Finite Group Databasebeta

← #46850372

Subgroups — #46850372

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#31774618C213yesmaximal, complemented
3#31774617C213yesmaximal, complemented
4#31774616C213yesmaximal, complemented
5#31774611C213yesmaximal, complemented
6Q8×C22⋊D271C211yesmaximal, characteristic, complemented
7#31774554C211yesmaximal, characteristic, complemented
8#31774516C213yesmaximal, complemented
9#31774514C213yesmaximal, complemented
10#31774449C213yesmaximal, complemented
11#31774427C213yesmaximal, complemented
12#31774375C211yesmaximal, characteristic, complemented
13#31774373C211yesmaximal, characteristic, complemented
14#41×Dic271C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17#22576160C2213yes—
18#22576159C2213yes—
19#22576159C2213yes—
20#22576158C2213yes—
21#22576155C2213yes—
22#22576153C2216yes—
23#22576153C2216yes—
24#22576153C2213yes—
25#22576150C2213yes—
26#22576147C2213yes—
27#22576147C2216yes—
28#22576146C2216yes—
29#22576146C2211yescharacteristic
30#22576144C2213yes—
31#22576141C2213yes—
32#22576139C2213yes—
33#22576139C2213yes—
34#22576137C2216yes—
35#22576137C2213yes—
36#22576136C2213yes—
37#22576132C2213yes—
38#22576131C2213yes—
39Q8×Dic271C2213yes—
40Q8×Dic271C2211yescharacteristic
41Q8×Dic271C2211yescharacteristic
42Q8×Dic271C2213yes—
43Q8×Dic271C2211yescharacteristic
44C4×Dic542C2213yes—
45C4×Dic542C2213yes—
46#22576125C2213yes—
47#22576125C2216yes—
48D4×Dic271C2213yes—
49D4×Dic271C2213yes—
50C4×C22⋊D271C2213yes—
51C4×C22⋊D271C2213yes—
52#22576119C2213yes—
53#22576119C2213yes—
54#22576119C2213yes—
55#22576118C2213yes—
56#22576117C2213yes—
57#22576116C2216yes—
58C2×C4×Dic271C2213yes—
59C2×#41×C271C2211yescharacteristic
60C22×Q8×C271C2211yescharacteristic
61#124×C271C2213yes—
62C22⋊Q8×C271C2213yes—
63#121×C271C2213yes—
64C4×Q8×C271C2211yescharacteristic
65C4×D4×C271C2213yes—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2313yes—
68C2×Dic542C2316yes—
69Q8×D271—24no—
70C2×C22⋊D271C2311yescharacteristic
71C2×C22⋊D271C2313yes—
72C22×Dic271C2313yes—
73C22×Dic271C2313yes—
74C2×C4×D271C2313yes—
75C4⋊Dic271C2313yes—
76C4⋊Dic271C2313yes—
77C4⋊Dic271C2313yes—
78C4⋊Dic271C2313yes—
79C4⋊Dic271C2313yes—
80C4⋊Dic271C2313yes—
81Dic271⋊C4C2313yes—
82Dic271⋊C4C2316yes—
83Dic271⋊C4C2313yes—
84Dic271⋊C4C2313yes—
85Dic271⋊C4C2313yes—
86Dic271⋊C4C2313yes—
87Dic271⋊C4C2313yes—
88C22⋊Dic271C2316yes—
89C22⋊Dic271C2313yes—
90C22⋊Dic271C2313yes—
91C22⋊Dic271C2313yes—
92C22⋊Dic271C2311yescharacteristic
93C22⋊Dic271C2316yes—
94(C2×D271)⋊C4C2313yes—
95(C2×D271)⋊C4C2313yes—
96(C2×D271)⋊C4C2311yescharacteristic
97(C2×D271)⋊C4C2311yescharacteristic
98C4×Dic271C2311yescharacteristic
99C4×Dic271C2316yes—
100C4×Dic271C2316yes—
101C4×Dic271C2313yes—
102C4×Dic271C2313yes—
103C4×Dic271C2313yes—
104C4×Dic271C2313yes—
105#41×C271—24no—
106C2×Q8×C271—24no—
107C2×Q8×C271C2313yes—
108C2×Q8×C271C2311yescharacteristic
109C2×D4×C271C2313yes—
110C2×D4×C271C2313yes—
111C22×C4×C271C2313yes—
112C22×C4×C271C2313yes—
113C4⋊C4×C271C2313yes—
114C4⋊C4×C271C2313yes—
115C22⋊C4×C271C2313yes—
116C22⋊C4×C271C2316yes—
117C22⋊C4×C271C2311yescharacteristic
118C42×C271C2313yes—
119C22×D271C2411yescharacteristic
120Dic542—26no—
121Dic542—22no—
122Dic542—46no—
123C22⋊D271—22no—
124C22⋊D271—43no—
125C2×Dic271—26no—
126C2×Dic271—43no—
127C2×Dic271C2413yes—
128C2×Dic271C2411yescharacteristic
129C2×Dic271C2411yescharacteristic
130C2×Dic271C2413yes—
131C2×Dic271C2413yes—
132C2×Dic271C2413yes—
133C2×Dic271C2411yescharacteristic
134C4×D271—26no—
135C23×C271C2413yes—
136C23×C271C2411yescharacteristic
137Q8×C271—26no—
138Q8×C271#4114yes—
139D4×C271—26no—
140D4×C271—43no—
141C2×C4×C271—26no—
142C2×C4×C271—26no—
143C2×C4×C271C2411yescharacteristic
144C2×C4×C271C2413yes—
145C2×C4×C271C2413yes—
146C2×C4×C271C2411yescharacteristic
147C2×C4×C271C2413yes—
148C22×C271—21no—
149C22×C271—23no—
150C22×C271—23no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271—23no—
154C22×C271C2511yescharacteristic
155C2×D271—21no—
156C2×D271—21no—
157C2×D271—21no—
158C4×C271—21no—
159C4×C271—23no—
160C4×C271—21no—
161C4×C271—23no—
162C4×C271C2×#4116yes—
163Dic271—23no—
164Dic271—23no—
165Dic271—21no—
166Dic271—21no—
167Dic271—23no—
168Dic271—23no—
169Dic2712−1+412yes—
170C2×C271—41no—
171C2×C271—43no—
172C2×C271C22×#4111yescharacteristic
173C2×C271C2×2−1+411yescharacteristic
174C2×C271#58511yescharacteristic
175D271—41no—
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#548D27111yescharacteristic, complemented
186#542—2713no—
187#542—2713no—
188#533—2713no—
189#529—2713no—
190#519—2711no—
191#519—2711no—
192C4×#41—2711no—
193C2×#41C2×D27111yescharacteristic
194C22×Q8—2711no—
195C22×Q8C2×D27111yescharacteristic
196C4⋊Q8—2713no—
197C4⋊Q8—2713no—
198C4⋊Q8—2713no—
199C4⋊Q8—2713no—
200#126—2713no—
201#126—2713no—
202#126—2716no—
203#126—2716no—
204#125—2713no—
205#124—2713no—
206#124—2716no—
207#124—2711no—
208#124—2716no—
209#124C2×D27113yes—
210#123—2713no—
211#123—2713no—
212C22⋊Q8—2713no—
213C22⋊Q8—2713no—
214C22⋊Q8—2713no—
215C22⋊Q8—2713no—
216C22⋊Q8—2716no—
217C22⋊Q8C2×D27113yes—
218#121—2713no—
219#121—2713no—
220#121C2×D27113yes—
221C4×Q8—2711no—
222C4×Q8—2711no—
223C4×Q8—2713no—
224C4×Q8—2711no—
225C4×Q8—2713no—
226C4×Q8—2713no—
227C4×Q8—2716no—
228C4×Q8—2713no—
229C4×Q8—2713no—
230C4×Q8C2×D27111yescharacteristic
231C4×D4—2713no—
232C4×D4—2713no—
233C4×D4—2713no—
234C4×D4—2713no—
235C4×D4C2×D27113yes—
236#117—2713no—
237#117—2713no—
238#117—2713no—
239#117—2713no—
240#117—2716no—
241#117—2713no—
242C2×C42—2713no—
243#41—24no—
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×D4C22×D27113yes—
254C2×D4C22×D27113yes—
255C22×C4—2713no—
256C22×C4—2713no—
257C22×C4—2713no—
258C22×C4C22×D27113yes—
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—2713no—
269C4⋊C4—2713no—
270C4⋊C4—2716no—
271C4⋊C4—2713no—
272C4⋊C4—2713no—
273C4⋊C4C22×D27113yes—
274C4⋊C4C22×D27113yes—
275C22⋊C4—2713no—
276C22⋊C4—2711no—
277C22⋊C4—2716no—
278C22⋊C4—2716no—
279C22⋊C4—2713no—
280C22⋊C4—2713no—
281C22⋊C4—2713no—
282C22⋊C4—2711no—
283C22⋊C4—2711no—
284C22⋊C4—2713no—
285C22⋊C4C22×D27116yes—
286C22⋊C4C22×D27111yescharacteristic
287C22⋊C4C22×D27113yes—
288C42—2711no—
289C42—2713no—
290C42—2713no—
291C42—2713no—
292C42—2716no—
293C42—2716no—
294C42—2713no—
295C42C22×D27113yes—
296C23—2711no—
297C23C23×D27111yescharacteristic
298C23C23×D27113yes—
299Q8—26no—
300Q8—5422no—
301Q8—5426no—
302Q8—10846no—
303Q8#1686465314yes—
304D4—26no—
305D4—43no—
306D4—5422no—
307D4—10843no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—2711no—
311C2×C4—2713no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—2713no—
315C2×C4—2713no—
316C2×C4—2713no—
317C2×C4—5426no—
318C2×C4—5426no—
319C2×C4—10843no—
320C2×C4C23×D27111yescharacteristic
321C2×C4C23×D27113yes—
322C2×C4C23×D27113yes—
323C2×C4C23×D27111yescharacteristic
324C2×C4C23×D27113yes—
325C22—21no—
326C22—23no—
327C22—23no—
328C22—21no—
329C22—23no—
330C22—21no—
331C22—5421no—
332C22—5421no—
333C22—5421no—
334C22C24×D27111yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—2712no—
340C4—5423no—
341C4—5423no—
342C4—5421no—
343C4—5421no—
344C4—5423no—
345C4—5423no—
346C4C2×#1686465316yes—
347C2—41no—
348C2—43no—
349C2—10841no—
350C2C22×#1686465311yescharacteristic
351C22−1+4×D27111yescharacteristic
352C2#3177473311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
Fitting#548×C271C2
hypercenter#548D271
hypocenterC271#3518
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. Q8×Dic542C2semidirect
2. #31774618C2semidirect
3. #31774617C2semidirect
4. #31774616C2semidirect
5. #31774611C2semidirect
6. Q8×C22⋊D271C2semidirect
7. #31774554C2semidirect
8. #31774516C2semidirect
9. #31774514C2semidirect
10. #31774449C2semidirect
11. #31774427C2semidirect
12. #31774375C2semidirect
13. #31774373C2semidirect
14. #41×Dic271C2semidirect
15. #548×C271C2semidirect
176. C271#3518semidirect
185. #548D271semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic5422
#317746182
#317746172
#317746162
#317746112
Q8×C22⋊D2712
#317745542
#317745162
#317745142
#317744492
#317744272
#317743752
#317743732
#41×Dic2712
#548×C2712
#3518271