Finite Group Databasebeta

← #46849981

Subgroups — #46849981

Order 34688.

1094 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#31774623C211yesmaximal, characteristic, complemented
2#31774619C213yesmaximal, complemented
3#31774588C213yesmaximal, complemented
4#31774582C213yesmaximal, complemented
5Q8×C22⋊D271C211yesmaximal, characteristic, complemented
6#31774559C211yesmaximal, characteristic, complemented
7#31774558C211yesmaximal, characteristic, complemented
8#31774548C213yesmaximal, complemented
9#31774451C213yesmaximal, complemented
10#31774434C213yesmaximal, complemented
11#31774431C213yesmaximal, complemented
12C4×#16864655C211yesmaximal, characteristic, complemented
13C4×#16864654C213yesmaximal, complemented
14C4×Q8×D271C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864654C2213yes—
18C2×Q8×D271C2211yescharacteristic
19#22576159C2213yes—
20#22576158C2213yes—
21#22576155C2216yes—
22#22576154C2216yes—
23#22576152C2213yes—
24#22576151C2213yes—
25#22576150C2216yes—
26#22576149C2213yes—
27#22576148C2213yes—
28#22576145C2216yes—
29#22576143C2216yes—
30#22576140C2213yes—
31#22576139C2213yes—
32#22576138C2216yes—
33#22576136C2213yes—
34#22576136C2213yes—
35#22576135C2213yes—
36#22576134C2213yes—
37Q8×Dic271C2211yescharacteristic
38Q8×Dic271C2211yescharacteristic
39C4×Dic542C2213yes—
40C4×Dic542C2213yes—
41C4×Dic542C2213yes—
42#22576125C2213yes—
43C4×D1084C2213yes—
44C4×D1084C2213yes—
45C4×D1084C2213yes—
46#22576122C2213yes—
47C4×C22⋊D271C2213yes—
48C4×C22⋊D271C2213yes—
49C4×C22⋊D271C2211yescharacteristic
50C4×C22⋊D271C2213yes—
51#22576119C2213yes—
52#22576118C2213yes—
53#22576117C2213yes—
54#22576117C2213yes—
55#22576117C2216yes—
56C4⋊C4×D271C2213yes—
57C42×D271C2213yes—
58C42×D271C2213yes—
59C22×Q8×C271C2211yescharacteristic
60C4×Q8×C271C2211yescharacteristic
61C4×Q8×C271C2213yes—
62C4×Q8×C271C2211yescharacteristic
63C4×Q8×C271C2213yes—
64C2×C4⋊C4×C271C2213yes—
65C2×C42×C271C2213yes—
66#16864655—24no—
67#16864654—212no—
68C2×Dic542C2313yes—
69C2×Dic542C2313yes—
70Q8×D271—24no—
71C2×D1084C2313yes—
72C2×D1084C2313yes—
73C2×C22⋊D271C2313yes—
74C2×C22⋊D271C2311yescharacteristic
75C2×C4×D271C2313yes—
76C2×C4×D271C2313yes—
77C2×C4×D271C2313yes—
78C2×C4×D271C2313yes—
79C2×C4×D271C2311yescharacteristic
80C2×C4×D271C2313yes—
81C4⋊Dic271C2313yes—
82C4⋊Dic271C2313yes—
83C4⋊Dic271C2313yes—
84C4⋊Dic271C2313yes—
85Dic271⋊C4C2313yes—
86Dic271⋊C4C2316yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2316yes—
89Dic271⋊C4C2316yes—
90Dic271⋊C4C2313yes—
91Dic271⋊C4C2311yescharacteristic
92C22⋊Dic271C2313yes—
93C22⋊Dic271C2311yescharacteristic
94(C2×D271)⋊C4C2316yes—
95(C2×D271)⋊C4C2316yes—
96(C2×D271)⋊C4C2313yes—
97(C2×D271)⋊C4C2316yes—
98(C2×D271)⋊C4C2313yes—
99(C2×D271)⋊C4C2313yes—
100(C2×D271)⋊C4C2311yescharacteristic
101C4×Dic271C2313yes—
102C4×Dic271C2313yes—
103C4×Dic271C2313yes—
104C4×Dic271C2313yes—
105C4×Dic271C2311yescharacteristic
106C4×Dic271C2313yes—
107C2×Q8×C271—24no—
108C2×Q8×C271C2313yes—
109C2×Q8×C271C2311yescharacteristic
110C22×C4×C271C2313yes—
111C22×C4×C271C2313yes—
112C22×C4×C271C2311yescharacteristic
113C4⋊C4×C271C2313yes—
114C4⋊C4×C271C2316yes—
115C4⋊C4×C271C2313yes—
116C42×C271C2313yes—
117C42×C271C2313yes—
118C42×C271C2313yes—
119C42×C271C2313yes—
120C22×D271C2413yes—
121C22×D271C2411yescharacteristic
122Dic542—26no—
123Dic542—26no—
124D1084—26no—
125D1084—26no—
126C22⋊D271—26no—
127C22⋊D271#4114yescomplemented
128C2×Dic271C2411yescharacteristic
129C2×Dic271C2411yescharacteristic
130C2×Dic271C2413yes—
131C2×Dic271C2413yes—
132C2×Dic271C2413yes—
133C2×Dic271C2411yescharacteristic
134C4×D271—26no—
135C4×D271—26no—
136C4×D271—26no—
137C4×D271—22no—
138C4×D271—26no—
139C4×D271—26no—
140C23×C271C2411yescharacteristic
141Q8×C271—26no—
142Q8×C271#4114yescomplemented
143C2×C4×C271—26no—
144C2×C4×C271—26no—
145C2×C4×C271—22no—
146C2×C4×C271C2411yescharacteristic
147C2×C4×C271C2413yes—
148C2×C4×C271C2413yes—
149C2×C4×C271C2413yes—
150C2×C4×C271C2413yes—
151C2×C4×C271C2411yescharacteristic
152C22×C271—21no—
153C22×C271—21no—
154C22×C271C2×#4112yes—
155C22×C271C2511yescharacteristic
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—23no—
159C2×D271—23no—
160C2×D271—23no—
161C2×D271C2×#4112yes—
162C4×C271—23no—
163C4×C271—23no—
164C4×C271—23no—
165C4×C271—21no—
166C4×C271C2×#4116yes—
167C4×C2712−1+412yes—
168Dic271—21no—
169Dic271—23no—
170Dic271—23no—
171Dic271—21no—
172Dic271—23no—
173Dic271C2×#4112yes—
174C2×C271—22no—
175C2×C271C22×#4111yescharacteristic
176C2×C271C22×#4111yescharacteristic
177C2×C271C2×2−1+411yescharacteristic
178D271—22no—
179D271—43no—
180C271#348211yescharacteristic, complemented, Sylow
181#3482—2711nomaximal, Sylow
182#557—2711no—
183#556—2713no—
184#552—2713no—
185#552—2713no—
186D4×Q8—2711no—
187#548—2711no—
188#548—2711no—
189#547—2713no—
190#533—2713no—
191#529—2713no—
192#529—2713no—
193C4×#41—2711no—
194C4×#41—2713no—
195C2×C4×Q8—2711no—
196C2×C4×Q8D27111yescharacteristic, complemented
197C2×#41—2711no—
198C2×#41—2713no—
199C22×Q8—2711no—
200C22×Q8C2×D27111yescharacteristic
201C4⋊Q8—2713no—
202C4⋊Q8—2713no—
203#126—2716no—
204#126—2716no—
205#125—2713no—
206#125—2716no—
207#125—2713no—
208#124—2713no—
209#124—2713no—
210#123—2716no—
211#123—2716no—
212C22⋊Q8—2713no—
213C22⋊Q8—2713no—
214C22⋊Q8—2713no—
215C22⋊Q8—2716no—
216C22⋊Q8—2713no—
217#121—2713no—
218#121—2713no—
219C4×Q8—2711no—
220C4×Q8—2711no—
221C4×Q8—2713no—
222C4×Q8—2713no—
223C4×Q8—2713no—
224C4×Q8—2713no—
225C4×Q8C2×D27111yescharacteristic
226C4×Q8C2×D27113yes—
227C4×Q8C2×D27111yescharacteristic
228C4×Q8C2×D27113yes—
229C4×D4—2711no—
230C4×D4—2713no—
231C4×D4—2713no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4—2713no—
236C4×D4—2713no—
237#117—2713no—
238#117—2713no—
239#117—2713no—
240#117—2713no—
241#117—2716no—
242C2×C4⋊C4—2713no—
243C2×C4⋊C4C2×D27113yes—
244C2×C42—2713no—
245C2×C42—2713no—
246C2×C42C2×D27113yes—
247#41—54212no—
248#41—5424no—
249C2×Q8—24no—
250C2×Q8—2713no—
251C2×Q8—2713no—
252C2×Q8—5424no—
253C2×Q8C22×D27113yes—
254C2×Q8C22×D27111yescharacteristic
255C2×D4—2711no—
256C2×D4—2713no—
257C2×D4—2713no—
258C2×D4—2713no—
259C22×C4—2713no—
260C22×C4—2713no—
261C22×C4—2713no—
262C22×C4—2713no—
263C22×C4—2711no—
264C22×C4—2713no—
265C22×C4C22×D27113yes—
266C22×C4C22×D27113yes—
267C22×C4C22×D27111yescharacteristic
268C4⋊C4—2713no—
269C4⋊C4—2716no—
270C4⋊C4—2711no—
271C4⋊C4—2713no—
272C4⋊C4—2713no—
273C4⋊C4—2713no—
274C4⋊C4—2713no—
275C4⋊C4—2713no—
276C4⋊C4—2713no—
277C4⋊C4—2716no—
278C4⋊C4—2716no—
279C4⋊C4C22×D27113yes—
280C4⋊C4C22×D27113yes—
281C4⋊C4C22×D27116yes—
282C22⋊C4—2711no—
283C22⋊C4—2713no—
284C22⋊C4—2711no—
285C22⋊C4—2713no—
286C22⋊C4—2716no—
287C22⋊C4—2713no—
288C22⋊C4—2713no—
289C22⋊C4—2716no—
290C22⋊C4—2716no—
291C42—2713no—
292C42—2713no—
293C42—2713no—
294C42—2713no—
295C42—2711no—
296C42—2713no—
297C42C22×D27113yes—
298C42C22×D27113yes—
299C42C22×D27113yes—
300C42C22×D27113yes—
301C23—2711no—
302C23—2713no—
303C23C23×D27111yescharacteristic
304Q8—26no—
305Q8—5426no—
306Q8—5426no—
307Q8#1686465414yescomplemented
308D4—2714no—
309D4—5426no—
310D4—5426no—
311D4—5426no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—2711no—
316C2×C4—2711no—
317C2×C4—2711no—
318C2×C4—2713no—
319C2×C4—2713no—
320C2×C4—2713no—
321C2×C4—5426no—
322C2×C4—5426no—
323C2×C4—5426no—
324C2×C4—5426no—
325C2×C4—5422no—
326C2×C4—5426no—
327C2×C4C23×D27113yes—
328C2×C4C23×D27111yescharacteristic
329C2×C4C23×D27113yes—
330C2×C4C23×D27113yes—
331C2×C4C23×D27113yes—
332C2×C4C23×D27111yescharacteristic
333C22—21no—
334C22—21no—
335C22—2712no—
336C22—5421no—
337C22—5421no—
338C22—5423no—
339C22—5423no—
340C22—5423no—
341C22#41×D27112yes—
342C22C24×D27111yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—2712no—
348C4—5423no—
349C4—5421no—
350C4—5423no—
351C4—5421no—
352C4—5423no—
353C4C2×#1686465416yes—
354C4#2257620812yes—
355C2—22no—
356C2—5422no—
357C2—10843no—
358C2C2×#41×D27111yescharacteristic
359C2C22×#1686465411yescharacteristic
360C2C2×#2257620811yescharacteristic
subgroupG/H
centerC2×C4C23×D271
commutator, socle, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC2×C4×Q8×C271C2
hypercenterC2×C4×Q8D271
hypocenterC271#3482
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #31774623C2semidirect
2. #31774619C2semidirect
3. #31774588C2semidirect
4. #31774582C2semidirect
5. Q8×C22⋊D271C2semidirect
6. #31774559C2semidirect
7. #31774558C2semidirect
8. #31774548C2semidirect
9. #31774451C2semidirect
10. #31774434C2semidirect
11. #31774431C2semidirect
12. C4×#16864655C2semidirect
13. C4×#16864654C2semidirect
14. C4×Q8×D271C2semidirect
15. C2×C4×Q8×C271C2semidirect
127. C22⋊D271#41semidirect
142. Q8×C271#41semidirect
180. C271#3482semidirect
196. C2×C4×Q8D271semidirect
307. Q8#16864654semidirect

Maximal subgroups

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

subgroupindex
#317746232
#317746192
#317745882
#317745822
Q8×C22⋊D2712
#317745592
#317745582
#317745482
#317744512
#317744342
#317744312
C4×#168646552
C4×#168646542
C4×Q8×D2712
C2×C4×Q8×C2712
#3482271