Finite Group Databasebeta

← #46850443

Subgroups — #46850443

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#31774630C213yesmaximal, complemented
2Q8×Dic542C211yesmaximal, characteristic, complemented
3#31774602C213yesmaximal, complemented
4#31774566C211yesmaximal, characteristic, complemented
5#31774559C211yesmaximal, characteristic, complemented
6#31774554C211yesmaximal, characteristic, complemented
7#31774552C213yesmaximal, complemented
8#31774548C213yesmaximal, complemented
9#31774518C213yesmaximal, complemented
10#31774512C213yesmaximal, complemented
11#31774444C213yesmaximal, complemented
12#31774425C213yesmaximal, complemented
13#31774379C211yesmaximal, characteristic, complemented
14C4×Q8×D271C211yesmaximal, characteristic, complemented
15#520×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2216yes—
18C2×Q8×D271C2211yescharacteristic
19#22576160C2213yes—
20#22576159C2213yes—
21#22576159C2213yes—
22#22576156C2213yes—
23#22576155C2213yes—
24#22576153C2213yes—
25#22576150C2213yes—
26#22576148C2213yes—
27#22576147C2213yes—
28#22576146C2216yes—
29#22576146C2213yes—
30#22576145C2213yes—
31#22576143C2216yes—
32#22576142C2216yes—
33#22576142C2213yes—
34#22576140C2213yes—
35#22576139C2216yes—
36#22576139C2213yes—
37#22576138C2216yes—
38#22576138C2211yescharacteristic
39#22576135C2213yes—
40#22576134C2213yes—
41#22576133C2213yes—
42#22576131C2213yes—
43#22576131C2213yes—
44Q8×Dic271C2211yescharacteristic
45Q8×Dic271C2211yescharacteristic
46C4×Dic542C2213yes—
47C4×Dic542C2213yes—
48#22576125C2213yes—
49C4×D1084C2213yes—
50C4×D1084C2213yes—
51#22576122C2213yes—
52C4×C22⋊D271C2213yes—
53C4×C22⋊D271C2213yes—
54C4×C22⋊D271C2216yes—
55#22576117C2213yes—
56C4⋊C4×D271C2213yes—
57C4⋊C4×D271C2213yes—
58C42×D271C2213yes—
59C2×#41×C271C2211yescharacteristic
60C4×Q8×C271C2211yescharacteristic
61C4×Q8×C271C2211yescharacteristic
62C4×D4×C271C2213yes—
63C4×D4×C271C2213yes—
64#117×C271C2213yes—
65C2×C4⋊C4×C271C2213yes—
66#16864655—24no—
67#16864654—412no—
68C2×Dic542C2311yescharacteristic
69C2×Dic542C2313yes—
70C2×Dic542C2316yes—
71Q8×D271—24no—
72C2×D1084C2313yes—
73C2×D1084C2313yes—
74C2×C22⋊D271C2313yes—
75C2×C22⋊D271C2313yes—
76C2×C22⋊D271C2316yes—
77C2×C4×D271C2313yes—
78C2×C4×D271C2313yes—
79C2×C4×D271C2311yescharacteristic
80C2×C4×D271C2313yes—
81C2×C4×D271C2316yes—
82C4⋊Dic271C2313yes—
83C4⋊Dic271C2313yes—
84C4⋊Dic271C2313yes—
85C4⋊Dic271C2313yes—
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2316yes—
89Dic271⋊C4C2311yescharacteristic
90Dic271⋊C4C2311yescharacteristic
91Dic271⋊C4C2313yes—
92Dic271⋊C4C2313yes—
93C22⋊Dic271C2313yes—
94C22⋊Dic271C2316yes—
95C22⋊Dic271C2313yes—
96(C2×D271)⋊C4C2313yes—
97(C2×D271)⋊C4C2313yes—
98(C2×D271)⋊C4C2316yes—
99(C2×D271)⋊C4C2311yescharacteristic
100(C2×D271)⋊C4C2311yescharacteristic
101(C2×D271)⋊C4C2313yes—
102(C2×D271)⋊C4C2313yes—
103C4×Dic271C2313yes—
104C4×Dic271C2313yes—
105C4×Dic271C2313yes—
106C4×Dic271C2313yes—
107#41×C271—24no—
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271C2313yes—
110C22×C4×C271C2313yes—
111C22×C4×C271C2316yes—
112C4⋊C4×C271C2313yes—
113C4⋊C4×C271C2313yes—
114C4⋊C4×C271C2311yescharacteristic
115C4⋊C4×C271C2313yes—
116C22⋊C4×C271C2313yes—
117C22⋊C4×C271C2313yes—
118C42×C271C2313yes—
119C42×C271C2313yes—
120C22×D271C2413yes—
121C22×D271C2411yescharacteristic
122Dic542—26no—
123Dic542—22no—
124Dic542—46no—
125D1084—26no—
126D1084—26no—
127C22⋊D271—26no—
128C22⋊D271—26no—
129C22⋊D271—46no—
130C2×Dic271C2411yescharacteristic
131C2×Dic271C2413yes—
132C2×Dic271C2413yes—
133C2×Dic271C2413yes—
134C2×Dic271C2411yescharacteristic
135C2×Dic271C2411yescharacteristic
136C4×D271—26no—
137C4×D271—26no—
138C4×D271—26no—
139C4×D271—22no—
140C4×D271—46no—
141C23×C271C2413yes—
142Q8×C271#4114yescomplemented
143D4×C271—26no—
144C2×C4×C271—26no—
145C2×C4×C271—46no—
146C2×C4×C271C2411yescharacteristic
147C2×C4×C271C2411yescharacteristic
148C2×C4×C271C2413yes—
149C2×C4×C271C2413yes—
150C2×C4×C271C2411yescharacteristic
151C2×C4×C271C2413yes—
152C22×C271—23no—
153C22×C271—23no—
154C22×C271—23no—
155C22×C271C2511yescharacteristic
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—23no—
159C2×D271—23no—
160C2×D271—23no—
161C2×D271H(5,𝔽2)12yes—
162C4×C271—21no—
163C4×C271—23no—
164C4×C271—23no—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271C2×#4116yes—
168Dic271—21no—
169Dic271—23no—
170Dic271—23no—
171Dic271—23no—
172Dic271—21no—
173Dic2712−1+412yes—
174C2×C271—43no—
175C2×C271C22×#4111yescharacteristic
176C2×C271C2×H(5,𝔽2)11yescharacteristic
177C2×C271C2×2−1+411yescharacteristic
178D271—22no—
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#548—2711no—
188#547—2713no—
189#547—2713no—
190#542—2713no—
191#542—2713no—
192#532—2713no—
193#529—2713no—
194#520—2711no—
195#520D27111yescharacteristic, complemented
196C2×C4×Q8—2711no—
197C2×#41—2711no—
198C2×#41—2716no—
199C2×#41C2×D27111yescharacteristic
200C22×Q8—2711no—
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#124—2713no—
212#123—2716no—
213#123—2713no—
214#123—2713no—
215#123—2716no—
216C22⋊Q8—2716no—
217C22⋊Q8—2713no—
218C22⋊Q8—2711no—
219C22⋊Q8—2716no—
220C22⋊Q8—2713no—
221#121—2713no—
222#121—2713no—
223#121—2713no—
224#121—2713no—
225#121—2713no—
226C4×Q8—2711no—
227C4×Q8—2711no—
228C4×Q8—2713no—
229C4×Q8—2713no—
230C4×Q8—2713no—
231C4×Q8C2×D27111yescharacteristic
232C4×Q8C2×D27111yescharacteristic
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4—2716no—
236C4×D4—2713no—
237C4×D4—2713no—
238C4×D4—2713no—
239C4×D4C2×D27113yes—
240C4×D4C2×D27113yes—
241#117—2713no—
242#117C2×D27113yes—
243C2×C4⋊C4—2713no—
244C2×C4⋊C4—2713no—
245C2×C4⋊C4C2×D27113yes—
246C2×C42—2713no—
247#41—24no—
248#41—5424no—
249#41—108412no—
250C2×Q8—2711no—
251C2×Q8—2713no—
252C2×Q8—2716no—
253C2×Q8—5424no—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—2713no—
256C2×D4—2713no—
257C2×D4—2716no—
258C2×D4—2713no—
259C2×D4—2713no—
260C2×D4C22×D27113yes—
261C22×C4—2711no—
262C22×C4—2713no—
263C22×C4—2713no—
264C22×C4—2713no—
265C22×C4—2716no—
266C22×C4C22×D27113yes—
267C22×C4C22×D27116yes—
268C4⋊C4—2713no—
269C4⋊C4—2713no—
270C4⋊C4—2713no—
271C4⋊C4—2716no—
272C4⋊C4—2711no—
273C4⋊C4—2711no—
274C4⋊C4—2713no—
275C4⋊C4—2713no—
276C4⋊C4—2713no—
277C4⋊C4—2713no—
278C4⋊C4—2713no—
279C4⋊C4C22×D27113yes—
280C4⋊C4C22×D27113yes—
281C4⋊C4C22×D27113yes—
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—2713no—
284C22⋊C4—2716no—
285C22⋊C4—2713no—
286C22⋊C4—2713no—
287C22⋊C4—2713no—
288C22⋊C4—2711no—
289C22⋊C4—2711no—
290C22⋊C4—2713no—
291C22⋊C4—2716no—
292C22⋊C4—2713no—
293C22⋊C4C22×D27113yes—
294C22⋊C4C22×D27113yes—
295C42—2713no—
296C42—2713no—
297C42—2713no—
298C42—2713no—
299C42C22×D27113yes—
300C42C22×D27113yes—
301C23—2711no—
302C23—2713no—
303C23C23×D27113yes—
304Q8—5422no—
305Q8—5426no—
306Q8—10846no—
307Q8#1686465414yescomplemented
308D4—26no—
309D4—5426no—
310D4—5426no—
311D4—5426no—
312D4—5426no—
313D4—10846no—
314C2×C4—26no—
315C2×C4—46no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—2713no—
319C2×C4—2713no—
320C2×C4—2713no—
321C2×C4—2711no—
322C2×C4—5422no—
323C2×C4—5426no—
324C2×C4—5426no—
325C2×C4—5426no—
326C2×C4—10846no—
327C2×C4C23×D27111yescharacteristic
328C2×C4C23×D27113yes—
329C2×C4C23×D27111yescharacteristic
330C2×C4C23×D27113yes—
331C2×C4C23×D27111yescharacteristic
332C2×C4C23×D27113yes—
333C22—23no—
334C22—23no—
335C22—23no—
336C22—2712no—
337C22—5421no—
338C22—5421no—
339C22—5423no—
340C22—5423no—
341C22—5423no—
342C22C24×D27111yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—21no—
348C4—2712no—
349C4—5421no—
350C4—5423no—
351C4—5423no—
352C4—5423no—
353C4—5421no—
354C4C2×#1686465416yes—
355C2—43no—
356C2—5422no—
357C2—10843no—
358C2C22×#1686465411yescharacteristic
359C2H(5,𝔽2)×D27111yescharacteristic
360C22−1+4×D27111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#520×C271C2
hypercenter#520D271
hypocenterC271#3522
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #31774630C2semidirect
2. Q8×Dic542C2semidirect
3. #31774602C2semidirect
4. #31774566C2semidirect
5. #31774559C2semidirect
6. #31774554C2semidirect
7. #31774552C2semidirect
8. #31774548C2semidirect
9. #31774518C2semidirect
10. #31774512C2semidirect
11. #31774444C2semidirect
12. #31774425C2semidirect
13. #31774379C2semidirect
14. C4×Q8×D271C2semidirect
15. #520×C271C2semidirect
142. Q8×C271#41semidirect
180. C271#3522semidirect
195. #520D271semidirect
307. Q8#16864654semidirect

Maximal subgroups

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

subgroupindex
#317746302
Q8×Dic5422
#317746022
#317745662
#317745592
#317745542
#317745522
#317745482
#317745182
#317745122
#317744442
#317744252
#317743792
C4×Q8×D2712
#520×C2712
#3522271