Finite Group Databasebeta

← #46848411

Subgroups — #46848411

Order 34688.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774582C211yesmaximal, characteristic
2Q8×C22⋊D271C211yesmaximal, characteristic
3#31774264C211yesmaximal, characteristic, complemented
4#31774237C211yesmaximal, characteristic, complemented
5#31774227C211yesmaximal, characteristic, complemented
6#31774206C211yesmaximal, characteristic, complemented
7#31774154C211yesmaximal, characteristic, complemented
8#31774138C211yesmaximal, characteristic, complemented
9#31774106C211yesmaximal, characteristic, complemented
10#31774054C211yesmaximal, characteristic, complemented
11#31774053C211yesmaximal, characteristic, complemented
12#31774001C211yesmaximal, characteristic, complemented
13#31773968C211yesmaximal, characteristic
14#31773904C211yesmaximal, characteristic, complemented
15C2×#103×C271C211yesmaximal, characteristic, complemented
16C2×Q8×D271C2211yescharacteristic
17C2×Dic1084C2211yescharacteristic
18C2×#16864644C2211yescharacteristic
19C2×#16864644C2211yescharacteristic
20C2×#16864637C2211yescharacteristic
21#22576158—21no—
22#22576158C2211yescharacteristic
23#22576155—21no—
24#22576151C2211yescharacteristic
25#22576150—21no—
26#22576143—21no—
27#22576140C2211yescharacteristic
28#22576139—21no—
29#22576139C2211yescharacteristic
30#22576139C2211yescharacteristic
31#22576136—21no—
32#22576136C2211yescharacteristic
33Q8×Dic271C2211yescharacteristic
34#22576125C2211yescharacteristic
35C4×C22⋊D271—21no—
36C4×C22⋊D271C2211yescharacteristic
37#22576119—21no—
38#22576118C2211yescharacteristic
39#22576094C2211yescharacteristic
40#22576091C2211yescharacteristic
41#22576091C2211yescharacteristic
42Dic271⋊C8C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576083C2211yescharacteristic
48#22576083C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51#22576070C2211yescharacteristic
52C22×Q8×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C2×C4⋊C4×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542C2311yescharacteristic
61C2×Dic542C2311yescharacteristic
62Q8×D271—22no—
63Q8×D271—41no—
64C2×C22⋊D271D411yescharacteristic
65C2×C4×D271—21no—
66C2×C4×D271C2311yescharacteristic
67Dic1084—41no—
68#16864644—22no—
69#16864644—41no—
70#16864637—41no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271—21no—
74C4⋊Dic271—21no—
75C4⋊Dic271—21no—
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4D411yescharacteristic
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4D411yescharacteristic
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271C2311yescharacteristic
97C2×Q8×C271—21no—
98C2×Q8×C271—22no—
99C2×Q8×C271—41no—
100C2×Q8×C271C2311yescharacteristic
101C2×Q8×C271C2311yescharacteristic
102C22×C4×C271—21no—
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105C2×C8×C271—41no—
106C2×C8×C271C2311yescharacteristic
107C2×C8×C271C2311yescharacteristic
108C4⋊C4×C271—21no—
109C4⋊C4×C271C2311yescharacteristic
110C4⋊C4×C271C2311yescharacteristic
111C22×D271C2×D411yescharacteristic
112Dic542—21no—
113Dic542—21no—
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117C22⋊D271—22no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C2×Dic271C2×D411yescharacteristic
125C4×D271—21no—
126C4×D271—21no—
127C4×D271—42no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C23×C271C2×D411yescharacteristic
131Q8×C271—21no—
132Q8×C271—21no—
133Q8×C271—21no—
134Q8×C271—42no—
135Q8×C271#4112yes—
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—42no—
143C2×C4×C271—42no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2411yescharacteristic
146C8×C271—21no—
147C8×C271—21no—
148C22×C271—21no—
149C22×C271—21no—
150C22×C271#13712yes—
151C22×C271C22×D411yescharacteristic
152C2×D271—21no—
153C2×D271—21no—
154C2×D271—21no—
155C4×C271—21no—
156C4×C271—22no—
157C4×C271—41no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271C2×#4111yescharacteristic
161C4×C271H(5,𝔽2)11yescharacteristic
162Dic271—21no—
163Dic271—21no—
164Dic271—21no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168Dic271—41no—
169C2×C271—22no—
170C2×C271#54611yescharacteristic
171C2×C271C2×#13711yescharacteristic
172C2×C271#57811yescharacteristic
173D271—41no—
174C271#331211yescharacteristic, complemented, Sylow
175#3312—2711nomaximal, Sylow
176#552—2711no—
177D4×Q8—2711no—
178#491—2711no—
179#485—2711no—
180#483—2711no—
181#479—2711no—
182#470—2711no—
183#467—2711no—
184#462—2711no—
185#451—2711no—
186#451—2711no—
187#441—2711no—
188#435—2711no—
189#419—2711no—
190C2×#103D27111yescharacteristic, complemented
191C22×Q8—2711no—
192C22×Q8C2×D27111yescharacteristic
193C2×Q16—2711no—
194C2×Q16—2711no—
195C2×Q16—2711no—
196C2×M16—2711no—
197C22×C8C2×D27111yescharacteristic
198C4⋊Q8—2711no—
199C4⋊Q8—5421no—
200#126—5421no—
201#125—2711no—
202#125—5421no—
203#123—5421no—
204C22⋊Q8—2711no—
205C22⋊Q8—2711no—
206C22⋊Q8—2711no—
207C22⋊Q8—2711no—
208C22⋊Q8—5421no—
209C22⋊Q8—5421no—
210C4×Q8—2711no—
211C4×Q8—2711no—
212C4×D4—2711no—
213C4×D4—5421no—
214#117—2711no—
215#117—5421no—
216C2×C4⋊C4C2×D27111yescharacteristic
217#107—2711no—
218#106—2711no—
219#106—2711no—
220C4⋊C8—2711no—
221#103—2711no—
222#103—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#103—2711no—
227#103C2×D27111yescharacteristic
228#103C2×D27111yescharacteristic
229#103C2×D27111yescharacteristic
230#103C2×D27111yescharacteristic
231C22⋊C8—2711no—
232C22⋊C8—2711no—
233#97—2711no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—2711no—
238C2×Q8—2711no—
239C2×Q8—5421no—
240C2×Q8—5422no—
241C2×Q8—10841no—
242C2×Q8C22×D27111yescharacteristic
243C2×Q8C22×D27111yescharacteristic
244C2×D4—2711no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—2711no—
248C22×C4—5421no—
249C22×C4C22×D27111yescharacteristic
250Q16—5422no—
251Q16—10841no—
252Q16—10841no—
253M16—10841no—
254C2×C8—41no—
255C2×C8—2711no—
256C2×C8—2711no—
257C2×C8C22×D27111yescharacteristic
258C2×C8C22×D27111yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4—5421no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4C22×D27111yescharacteristic
274C4⋊C4C22×D27111yescharacteristic
275C22⋊C4—2711no—
276C22⋊C4—2711no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C42—2711no—
283C42—5421no—
284C42—5421no—
285C42—5421no—
286C23—2711no—
287C23D4×D27111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—5421no—
293Q8—5421no—
294Q8—5421no—
295Q8—5421no—
296Q8—10842no—
297Q8#1686465412yes—
298D4—5422no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—2711no—
308C2×C4—2711no—
309C2×C4—2711no—
310C2×C4—5421no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—10842no—
317C2×C4D4×D27111yescharacteristic
318C2×C4C23×D27111yescharacteristic
319C8—21no—
320C8—21no—
321C8—5421no—
322C8—5421no—
323C22—21no—
324C22—21no—
325C22—5421no—
326C22—5421no—
327C22—5421no—
328C22#2257618912yes—
329C22C2×D4×D27111yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—5421no—
336C4—5421no—
337C4—5421no—
338C4—10841no—
339C4—10841no—
340C4—10841no—
341C4—10841no—
342C4C2×#1686465411yescharacteristic
343C4#2257620511yescharacteristic
344C2—22no—
345C2—10841no—
346C2#3177453711yescharacteristic
347C2C2×#2257618911yescharacteristic
348C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#103×C271C2
socleC22×C271C22×D4
prime radicalC2×C22⋊D271D4
hypercenterC2×#103D271
hypocenterC271#3312
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. #31774264C2semidirect
4. #31774237C2semidirect
5. #31774227C2semidirect
6. #31774206C2semidirect
7. #31774154C2semidirect
8. #31774138C2semidirect
9. #31774106C2semidirect
10. #31774054C2semidirect
11. #31774053C2semidirect
12. #31774001C2semidirect
14. #31773904C2semidirect
15. C2×#103×C271C2semidirect
174. C271#3312semidirect
190. C2×#103D271semidirect

Maximal subgroups

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

subgroupindex
#317745822
Q8×C22⋊D2712
#317742642
#317742372
#317742272
#317742062
#317741542
#317741382
#317741062
#317740542
#317740532
#317740012
#317739682
#317739042
C2×#103×C2712
#3312271