. #448×C271C2semidirect
118. C22⋊D271SD16semidirect
140. Q8×C271#41semidirect
183. C271#3298semidirect
195. #448D271semidirect
307. Q8#16864653semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D2712
#317745472
#317742462
#317742282
#317742262
#317741802
#317741242
#317741232
#317741002
#317740342
SD16×Dic2712
(C22⋊D271)⋊C82
C2×#225760842
#103×D2712
#448×C2712
#3298271