Finite Group Databasebeta

← #46849954

Subgroups — #46849954

Order 34688.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774629C213yesmaximal, complemented
2#31774623C211yesmaximal, characteristic, complemented
3#31774618C213yesmaximal, complemented
4#31774599C213yesmaximal, complemented
5#31774593C213yesmaximal, complemented
6#31774566C211yesmaximal, characteristic, complemented
7#31774558C211yesmaximal, characteristic, complemented
8#31774442C213yesmaximal, complemented
9#31774433C213yesmaximal, complemented
10#31774426C213yesmaximal, complemented
11#41×Dic271C211yesmaximal, characteristic, complemented
12C4×#16864655C211yesmaximal, characteristic, complemented
13C4×#16864653C213yesmaximal, complemented
14C2×Q8×Dic271C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2213yes—
18#22576159C2213yes—
19#22576156C2213yes—
20#22576155C2216yes—
21#22576153C2216yes—
22#22576152C2213yes—
23#22576151C2213yes—
24#22576149C2213yes—
25#22576146C2216yes—
26#22576144C2216yes—
27#22576142C2216yes—
28#22576136C2213yes—
29#22576134C2213yes—
30#22576134C2213yes—
31#22576132C2216yes—
32#22576131C2213yes—
33Q8×Dic271C2211yescharacteristic
34Q8×Dic271C2211yescharacteristic
35Q8×Dic271C2213yes—
36Q8×Dic271C2211yescharacteristic
37Q8×Dic271C2213yes—
38C4×Dic542C2213yes—
39#22576125C2213yes—
40C4×D1084C2213yes—
41D4×Dic271C2213yes—
42D4×Dic271C2213yes—
43D4×Dic271C2213yes—
44#22576122C2213yes—
45C4×C22⋊D271C2213yes—
46C4×C22⋊D271C2213yes—
47(C22⋊D271)⋊C4C2213yes—
48(C22⋊D271)⋊C4C2213yes—
49(C22⋊D271)⋊C4C2216yes—
50#22576119C2213yes—
51#22576118C2213yes—
52#22576118C2213yes—
53#22576116C2211yescharacteristic
54#22576116C2213yes—
55C2×C4⋊Dic271C2213yes—
56C2×C4×Dic271C2213yes—
57C2×C4×Dic271C2213yes—
58C42×D271C2213yes—
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—212no—
68C2×Dic542C2313yes—
69C2×D1084C2313yes—
70C2×C22⋊D271C2313yes—
71C2×C22⋊D271C2313yes—
72C2×C22⋊D271C2316yes—
73C22×Dic271C2313yes—
74C22×Dic271C2313yes—
75C22×Dic271C2313yes—
76C22×Dic271C2311yescharacteristic
77C2×C4×D271C2313yes—
78C2×C4×D271C2313yes—
79C2×C4×D271C2313yes—
80C4⋊Dic271C2313yes—
81C4⋊Dic271C2313yes—
82C4⋊Dic271C2316yes—
83C4⋊Dic271C2313yes—
84C4⋊Dic271C2313yes—
85Dic271⋊C4C2316yes—
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2311yescharacteristic
89Dic271⋊C4C2311yescharacteristic
90C22⋊Dic271C2313yes—
91C22⋊Dic271C2316yes—
92C22⋊Dic271C2311yescharacteristic
93(C2×D271)⋊C4C2316yes—
94(C2×D271)⋊C4C2313yes—
95(C2×D271)⋊C4C2316yes—
96(C2×D271)⋊C4C2313yes—
97C4×Dic271C2313yes—
98C4×Dic271C2313yes—
99C4×Dic271C2313yes—
100C4×Dic271C2313yes—
101C4×Dic271C2313yes—
102C4×Dic271C2311yescharacteristic
103C4×Dic271C2313yes—
104C4×Dic271C2311yescharacteristic
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×C271C2313yes—
117C22⋊C4×C271C2316yes—
118C22⋊C4×C271C2311yescharacteristic
119C42×C271C2313yes—
120C22×D271C2413yes—
121Dic542—26no—
122D1084—26no—
123C22⋊D271—26no—
124C22⋊D271—26no—
125C22⋊D271—46no—
126C2×Dic271—26no—
127C2×Dic271—43no—
128C2×Dic271—43no—
129C2×Dic271#4114yescomplemented
130C2×Dic271C2411yescharacteristic
131C2×Dic271C2413yes—
132C2×Dic271C2411yescharacteristic
133C2×Dic271C2413yes—
134C2×Dic271C2413yes—
135C2×Dic271C2411yescharacteristic
136C2×Dic271C2411yescharacteristic
137C4×D271—26no—
138C4×D271—26no—
139C4×D271—43no—
140C23×C271C2413yes—
141C23×C271C2411yescharacteristic
142Q8×C271—26no—
143Q8×C271#4114yescomplemented
144D4×C271—26no—
145D4×C271—26no—
146C2×C4×C271—26no—
147C2×C4×C271—26no—
148C2×C4×C271C2411yescharacteristic
149C2×C4×C271C2413yes—
150C2×C4×C271C2413yes—
151C2×C4×C271C2411yescharacteristic
152C2×C4×C271C2413yes—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—23no—
156C22×C271—23no—
157C22×C271—23no—
158C22×C271C2×#4112yes—
159C22×C271C2511yescharacteristic
160C2×D271—23no—
161C2×D271—23no—
162C2×D271—23no—
163C4×C271—21no—
164C4×C271—23no—
165C4×C271—23no—
166C4×C271—21no—
167C4×C271C2×#4116yes—
168Dic271—21no—
169Dic271—23no—
170Dic271—23no—
171Dic271—21no—
172Dic271—23no—
173Dic271C2×#4112yes—
174Dic271C2×#4112yes—
175C2×C271—22no—
176C2×C271—43no—
177C2×C271C22×#4111yescharacteristic
178C2×C271C22×#4111yescharacteristic
179C2×C271#58511yescharacteristic
180D271—43no—
181C271#348011yescharacteristic, complemented, Sylow
182#3480—2711nomaximal, Sylow
183#559—2713no—
184#557—2711no—
185#556—2713no—
186#553—2713no—
187#553—2713no—
188#550—2711no—
189#548—2711no—
190#548D27111yescharacteristic, complemented
191#532—2713no—
192#529—2713no—
193#529—2713no—
194C4×#41—2711no—
195C4×#41—2711no—
196C4×#41—2713no—
197C2×C4×Q8—2711no—
198C2×#41—2711no—
199C2×#41—2713no—
200C2×#41C2×D27111yescharacteristic
201C22×Q8C2×D27111yescharacteristic
202C4⋊Q8—2713no—
203#127—2713no—
204#126—2716no—
205#126—2716no—
206#125—2713no—
207#125—2713no—
208#124—2713no—
209#124—2716no—
210#124C2×D27113yes—
211#123—2716no—
212#123—2716no—
213C22⋊Q8—2713no—
214C22⋊Q8C2×D27113yes—
215#121—2713no—
216#121—2716no—
217#121—2713no—
218#121—2713no—
219#121C2×D27113yes—
220C4×Q8—2711no—
221C4×Q8—2713no—
222C4×Q8—2711no—
223C4×Q8—2713no—
224C4×Q8—2711no—
225C4×Q8—2713no—
226C4×Q8—2713no—
227C4×Q8C2×D27111yescharacteristic
228C4×D4—2713no—
229C4×D4—2713no—
230C4×D4—2713no—
231C4×D4—2713no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4—2713no—
236C4×D4—2716no—
237C4×D4—2713no—
238C4×D4C2×D27113yes—
239#117—2713no—
240#117—2713no—
241#117—2713no—
242#117—2711no—
243#117—2713no—
244C2×C4⋊C4—2713no—
245C2×C42—2713no—
246C2×C42—2713no—
247C2×C42—2713no—
248#41—24no—
249#41—54212no—
250#41—5424no—
251C2×Q8—24no—
252C2×Q8—2713no—
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—2713no—
264C22×C4—2713no—
265C22×C4—2713no—
266C22×C4—2713no—
267C22×C4—2713no—
268C22×C4C22×D27113yes—
269C22×C4C22×D27113yes—
270C4⋊C4—2713no—
271C4⋊C4—2713no—
272C4⋊C4—2716no—
273C4⋊C4—2711no—
274C4⋊C4—2711no—
275C4⋊C4—2713no—
276C4⋊C4—2713no—
277C4⋊C4—2713no—
278C4⋊C4—2713no—
279C4⋊C4—2716no—
280C4⋊C4C22×D27113yes—
281C4⋊C4C22×D27113yes—
282C22⋊C4—2711no—
283C22⋊C4—2713no—
284C22⋊C4—2716no—
285C22⋊C4—2713no—
286C22⋊C4—2716no—
287C22⋊C4—2713no—
288C22⋊C4—2716no—
289C22⋊C4C22×D27111yescharacteristic
290C22⋊C4C22×D27113yes—
291C22⋊C4C22×D27116yes—
292C42—2711no—
293C42—2711no—
294C42—2713no—
295C42—2713no—
296C42—2713no—
297C42—2713no—
298C42—2713no—
299C42—2713no—
300C42—2713no—
301C42C22×D27113yes—
302C23—2713no—
303C23C23×D27113yes—
304C23C23×D27111yescharacteristic
305Q8—26no—
306Q8—5426no—
307Q8#1686465314yescomplemented
308D4—26no—
309D4—26no—
310D4—5426no—
311D4—5426no—
312D4—5426no—
313D4—10846no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—2714no—
317C2×C4—2711no—
318C2×C4—2711no—
319C2×C4—2711no—
320C2×C4—2711no—
321C2×C4—2713no—
322C2×C4—2713no—
323C2×C4—2713no—
324C2×C4—5426no—
325C2×C4—5426no—
326C2×C4—5426no—
327C2×C4—10843no—
328C2×C4—10843no—
329C2×C4—10843no—
330C2×C4C23×D27111yescharacteristic
331C2×C4C23×D27113yes—
332C2×C4C23×D27111yescharacteristic
333C2×C4C23×D27113yes—
334C2×C4C23×D27113yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—5423no—
341C22—5423no—
342C22—5423no—
343C22#41×D27112yes—
344C22C24×D27111yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—2712no—
350C4—2712no—
351C4—5423no—
352C4—5421no—
353C4—5423no—
354C4—5421no—
355C4—5423no—
356C4C2×#1686465316yes—
357C2—22no—
358C2—43no—
359C2—10843no—
360C2C2×#41×D27111yescharacteristic
361C2C22×#1686465311yescharacteristic
362C2#3177473411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#548×C271C2
hypercenter#548D271
hypocenterC271#3480
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31774629C2semidirect
2. #31774623C2semidirect
3. #31774618C2semidirect
4. #31774599C2semidirect
5. #31774593C2semidirect
6. #31774566C2semidirect
7. #31774558C2semidirect
8. #31774442C2semidirect
9. #31774433C2semidirect
10. #31774426C2semidirect
11. #41×Dic271C2semidirect
12. C4×#16864655C2semidirect
13. C4×#16864653C2semidirect
14. C2×Q8×Dic271C2semidirect
15. #548×C271C2semidirect
129. C2×Dic271#41semidirect
143. Q8×C271#41semidirect
181. C271#3480semidirect
190. #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
#317746232
#317746182
#317745992
#317745932
#317745662
#317745582
#317744422
#317744332
#317744262
#41×Dic2712
C4×#168646552
C4×#168646532
C2×Q8×Dic2712
#548×C2712
#3480271