Finite Group Databasebeta

← #46850476

Subgroups — #46850476

Order 34688.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774632C213yesmaximal, complemented
2#31774613C213yesmaximal, complemented
3#31774600C213yesmaximal, complemented
4#31774585C213yesmaximal, complemented
5#31774569C211yesmaximal, characteristic, complemented
6#31774559C211yesmaximal, characteristic, complemented
7#31774559C211yesmaximal, characteristic, complemented
8#31774521C213yesmaximal, complemented
9#31774503C213yesmaximal, complemented
10#31774434C213yesmaximal, complemented
11#31774432C213yesmaximal, complemented
12#31774379C211yesmaximal, characteristic, complemented
13#31774373C211yesmaximal, characteristic, complemented
14C4×#16864655C211yesmaximal, characteristic, complemented
15#557×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576157C2213yes—
20#22576155C2216yes—
21#22576155C2213yes—
22#22576155C2211yescharacteristic
23#22576151C2213yes—
24#22576149C2216yes—
25#22576149C2213yes—
26#22576148C2213yes—
27#22576148C2213yes—
28#22576145C2216yes—
29#22576145C2216yes—
30#22576145C2213yes—
31#22576145C2213yes—
32#22576140C2213yes—
33#22576140C2213yes—
34#22576136C2213yes—
35#22576135C2213yes—
36#22576135C2213yes—
37#22576135C2213yes—
38#22576135C2216yes—
39#22576135C2213yes—
40#22576135C2213yes—
41Q8×Dic271C2211yescharacteristic
42C4×Dic542C2213yes—
43#22576125C2213yes—
44C4×D1084C2213yes—
45C4×D1084C2213yes—
46C4×D1084C2213yes—
47C4×D1084C2213yes—
48C4×D1084C2213yes—
49#22576122C2213yes—
50#22576122C2213yes—
51#22576122C2216yes—
52#22576118C2213yes—
53#22576118C2216yes—
54#22576118C2213yes—
55#22576117C2213yes—
56#22576117C2213yes—
57C4⋊C4×D271C2213yes—
58C42×D271C2213yes—
59C4⋊Q8×C271C2213yes—
60#125×C271C2213yes—
61#125×C271C2213yes—
62C4×Q8×C271C2211yescharacteristic
63C4×Q8×C271C2211yescharacteristic
64C4×Q8×C271C2211yescharacteristic
65C4×Q8×C271C2213yes—
66#16864655—24no—
67#16864655—24no—
68C2×Dic542C2313yes—
69Q8×D271—24no—
70C2×D1084C2313yes—
71C2×D1084C2313yes—
72C2×D1084C2313yes—
73C2×D1084C2313yes—
74C2×D1084C2316yes—
75C2×D1084C2313yes—
76C2×C4×D271C2313yes—
77C2×C4×D271C2313yes—
78C2×C4×D271C2313yes—
79C2×C4×D271C2313yes—
80C2×C4×D271C2316yes—
81C2×C4×D271C2313yes—
82C4⋊Dic271C2313yes—
83C4⋊Dic271C2313yes—
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2313yes—
86Dic271⋊C4C2311yescharacteristic
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2313yes—
89Dic271⋊C4C2313yes—
90(C2×D271)⋊C4C2311yescharacteristic
91(C2×D271)⋊C4C2313yes—
92(C2×D271)⋊C4C2313yes—
93(C2×D271)⋊C4C2313yes—
94(C2×D271)⋊C4C2316yes—
95(C2×D271)⋊C4C2316yes—
96(C2×D271)⋊C4C2313yes—
97(C2×D271)⋊C4C2316yes—
98(C2×D271)⋊C4C2313yes—
99(C2×D271)⋊C4C2311yescharacteristic
100(C2×D271)⋊C4C2313yes—
101(C2×D271)⋊C4C2313yes—
102(C2×D271)⋊C4C2311yescharacteristic
103C4×Dic271C2313yes—
104C4×Dic271C2316yes—
105C4×Dic271C2311yescharacteristic
106C4×Dic271C2313yes—
107C2×Q8×C271C2313yes—
108C2×Q8×C271C2311yescharacteristic
109C4⋊C4×C271C2313yes—
110C4⋊C4×C271C2313yes—
111C4⋊C4×C271C2313yes—
112C4⋊C4×C271C2313yes—
113C4⋊C4×C271C2316yes—
114C4⋊C4×C271C2311yescharacteristic
115C4⋊C4×C271C2313yes—
116C42×C271C2313yes—
117C42×C271C2313yes—
118C42×C271C2313yes—
119C22×D271C2411yescharacteristic
120C22×D271C2413yes—
121C22×D271C2413yes—
122Dic542—26no—
123D1084—26no—
124D1084—26no—
125D1084—43no—
126D1084—46no—
127D1084—43no—
128D1084—43no—
129C2×Dic271C2411yescharacteristic
130C2×Dic271C2413yes—
131C2×Dic271C2411yescharacteristic
132C2×Dic271C2413yes—
133C2×Dic271C2411yescharacteristic
134C4×D271—26no—
135C4×D271—26no—
136C4×D271—26no—
137C4×D271—43no—
138C4×D271—43no—
139C4×D271—46no—
140Q8×C271—43no—
141Q8×C271#4114yes—
142C2×C4×C271C2411yescharacteristic
143C2×C4×C271C2411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C2×C4×C271C2413yes—
146C2×C4×C271C2413yes—
147C2×C4×C271C2413yes—
148C2×C4×C271C2413yes—
149C22×C271C2511yescharacteristic
150C2×D271—21no—
151C2×D271—23no—
152C2×D271—23no—
153C2×D271—23no—
154C2×D271—23no—
155C2×D271—23no—
156C2×D271—21no—
157C2×D271—23no—
158C2×D271—21no—
159C4×C271—21no—
160C4×C271—23no—
161C4×C271—23no—
162C4×C271—21no—
163C4×C271—23no—
164C4×C271—21no—
165C4×C271C2×#4116yes—
166Dic271—21no—
167Dic271—21no—
168Dic271—23no—
169Dic271—21no—
170Dic271—23no—
171C2×C271C22×#4111yescharacteristic
172C2×C271#58511yescharacteristic
173C2×C271#58511yescharacteristic
174D271—41no—
175D271—43no—
176D271—43no—
177C271#352411yescharacteristic, complemented, Sylow
178#3524—2711nomaximal, Sylow
179#559—2713no—
180#557D27111yescharacteristic, complemented
181#555—2713no—
182#553—2713no—
183#552—2713no—
184#550—2711no—
185#548—2711no—
186#548—2711no—
187#542—2713no—
188#540—2713no—
189#529—2713no—
190#529—2713no—
191#520—2711no—
192#519—2711no—
193C4×#41—2711no—
194C2×#41—2711no—
195C2×#41—2711no—
196C22×Q8—2711no—
197C4⋊Q8C2×D27113yes—
198#127—2713no—
199#126—2711no—
200#126—2713no—
201#126—2716no—
202#125—2713no—
203#125C2×D27113yes—
204#125C2×D27113yes—
205#124—2713no—
206#124—2716no—
207#124—2713no—
208#124—2713no—
209#123—2716no—
210#123—2713no—
211#123—2713no—
212#123—2716no—
213C22⋊Q8—2713no—
214C22⋊Q8—2713no—
215C22⋊Q8—2713no—
216#121—2713no—
217#121—2713no—
218#121—2713no—
219#121—2713no—
220#121—2716no—
221#121—2713no—
222C4×Q8—2711no—
223C4×Q8—2713no—
224C4×Q8—2713no—
225C4×Q8C2×D27111yescharacteristic
226C4×Q8C2×D27111yescharacteristic
227C4×Q8C2×D27111yescharacteristic
228C4×Q8C2×D27113yes—
229C4×D4—2713no—
230C4×D4—2713no—
231C4×D4—2716no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4—2713no—
236C4×D4—2713no—
237#117—2713no—
238#117—2713no—
239#117—2716no—
240#117—2713no—
241#117—2713no—
242C2×C4⋊C4—2713no—
243C2×C42—2713no—
244#41—5424no—
245#41—5424no—
246C2×Q8—2713no—
247C2×Q8—5424no—
248C2×Q8C22×D27113yes—
249C2×Q8C22×D27111yescharacteristic
250C2×D4—2713no—
251C2×D4—2713no—
252C2×D4—2713no—
253C2×D4—2713no—
254C2×D4—2716no—
255C2×D4—2713no—
256C22×C4—2713no—
257C22×C4—2713no—
258C22×C4—2713no—
259C22×C4—2713no—
260C22×C4—2713no—
261C22×C4—2716no—
262C4⋊C4—2713no—
263C4⋊C4—2713no—
264C4⋊C4—2711no—
265C4⋊C4—2713no—
266C4⋊C4—2713no—
267C4⋊C4—2713no—
268C4⋊C4—2711no—
269C4⋊C4—2713no—
270C4⋊C4C22×D27113yes—
271C4⋊C4C22×D27113yes—
272C4⋊C4C22×D27113yes—
273C4⋊C4C22×D27116yes—
274C4⋊C4C22×D27113yes—
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27113yes—
277C22⋊C4—2711no—
278C22⋊C4—2713no—
279C22⋊C4—2713no—
280C22⋊C4—2713no—
281C22⋊C4—2716no—
282C22⋊C4—2713no—
283C22⋊C4—2716no—
284C22⋊C4—2711no—
285C22⋊C4—2713no—
286C22⋊C4—2711no—
287C22⋊C4—2713no—
288C22⋊C4—2716no—
289C22⋊C4—2713no—
290C42—2711no—
291C42—2713no—
292C42—2713no—
293C42—2716no—
294C42C22×D27113yes—
295C42C22×D27113yes—
296C42C22×D27113yes—
297C23—2711no—
298C23—2713no—
299C23—2713no—
300Q8—43no—
301Q8—5426no—
302Q8#1686465514yes—
303D4—5426no—
304D4—5426no—
305D4—10843no—
306D4—10846no—
307D4—10843no—
308D4—10843no—
309C2×C4—2711no—
310C2×C4—2711no—
311C2×C4—2713no—
312C2×C4—2713no—
313C2×C4—2711no—
314C2×C4—5426no—
315C2×C4—5426no—
316C2×C4—5426no—
317C2×C4—10843no—
318C2×C4—10843no—
319C2×C4—10846no—
320C2×C4C23×D27113yes—
321C2×C4C23×D27111yescharacteristic
322C2×C4C23×D27113yes—
323C2×C4C23×D27111yescharacteristic
324C2×C4C23×D27111yescharacteristic
325C2×C4C23×D27113yes—
326C2×C4C23×D27113yes—
327C22—5421no—
328C22—5423no—
329C22—5423no—
330C22—5423no—
331C22—5421no—
332C22—5423no—
333C22—5423no—
334C22—5423no—
335C22—5421no—
336C22C24×D27111yescharacteristic
337C4—23no—
338C4—23no—
339C4—21no—
340C4—21no—
341C4—23no—
342C4—21no—
343C4—5421no—
344C4—5421no—
345C4—5423no—
346C4—5421no—
347C4—5423no—
348C4C2×#1686465516yes—
349C2—10841no—
350C2—10843no—
351C2—10843no—
352C2C22×#1686465511yescharacteristic
353C2#3177473311yescharacteristic
354C2#3177473411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#557×C271C2
hypercenter#557D271
hypocenterC271#3524
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #31774632C2semidirect
2. #31774613C2semidirect
3. #31774600C2semidirect
4. #31774585C2semidirect
5. #31774569C2semidirect
6. #31774559C2semidirect
7. #31774559C2semidirect
8. #31774521C2semidirect
9. #31774503C2semidirect
10. #31774434C2semidirect
11. #31774432C2semidirect
12. #31774379C2semidirect
13. #31774373C2semidirect
14. C4×#16864655C2semidirect
15. #557×C271C2semidirect
177. C271#3524semidirect
180. #557D271semidirect

Maximal subgroups

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

subgroupindex
#317746322
#317746132
#317746002
#317745852
#317745692
#317745592
#317745212
#317745032
#317744342
#317744322
#317743792
#317743732
C4×#168646552
#557×C2712
#3524271