Finite Group Databasebeta

← #46848245

Subgroups — #46848245

Order 34688.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774556C211yesmaximal, characteristic
2#31774254C211yesmaximal, characteristic, complemented
3#31774238C211yesmaximal, characteristic, complemented
4#31774210C211yesmaximal, characteristic, complemented
5#31774201C211yesmaximal, characteristic, complemented
6#31774136C211yesmaximal, characteristic, complemented
7#31774128C211yesmaximal, characteristic, complemented
8#31774119C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774052C211yesmaximal, characteristic, complemented
11C4×#16864644C211yesmaximal, characteristic, complemented
12D4×C271⋊C8C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14C2×#22576083C211yesmaximal, characteristic, complemented
15#552×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C2×#16864644C2211yescharacteristic
19C2×#16864644C2211yescharacteristic
20C2×#16864641C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22C22×C271⋊C8C2211yescharacteristic
23#22576148C2211yescharacteristic
24#22576146—21no—
25#22576139C2211yescharacteristic
26#22576138—21no—
27#22576134C2211yescharacteristic
28#22576132—21no—
29C4×Dic542C2211yescharacteristic
30(C22⋊D271)⋊C4—21no—
31#22576093C2211yescharacteristic
32#22576093C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576084C2211yescharacteristic
36#22576083C2211yescharacteristic
37#22576083C2211yescharacteristic
38#22576083C2211yescharacteristic
39#22576083C2211yescharacteristic
40#22576083C2211yescharacteristic
41#22576083C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47C4×C271⋊C8C2211yescharacteristic
48#126×C271—21no—
49#125×C271—21no—
50#125×C271C2211yescharacteristic
51#123×C271—21no—
52C22⋊Q8×C271C2211yescharacteristic
53C22⋊Q8×C271C2211yescharacteristic
54C4×Q8×C271C2211yescharacteristic
55C4×D4×C271C2211yescharacteristic
56#117×C271—21no—
57#117×C271C2211yescharacteristic
58C2×C4⋊C4×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—41no—
61C2×Dic542—22no—
62C2×Dic542—21no—
63C2×Dic542—41no—
64C2×Dic542C2311yescharacteristic
65C2×Dic542C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C22×Dic271—21no—
70C2×C4×D271—21no—
71#16864644—22no—
72#16864644—41no—
73#16864641—41no—
74C2×C271⋊C8—41no—
75C2×C271⋊C8—41no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83C22⋊Dic271—21no—
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87C4×Dic271—21no—
88C2×Q8×C271C2311yescharacteristic
89C2×D4×C271D411yescharacteristic
90C22×C4×C271—21no—
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C4⋊C4×C271—21no—
94C4⋊C4×C271—21no—
95C4⋊C4×C271—21no—
96C4⋊C4×C271—21no—
97C4⋊C4×C271—21no—
98C4⋊C4×C271—21no—
99C4⋊C4×C271D411yescharacteristic
100C4⋊C4×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22⋊C4×C271D411yescharacteristic
107C22⋊C4×C271D411yescharacteristic
108C42×C271—21no—
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112Dic542—21no—
113Dic542—21no—
114Dic542—21no—
115Dic542—42no—
116Dic542#4112yes—
117D1084—21no—
118D1084—21no—
119C22⋊D271—42no—
120C22⋊D271—42no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—42no—
125C4×D271—42no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C23×C271C2×D411yescharacteristic
131C23×C271C2×D411yescharacteristic
132Q8×C271—21no—
133Q8×C271—21no—
134D4×C271—22no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—42no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2×D411yescharacteristic
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2411yescharacteristic
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271—21no—
153C22×C271—21no—
154C22×C271#13512yes—
155C22×C271C22×D411yescharacteristic
156C2×D271—41no—
157C2×D271—41no—
158C2×D271—41no—
159C4×C271—21no—
160C4×C271—21no—
161C4×C271—21no—
162C4×C271—21no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271C2×#4111yescharacteristic
168C4×C271H(5,𝔽2)11yescharacteristic
169Dic271—22no—
170Dic271—41no—
171Dic271—41no—
172C2×C271—22no—
173C2×C271—41no—
174C2×C271#54611yescharacteristic
175C2×C271C2×#13511yescharacteristic
176C2×C271#57811yescharacteristic
177D271—81no—
178C271#330011yescharacteristic, complemented, Sylow
179#3300—2711nomaximal, Sylow
180#552D27111yescharacteristic, complemented
181#548—2711no—
182#488—2711no—
183#485—2711no—
184#480—2711no—
185#479—2711no—
186#467—2711no—
187#465—2711no—
188#464—2711no—
189#452—2711no—
190#451—2711no—
191C4×Q16—2711no—
192C8×D4—2711no—
193#416—2711no—
194C2×#103—2711no—
195C2×#41—2711no—
196C22×Q8—2711no—
197C2×Q16—2711no—
198C2×Q16—2711no—
199C2×SD16—2711no—
200C22×C8—2711no—
201C22×C8—2711no—
202#126—21no—
203#125—21no—
204#125C2×D27111yescharacteristic
205#124—2711no—
206#124—5421no—
207#123—21no—
208C22⋊Q8—2711no—
209C22⋊Q8—5421no—
210C22⋊Q8C2×D27111yescharacteristic
211C22⋊Q8C2×D27111yescharacteristic
212#121—2711no—
213#121—5421no—
214C4×Q8—2711no—
215C4×Q8C2×D27111yescharacteristic
216C4×D4—5421no—
217C4×D4C2×D27111yescharacteristic
218#117—21no—
219#117C2×D27111yescharacteristic
220C2×C4⋊C4C2×D27111yescharacteristic
221#107—2711no—
222#107—2711no—
223#106—2711no—
224C4⋊C8—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103—2711no—
231#103—2711no—
232#102—2711no—
233#102—2711no—
234#102—2711no—
235C22⋊C8—2711no—
236C22⋊C8—2711no—
237C4×C8—2711no—
238#41—5422no—
239#41—10841no—
240C2×Q8—2711no—
241C2×Q8—2711no—
242C2×Q8—5422no—
243C2×Q8—5421no—
244C2×Q8—10841no—
245C2×Q8C22×D27111yescharacteristic
246C2×D4—2711no—
247C2×D4—5421no—
248C2×D4—5421no—
249C2×D4C22⋊D27111yescharacteristic
250C22×C4—21no—
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4C22×D27111yescharacteristic
254C22×C4C22×D27111yescharacteristic
255Q16—5422no—
256Q16—10841no—
257SD16—10841no—
258C2×C8—2711no—
259C2×C8—2711no—
260C2×C8—2711no—
261C2×C8—2711no—
262C2×C8—10841no—
263C2×C8—10841no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—2711no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4C22⋊D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4C22⋊D27111yescharacteristic
285C22⋊C4C22⋊D27111yescharacteristic
286C42—21no—
287C42—21no—
288C42—5421no—
289C42C22×D27111yescharacteristic
290C23—5421no—
291C23C2×C22⋊D27111yescharacteristic
292C23C2×C22⋊D27111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2712no—
296Q8—5421no—
297Q8—5421no—
298Q8—5421no—
299Q8—10842no—
300D4—22no—
301D4—5421no—
302D4—5421no—
303D4—10842no—
304D4—10842no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—10842no—
318C2×C4—10842no—
319C2×C4C2×C22⋊D27111yescharacteristic
320C2×C4C2×C22⋊D27111yescharacteristic
321C2×C4C2×C22⋊D27111yescharacteristic
322C2×C4C2×C22⋊D27111yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C8—5421no—
325C8—5421no—
326C8—5421no—
327C8—5421no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—10841no—
334C22—10841no—
335C22—10841no—
336C22#2257618012yes—
337C22C22×C22⋊D27111yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5422no—
347C4—10841no—
348C4—10841no—
349C4#41×D27111yescharacteristic
350C4#2257620611yescharacteristic
351C2—22no—
352C2—41no—
353C2—21681no—
354C2#3177453511yescharacteristic
355C2C2×#2257618011yescharacteristic
356C2#3177471211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#552×C271C2
socleC22×C271C22×D4
prime radical#31774556C2
hypercenter#552D271
hypocenterC271#3300
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31774254C2semidirect
3. #31774238C2semidirect
4. #31774210C2semidirect
5. #31774201C2semidirect
6. #31774136C2semidirect
7. #31774128C2semidirect
8. #31774119C2semidirect
9. #31774057C2semidirect
10. #31774052C2semidirect
11. C4×#16864644C2semidirect
12. D4×C271⋊C8C2semidirect
13. #31773891C2semidirect
14. C2×#22576083C2semidirect
15. #552×C271C2semidirect
178. C271#3300semidirect
180. #552D271semidirect

Maximal subgroups

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

subgroupindex
#317745562
#317742542
#317742382
#317742102
#317742012
#317741362
#317741282
#317741192
#317740572
#317740522
C4×#168646442
D4×C271⋊C82
#317738912
C2×#225760832
#552×C2712
#3300271