Finite Group Databasebeta

← #46848246

Subgroups — #46848246

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#31774581C211yesmaximal, characteristic, complemented
2#31774557C211yesmaximal, characteristic
3#31774253C211yesmaximal, characteristic, complemented
4#31774235C211yesmaximal, characteristic, complemented
5#31774212C211yesmaximal, characteristic, complemented
6#31774203C211yesmaximal, characteristic, complemented
7#31774129C211yesmaximal, characteristic, complemented
8#31774121C211yesmaximal, characteristic, complemented
9#31774057C211yesmaximal, characteristic, complemented
10#31774051C211yesmaximal, characteristic, complemented
11#31773989C211yesmaximal, characteristic, complemented
12C8×C22⋊D271C211yesmaximal, characteristic, complemented
13#31773894C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#467×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×#16864644C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21#22576153—21no—
22#22576151C2211yescharacteristic
23#22576150—21no—
24#22576149C2211yescharacteristic
25#22576146—21no—
26#22576142—21no—
27#22576138—21no—
28#22576136C2211yescharacteristic
29#22576135C2211yescharacteristic
30#22576133—21no—
31Q8×Dic271C2211yescharacteristic
32#22576125C2211yescharacteristic
33#22576122—21no—
34C4×C22⋊D271C2211yescharacteristic
35#22576119C2211yescharacteristic
36#22576118—21no—
37C4⋊C4×D271C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576091C2211yescharacteristic
40Dic271⋊C8C2211yescharacteristic
41#22576085C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576083C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576080C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×Q16×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C22⋊Q8×C271C2211yescharacteristic
55C22⋊Q8×C271C2211yescharacteristic
56#107×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#103×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—41no—
65C2×D1084—21no—
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271D411yescharacteristic
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73#16864644—22no—
74#16864641—41no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—41no—
78C4⋊Dic271—21no—
79C4⋊Dic271—21no—
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4—21no—
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4D411yescharacteristic
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4D411yescharacteristic
94C4×Dic271—21no—
95C4×Dic271—21no—
96C4×Dic271—21no—
97C4×Dic271C2311yescharacteristic
98C2×Q8×C271C2311yescharacteristic
99C2×Q8×C271C2311yescharacteristic
100C22×C4×C271C2311yescharacteristic
101Q16×C271—41no—
102C2×C8×C271—41no—
103C2×C8×C271C2311yescharacteristic
104C2×C8×C271C2311yescharacteristic
105C4⋊C4×C271—21no—
106C4⋊C4×C271—21no—
107C4⋊C4×C271C2311yescharacteristic
108C4⋊C4×C271C2311yescharacteristic
109C22⋊C4×C271—21no—
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—21no—
114Dic542—42no—
115Dic542#4112yes—
116D1084—21no—
117D1084—21no—
118D1084—42no—
119C22⋊D271—22no—
120C22⋊D271—42no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271C2×D411yescharacteristic
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—21no—
128C4×D271—21no—
129C4×D271—42no—
130C4×D271—42no—
131C4×D271—42no—
132C271⋊C8—21no—
133C271⋊C8—21no—
134C23×C271C2×D411yescharacteristic
135Q8×C271—21no—
136Q8×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271—21no—
149C22×C271—21no—
150C22×C271—21no—
151C22×C271—21no—
152C22×C271C22×D411yescharacteristic
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271#13512yes—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271—41no—
164C4×C271C2×#4111yescharacteristic
165C4×C271H(5,𝔽2)11yescharacteristic
166Dic271—21no—
167Dic271—22no—
168Dic271—21no—
169Dic271—21no—
170Dic271—41no—
171Dic271—41no—
172C2×C271—41no—
173C2×C271#54611yescharacteristic
174C2×C271C2×#13511yescharacteristic
175C2×C271#57811yescharacteristic
176D271—22no—
177D271—81no—
178C271#330011yescharacteristic, complemented, Sylow
179#3300—2711nomaximal, Sylow
180#552—2711no—
181#548—2711no—
182#488—2711no—
183#485—2711no—
184#480—2711no—
185#479—2711no—
186#467D27111yescharacteristic, complemented
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×Q16C2×D27111yescharacteristic
199C2×SD16—2711no—
200C22×C8—2711no—
201C22×C8C2×D27111yescharacteristic
202#126—5421no—
203#125—2711no—
204#125—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—2711no—
209C22⋊Q8—5421no—
210C22⋊Q8C2×D27111yescharacteristic
211C22⋊Q8C2×D27111yescharacteristic
212#121—2711no—
213#121—5421no—
214C4×Q8—2711no—
215C4×Q8—2711no—
216C4×D4—2711no—
217C4×D4—5421no—
218#117—2711no—
219#117—5421no—
220C2×C4⋊C4—2711no—
221#107—2711no—
222#107C2×D27111yescharacteristic
223#106—2711no—
224C4⋊C8—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#103—2711no—
229#103—2711no—
230#103C2×D27111yescharacteristic
231#103C2×D27111yescharacteristic
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—5421no—
242C2×Q8—5422no—
243C2×Q8—10841no—
244C2×Q8C22×D27111yescharacteristic
245C2×Q8C22×D27111yescharacteristic
246C2×D4—2711no—
247C2×D4—2711no—
248C2×D4—5421no—
249C2×D4—5421no—
250C22×C4—2711no—
251C22×C4—5421no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4C22×D27111yescharacteristic
255Q16—41no—
256Q16—5422no—
257SD16—10841no—
258C2×C8—41no—
259C2×C8—2711no—
260C2×C8—2711no—
261C2×C8—10841no—
262C2×C8C22×D27111yescharacteristic
263C2×C8C22×D27111yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—2711no—
267C4⋊C4—2711no—
268C4⋊C4—2711no—
269C4⋊C4—5421no—
270C4⋊C4—5421no—
271C4⋊C4—5421no—
272C4⋊C4—5421no—
273C4⋊C4—5421no—
274C4⋊C4—5421no—
275C4⋊C4C22×D27111yescharacteristic
276C4⋊C4C22×D27111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—2711no—
280C22⋊C4—2711no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—2711no—
287C42—5421no—
288C42—5421no—
289C42—5421no—
290C23—2711no—
291C23—5421no—
292C23D4×D27111yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—2712no—
298Q8—5421no—
299Q8—10842no—
300D4—5422no—
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—2711no—
312C2×C4—2711no—
313C2×C4—2711no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—10842no—
320C2×C4—10842no—
321C2×C4—10842no—
322C2×C4D4×D27111yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C8—21no—
325C8—21no—
326C8—5421no—
327C8—5421no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—2712no—
332C22—5421no—
333C22—5421no—
334C22—10841no—
335C22—10841no—
336C22—10841no—
337C22C2×D4×D27111yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—5421no—
344C4—5422no—
345C4—5421no—
346C4—5421no—
347C4—10841no—
348C4—10841no—
349C4#41×D27111yescharacteristic
350C4#2257620511yescharacteristic
351C2—41no—
352C2—5422no—
353C2—21681no—
354C2#3177453611yescharacteristic
355C2#135×D27111yescharacteristic
356C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#467×C271C2
socleC22×C271C22×D4
prime radical#31774557C2
hypercenter#467D271
hypocenterC271#3300
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #31774581C2semidirect
3. #31774253C2semidirect
4. #31774235C2semidirect
5. #31774212C2semidirect
6. #31774203C2semidirect
7. #31774129C2semidirect
8. #31774121C2semidirect
9. #31774057C2semidirect
10. #31774051C2semidirect
11. #31773989C2semidirect
12. C8×C22⋊D271C2semidirect
13. #31773894C2semidirect
14. #103×D271C2semidirect
15. #467×C271C2semidirect
178. C271#3300semidirect
186. #467D271semidirect

Maximal subgroups

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

subgroupindex
#317745812
#317745572
#317742532
#317742352
#317742122
#317742032
#317741292
#317741212
#317740572
#317740512
#317739892
C8×C22⋊D2712
#317738942
#103×D2712
#467×C2712
#3300271