Finite Group Databasebeta

← #46850440

Subgroups — #46850440

Order 34688.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774629C213yesmaximal, complemented
2Q8×Dic542C211yesmaximal, characteristic, complemented
3#31774605C213yesmaximal, complemented
4#31774566C211yesmaximal, characteristic, complemented
5#31774558C211yesmaximal, characteristic, complemented
6#31774551C213yesmaximal, complemented
7#31774547C213yesmaximal, complemented
8#31774520C213yesmaximal, complemented
9#31774516C213yesmaximal, complemented
10#31774442C213yesmaximal, complemented
11#31774427C213yesmaximal, complemented
12#31774380C211yesmaximal, characteristic, complemented
13#31774379C211yesmaximal, characteristic, complemented
14C2×Q8×Dic271C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2216yes—
18#22576160C2213yes—
19#22576159C2213yes—
20#22576159C2213yes—
21#22576156C2213yes—
22#22576155C2213yes—
23#22576153C2213yes—
24#22576150C2213yes—
25#22576149C2213yes—
26#22576146C2216yes—
27#22576146C2213yes—
28#22576144C2216yes—
29#22576144C2213yes—
30#22576142C2216yes—
31#22576141C2213yes—
32#22576140C2216yes—
33#22576137C2216yes—
34#22576137C2211yescharacteristic
35#22576136C2213yes—
36#22576134C2213yes—
37#22576134C2213yes—
38#22576132C2213yes—
39#22576131C2213yes—
40Q8×Dic271C2213yes—
41Q8×Dic271C2211yescharacteristic
42Q8×Dic271C2211yescharacteristic
43Q8×Dic271C2213yes—
44Q8×Dic271C2211yescharacteristic
45C4×Dic542C2213yes—
46C4×D1084C2213yes—
47D4×Dic271C2213yes—
48D4×Dic271C2213yes—
49#22576122C2213yes—
50C4×C22⋊D271C2213yes—
51C4×C22⋊D271C2213yes—
52(C22⋊D271)⋊C4C2216yes—
53#22576119C2213yes—
54#22576117C2213yes—
55C2×C4⋊Dic271C2213yes—
56C2×C4⋊Dic271C2213yes—
57C4⋊C4×D271C2213yes—
58C2×C4×Dic271C2213yes—
59C2×#41×C271C2211yescharacteristic
60C22×Q8×C271C2211yescharacteristic
61#124×C271C2213yes—
62C22⋊Q8×C271C2213yes—
63#121×C271C2213yes—
64C4×Q8×C271C2211yescharacteristic
65C4×D4×C271C2213yes—
66#16864655—24no—
67#16864653—412no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2316yes—
70C2×D1084C2313yes—
71C2×C22⋊D271C2313yes—
72C2×C22⋊D271C2313yes—
73C2×C22⋊D271C2316yes—
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2313yes—
76C22×Dic271C2316yes—
77C2×C4×D271C2313yes—
78C2×C4×D271C2316yes—
79C4⋊Dic271C2313yes—
80C4⋊Dic271C2313yes—
81C4⋊Dic271C2316yes—
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2313yes—
84C4⋊Dic271C2313yes—
85C4⋊Dic271C2313yes—
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2313yes—
89Dic271⋊C4C2311yescharacteristic
90Dic271⋊C4C2311yescharacteristic
91Dic271⋊C4C2313yes—
92C22⋊Dic271C2313yes—
93C22⋊Dic271C2313yes—
94C22⋊Dic271C2311yescharacteristic
95C22⋊Dic271C2313yes—
96(C2×D271)⋊C4C2313yes—
97(C2×D271)⋊C4C2313yes—
98(C2×D271)⋊C4C2316yes—
99(C2×D271)⋊C4C2313yes—
100(C2×D271)⋊C4C2313yes—
101C4×Dic271C2313yes—
102C4×Dic271C2313yes—
103C4×Dic271C2313yes—
104C4×Dic271C2313yes—
105C4×Dic271C2313yes—
106#41×C271—24no—
107C2×Q8×C271—24no—
108C2×Q8×C271C2313yes—
109C2×Q8×C271C2311yescharacteristic
110C2×D4×C271C2313yes—
111C2×D4×C271C2313yes—
112C22×C4×C271C2313yes—
113C22×C4×C271C2313yes—
114C4⋊C4×C271C2313yes—
115C4⋊C4×C271C2313yes—
116C22⋊C4×C271C2316yes—
117C22⋊C4×C271C2311yescharacteristic
118C22⋊C4×C271C2313yes—
119C42×C271C2313yes—
120C22×D271C2413yes—
121Dic542—22no—
122Dic542—46no—
123D1084—26no—
124C22⋊D271—26no—
125C22⋊D271—26no—
126C22⋊D271—46no—
127C2×Dic271—26no—
128C2×Dic271—22no—
129C2×Dic271—46no—
130C2×Dic271C2411yescharacteristic
131C2×Dic271C2411yescharacteristic
132C2×Dic271C2413yes—
133C2×Dic271C2413yes—
134C2×Dic271C2413yes—
135C2×Dic271C2411yescharacteristic
136C2×Dic271C2411yescharacteristic
137C4×D271—26no—
138C4×D271—46no—
139C23×C271C2413yes—
140C23×C271C2411yescharacteristic
141Q8×C271—26no—
142Q8×C271#4114yescomplemented
143D4×C271—26no—
144D4×C271—26no—
145C2×C4×C271—26no—
146C2×C4×C271—26no—
147C2×C4×C271C2411yescharacteristic
148C2×C4×C271C2413yes—
149C2×C4×C271C2413yes—
150C2×C4×C271C2411yescharacteristic
151C2×C4×C271C2413yes—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—23no—
155C22×C271—23no—
156C22×C271—23no—
157C22×C271H(5,𝔽2)12yes—
158C22×C271C2511yescharacteristic
159C2×D271—23no—
160C2×D271—23no—
161C2×D271—23no—
162C4×C271—21no—
163C4×C271—23no—
164C4×C271—23no—
165C4×C271—21no—
166C4×C271C2×#4116yes—
167Dic271—21no—
168Dic271—23no—
169Dic271—23no—
170Dic271—23no—
171Dic271—21no—
172Dic271—21no—
173Dic2712−1+412yes—
174C2×C271—22no—
175C2×C271—43no—
176C2×C271C22×#4111yescharacteristic
177C2×C271C2×H(5,𝔽2)11yescharacteristic
178C2×C271C2×2−1+411yescharacteristic
179D271—43no—
180C271#352211yescharacteristic, complemented, Sylow
181#3522—2711nomaximal, Sylow
182#559—2713no—
183Q82—2711no—
184#554—2713no—
185#550—2711no—
186#548—2711no—
187#548D27111yescharacteristic, complemented
188#547—2713no—
189#547—2713no—
190#542—2713no—
191#542—2713no—
192#532—2713no—
193#529—2713no—
194#520—2711no—
195#520—2711no—
196C2×C4×Q8—2711no—
197C2×#41—2711no—
198C2×#41—2716no—
199C2×#41C2×D27111yescharacteristic
200C22×Q8C2×D27111yescharacteristic
201C4⋊Q8—2713no—
202C4⋊Q8—2713no—
203C4⋊Q8—2713no—
204#127—2713no—
205#126—2713no—
206#126—2713no—
207#125—2713no—
208#124—2713no—
209#124—2713no—
210#124—2716no—
211#124C2×D27113yes—
212#123—2713no—
213#123—2716no—
214#123—2713no—
215#123—2716no—
216C22⋊Q8—2713no—
217C22⋊Q8—2716no—
218C22⋊Q8—2711no—
219C22⋊Q8—2716no—
220C22⋊Q8C2×D27113yes—
221#121—2713no—
222#121—2713no—
223#121—2713no—
224#121—2713no—
225#121C2×D27113yes—
226C4×Q8—2711no—
227C4×Q8—2713no—
228C4×Q8—2711no—
229C4×Q8—2713no—
230C4×Q8—2711no—
231C4×Q8—2713no—
232C4×Q8C2×D27111yescharacteristic
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4—2713no—
236C4×D4—2713no—
237C4×D4—2713no—
238C4×D4—2716no—
239C4×D4—2713no—
240C4×D4C2×D27113yes—
241#117—2713no—
242#117—2713no—
243C2×C4⋊C4—2713no—
244C2×C4⋊C4—2713no—
245C2×C4⋊C4—2713no—
246C2×C42—2713no—
247#41—24no—
248#41—5424no—
249#41—108412no—
250C2×Q8—24no—
251C2×Q8—2711no—
252C2×Q8—2716no—
253C2×Q8C22×D27113yes—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—2713no—
256C2×D4—2713no—
257C2×D4—2716no—
258C2×D4—2713no—
259C2×D4C22×D27113yes—
260C2×D4C22×D27113yes—
261C22×C4—2711no—
262C22×C4—2713no—
263C22×C4—2716no—
264C22×C4—2713no—
265C22×C4—2716no—
266C22×C4C22×D27113yes—
267C22×C4C22×D27113yes—
268C4⋊C4—2713no—
269C4⋊C4—2713no—
270C4⋊C4—2713no—
271C4⋊C4—2713no—
272C4⋊C4—2713no—
273C4⋊C4—2716no—
274C4⋊C4—2711no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—2713no—
278C4⋊C4—2713no—
279C4⋊C4—2713no—
280C4⋊C4—2713no—
281C4⋊C4C22×D27113yes—
282C4⋊C4C22×D27113yes—
283C22⋊C4—2713no—
284C22⋊C4—2713no—
285C22⋊C4—2711no—
286C22⋊C4—2713no—
287C22⋊C4—2713no—
288C22⋊C4—2713no—
289C22⋊C4—2713no—
290C22⋊C4—2716no—
291C22⋊C4—2713no—
292C22⋊C4C22×D27116yes—
293C22⋊C4C22×D27111yescharacteristic
294C22⋊C4C22×D27113yes—
295C42—2713no—
296C42—2713no—
297C42—2713no—
298C42—2713no—
299C42—2713no—
300C42C22×D27113yes—
301C23—2713no—
302C23C23×D27113yes—
303C23C23×D27111yescharacteristic
304Q8—26no—
305Q8—5422no—
306Q8—10846no—
307Q8#1686465314yescomplemented
308D4—26no—
309D4—26no—
310D4—5426no—
311D4—5426no—
312D4—5426no—
313D4—10846no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—2711no—
319C2×C4—2713no—
320C2×C4—2713no—
321C2×C4—2713no—
322C2×C4—2711no—
323C2×C4—5422no—
324C2×C4—5426no—
325C2×C4—5426no—
326C2×C4—10846no—
327C2×C4—10846no—
328C2×C4C23×D27113yes—
329C2×C4C23×D27111yescharacteristic
330C2×C4C23×D27113yes—
331C2×C4C23×D27111yescharacteristic
332C2×C4C23×D27113yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—5423no—
339C22—5423no—
340C22—5423no—
341C22#2257620612yes—
342C22C24×D27111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—2712no—
348C4—5421no—
349C4—5423no—
350C4—5423no—
351C4—5423no—
352C4—5421no—
353C4—5421no—
354C4C2×#1686465316yes—
355C2—22no—
356C2—43no—
357C2—10843no—
358C2C22×#1686465311yescharacteristic
359C2C2×#2257620611yescharacteristic
360C22−1+4×D27111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#548×C271C2
hypercenter#548D271
hypocenterC271#3522
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31774629C2semidirect
2. Q8×Dic542C2semidirect
3. #31774605C2semidirect
4. #31774566C2semidirect
5. #31774558C2semidirect
6. #31774551C2semidirect
7. #31774547C2semidirect
8. #31774520C2semidirect
9. #31774516C2semidirect
10. #31774442C2semidirect
11. #31774427C2semidirect
12. #31774380C2semidirect
13. #31774379C2semidirect
14. C2×Q8×Dic271C2semidirect
15. #548×C271C2semidirect
142. Q8×C271#41semidirect
180. C271#3522semidirect
187. #548D271semidirect
307. Q8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317746292
Q8×Dic5422
#317746052
#317745662
#317745582
#317745512
#317745472
#317745202
#317745162
#317744422
#317744272
#317743802
#317743792
C2×Q8×Dic2712
#548×C2712
#3522271