Finite Group Databasebeta

← #46850375

Subgroups — #46850375

Order 34688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic542C211yesmaximal, characteristic, complemented
2#31774620C213yesmaximal, complemented
3#31774619C213yesmaximal, complemented
4#31774615C213yesmaximal, complemented
5#31774610C213yesmaximal, complemented
6Q8×C22⋊D271C211yesmaximal, characteristic, complemented
7#31774559C211yesmaximal, characteristic, complemented
8#31774558C211yesmaximal, characteristic, complemented
9#31774519C213yesmaximal, complemented
10#31774518C213yesmaximal, complemented
11#31774451C213yesmaximal, complemented
12#31774425C213yesmaximal, complemented
13#31774373C211yesmaximal, characteristic, complemented
14C4×#16864655C211yesmaximal, characteristic, complemented
15#519×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18#22576160C2213yes—
19#22576159C2213yes—
20#22576159C2213yes—
21#22576158C2213yes—
22#22576155C2216yes—
23#22576155C2213yes—
24#22576154C2216yes—
25#22576153C2213yes—
26#22576150C2213yes—
27#22576149C2213yes—
28#22576148C2213yes—
29#22576148C2216yes—
30#22576146C2216yes—
31#22576146C2211yescharacteristic
32#22576145C2213yes—
33#22576142C2213yes—
34#22576140C2213yes—
35#22576139C2213yes—
36#22576138C2216yes—
37#22576138C2213yes—
38#22576136C2213yes—
39#22576136C2213yes—
40#22576135C2213yes—
41#22576134C2213yes—
42#22576133C2213yes—
43Q8×Dic271C2211yescharacteristic
44Q8×Dic271C2211yescharacteristic
45C4×Dic542C2213yes—
46C4×Dic542C2216yes—
47C4×Dic542C2213yes—
48#22576125C2213yes—
49C4×D1084C2213yes—
50C4×D1084C2213yes—
51#22576122C2213yes—
52C4×C22⋊D271C2213yes—
53C4×C22⋊D271C2213yes—
54#22576118C2213yes—
55#22576118C2213yes—
56#22576117C2213yes—
57#22576117C2216yes—
58C42×D271C2213yes—
59C22×Q8×C271C2211yescharacteristic
60C4×Q8×C271C2213yes—
61C4×Q8×C271C2211yescharacteristic
62C4×Q8×C271C2213yes—
63C4×Q8×C271C2211yescharacteristic
64#117×C271C2213yes—
65#117×C271C2213yes—
66#16864655—24no—
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2313yes—
69C2×Dic542C2316yes—
70Q8×D271—24no—
71C2×D1084C2313yes—
72C2×D1084C2313yes—
73C2×C22⋊D271C2311yescharacteristic
74C2×C22⋊D271C2313yes—
75C2×C4×D271C2313yes—
76C2×C4×D271C2313yes—
77C2×C4×D271C2313yes—
78C2×C4×D271C2313yes—
79C4⋊Dic271C2313yes—
80C4⋊Dic271C2313yes—
81C4⋊Dic271C2313yes—
82C4⋊Dic271C2313yes—
83Dic271⋊C4C2313yes—
84Dic271⋊C4C2313yes—
85Dic271⋊C4C2316yes—
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88Dic271⋊C4C2313yes—
89Dic271⋊C4C2313yes—
90C22⋊Dic271C2313yes—
91C22⋊Dic271C2311yescharacteristic
92(C2×D271)⋊C4C2313yes—
93(C2×D271)⋊C4C2316yes—
94(C2×D271)⋊C4C2316yes—
95(C2×D271)⋊C4C2313yes—
96(C2×D271)⋊C4C2313yes—
97(C2×D271)⋊C4C2313yes—
98(C2×D271)⋊C4C2311yescharacteristic
99(C2×D271)⋊C4C2316yes—
100(C2×D271)⋊C4C2311yescharacteristic
101C4×Dic271C2311yescharacteristic
102C4×Dic271C2316yes—
103C4×Dic271C2313yes—
104C4×Dic271C2313yes—
105C4×Dic271C2313yes—
106C2×Q8×C271—24no—
107C2×Q8×C271C2313yes—
108C2×Q8×C271C2311yescharacteristic
109C22×C4×C271C2313yes—
110C4⋊C4×C271C2313yes—
111C4⋊C4×C271C2313yes—
112C4⋊C4×C271C2313yes—
113C4⋊C4×C271C2313yes—
114C22⋊C4×C271C2313yes—
115C22⋊C4×C271C2311yescharacteristic
116C42×C271C2316yes—
117C42×C271C2313yes—
118C42×C271C2313yes—
119C22×D271C2411yescharacteristic
120C22×D271C2413yes—
121Dic542—26no—
122Dic542—22no—
123Dic542—46no—
124D1084—26no—
125D1084—43no—
126C22⋊D271—22no—
127C22⋊D271—43no—
128C2×Dic271C2411yescharacteristic
129C2×Dic271C2411yescharacteristic
130C2×Dic271C2413yes—
131C2×Dic271C2413yes—
132C2×Dic271C2413yes—
133C2×Dic271C2411yescharacteristic
134C4×D271—26no—
135C4×D271—26no—
136C4×D271—26no—
137C4×D271—43no—
138C23×C271C2411yescharacteristic
139Q8×C271—26no—
140Q8×C271#4114yes—
141C2×C4×C271—26no—
142C2×C4×C271C2413yes—
143C2×C4×C271C2411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C2×C4×C271C2413yes—
146C2×C4×C271C2413yes—
147C2×C4×C271C2413yes—
148C22×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271C2511yescharacteristic
152C2×D271—21no—
153C2×D271—23no—
154C2×D271—23no—
155C2×D271—21no—
156C2×D271—23no—
157C2×D271—21no—
158C4×C271—23no—
159C4×C271—23no—
160C4×C271—21no—
161C4×C271—21no—
162C4×C271—23no—
163C4×C271C2×#4116yes—
164Dic271—21no—
165Dic271—23no—
166Dic271—21no—
167Dic271—23no—
168Dic271—23no—
169Dic2712−1+412yes—
170C2×C271—41no—
171C2×C271C22×#4111yescharacteristic
172C2×C271C2×2−1+411yescharacteristic
173C2×C271#58511yescharacteristic
174D271—41no—
175D271—43no—
176C271#351811yescharacteristic, complemented, Sylow
177#3518—2711nomaximal, Sylow
178Q82—2711no—
179#556—2713no—
180#556—2713no—
181#556—2713no—
182#555—2713no—
183D4×Q8—2711no—
184#548—2711no—
185#548—2711no—
186#542—2713no—
187#542—2713no—
188#533—2713no—
189#529—2713no—
190#519—2711no—
191#519D27111yescharacteristic, complemented
192C4×#41—2711no—
193C2×#41—2711no—
194C22×Q8—2711no—
195C22×Q8C2×D27111yescharacteristic
196C4⋊Q8—2713no—
197C4⋊Q8—2713no—
198C4⋊Q8—2713no—
199C4⋊Q8—2713no—
200#126—2713no—
201#126—2716no—
202#126—2713no—
203#126—2716no—
204#125—2713no—
205#124—2713no—
206#124—2711no—
207#124—2716no—
208#124—2713no—
209#124—2716no—
210#123—2713no—
211#123—2713no—
212C22⋊Q8—2713no—
213C22⋊Q8—2713no—
214C22⋊Q8—2713no—
215C22⋊Q8—2713no—
216C22⋊Q8—2716no—
217C22⋊Q8—2713no—
218#121—2713no—
219#121—2713no—
220#121—2713no—
221C4×Q8—2711no—
222C4×Q8—2711no—
223C4×Q8—2713no—
224C4×Q8—2713no—
225C4×Q8—2713no—
226C4×Q8—2716no—
227C4×Q8C2×D27111yescharacteristic
228C4×Q8C2×D27113yes—
229C4×Q8C2×D27111yescharacteristic
230C4×Q8C2×D27113yes—
231C4×D4—2713no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4—2713no—
236#117—2713no—
237#117—2713no—
238#117—2713no—
239#117—2716no—
240#117C2×D27113yes—
241#117C2×D27113yes—
242C2×C42—2713no—
243#41—5424no—
244C2×Q8—24no—
245C2×Q8—2711no—
246C2×Q8—2713no—
247C2×Q8—2716no—
248C2×Q8—5424no—
249C2×Q8C22×D27113yes—
250C2×Q8C22×D27111yescharacteristic
251C2×D4—2711no—
252C2×D4—2713no—
253C2×D4—2713no—
254C2×D4—2713no—
255C22×C4—2713no—
256C22×C4—2713no—
257C22×C4—2713no—
258C22×C4—2713no—
259C22×C4C22×D27113yes—
260C4⋊C4—2713no—
261C4⋊C4—2713no—
262C4⋊C4—2713no—
263C4⋊C4—2713no—
264C4⋊C4—2713no—
265C4⋊C4—2713no—
266C4⋊C4—2713no—
267C4⋊C4—2713no—
268C4⋊C4—2716no—
269C4⋊C4—2713no—
270C4⋊C4—2713no—
271C4⋊C4C22×D27113yes—
272C4⋊C4C22×D27113yes—
273C4⋊C4C22×D27113yes—
274C4⋊C4C22×D27113yes—
275C22⋊C4—2711no—
276C22⋊C4—2713no—
277C22⋊C4—2713no—
278C22⋊C4—2711no—
279C22⋊C4—2716no—
280C22⋊C4—2716no—
281C22⋊C4—2716no—
282C22⋊C4—2711no—
283C22⋊C4—2713no—
284C22⋊C4—2713no—
285C22⋊C4—2713no—
286C22⋊C4C22×D27113yes—
287C22⋊C4C22×D27111yescharacteristic
288C42—2711no—
289C42—2713no—
290C42—2713no—
291C42—2716no—
292C42—2713no—
293C42C22×D27113yes—
294C42C22×D27116yes—
295C42C22×D27113yes—
296C23—2711no—
297C23—2713no—
298C23C23×D27111yescharacteristic
299Q8—26no—
300Q8—5422no—
301Q8—5426no—
302Q8—10846no—
303Q8#1686465414yes—
304D4—5422no—
305D4—5426no—
306D4—10843no—
307D4—10843no—
308C2×C4—26no—
309C2×C4—2711no—
310C2×C4—2711no—
311C2×C4—2711no—
312C2×C4—2713no—
313C2×C4—2713no—
314C2×C4—2713no—
315C2×C4—5426no—
316C2×C4—5426no—
317C2×C4—5426no—
318C2×C4—10843no—
319C2×C4C23×D27113yes—
320C2×C4C23×D27111yescharacteristic
321C2×C4C23×D27111yescharacteristic
322C2×C4C23×D27113yes—
323C2×C4C23×D27113yes—
324C2×C4C23×D27113yes—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—5421no—
329C22—5423no—
330C22—5423no—
331C22—5421no—
332C22—5423no—
333C22—5421no—
334C22C24×D27111yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—2712no—
341C4—5421no—
342C4—5423no—
343C4—5421no—
344C4—5423no—
345C4—5423no—
346C4C2×#1686465416yes—
347C2—41no—
348C2—10841no—
349C2—10843no—
350C2C22×#1686465411yescharacteristic
351C22−1+4×D27111yescharacteristic
352C2#3177473411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#519×C271C2
hypercenter#519D271
hypocenterC271#3518
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. Q8×Dic542C2semidirect
2. #31774620C2semidirect
3. #31774619C2semidirect
4. #31774615C2semidirect
5. #31774610C2semidirect
6. Q8×C22⋊D271C2semidirect
7. #31774559C2semidirect
8. #31774558C2semidirect
9. #31774519C2semidirect
10. #31774518C2semidirect
11. #31774451C2semidirect
12. #31774425C2semidirect
13. #31774373C2semidirect
14. C4×#16864655C2semidirect
15. #519×C271C2semidirect
176. C271#3518semidirect
191. #519D271semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic5422
#317746202
#317746192
#317746152
#317746102
Q8×C22⋊D2712
#317745592
#317745582
#317745192
#317745182
#317744512
#317744252
#317743732
C4×#168646552
#519×C2712
#3518271