Finite Group Databasebeta

← #46848256

Subgroups — #46848256

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#31774588C211yesmaximal, characteristic, complemented
2#31774559C211yesmaximal, characteristic
3#31774252C211yesmaximal, characteristic, complemented
4#31774240C211yesmaximal, characteristic, complemented
5#31774212C211yesmaximal, characteristic, complemented
6#31774204C211yesmaximal, characteristic, complemented
7#31774137C211yesmaximal, characteristic, complemented
8#31774129C211yesmaximal, characteristic, complemented
9#31774119C211yesmaximal, characteristic, complemented
10#31774058C211yesmaximal, characteristic, complemented
11#31774051C211yesmaximal, characteristic, complemented
12C8×D1084C211yesmaximal, characteristic, complemented
13#31773892C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15C4×Q16×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×#16864644C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×C8×D271C2211yescharacteristic
21C2×C8×D271C2211yescharacteristic
22#22576154—21no—
23#22576152C2211yescharacteristic
24#22576150—21no—
25#22576148—21no—
26#22576148C2211yescharacteristic
27#22576145—21no—
28#22576140—21no—
29#22576140C2211yescharacteristic
30#22576136C2211yescharacteristic
31#22576136C2211yescharacteristic
32#22576135—21no—
33#22576135C2211yescharacteristic
34C4×D1084—21no—
35C4×D1084C2211yescharacteristic
36#22576118C2211yescharacteristic
37#22576117—21no—
38C4⋊C4×D271C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576091C2211yescharacteristic
41C4⋊(C271⋊C8)C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576085C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576083C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576082C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52C2×Q16×C271C2211yescharacteristic
53C4×Q8×C271C2211yescharacteristic
54C4×Q8×C271C2211yescharacteristic
55#107×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58C4×C8×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—41no—
61C2×Dic542—21no—
62C2×Dic542C2311yescharacteristic
63Q8×D271—22no—
64Q8×D271—41no—
65C2×D1084—21no—
66C2×D1084—21no—
67C2×D1084D411yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×C4×D271—21no—
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73C2×C4×D271C2311yescharacteristic
74#16864644—41no—
75#16864641—41no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C8×D271—41no—
79C8×D271—41no—
80C4⋊Dic271—21no—
81C4⋊Dic271D411yescharacteristic
82C4⋊Dic271C2311yescharacteristic
83C4⋊Dic271C2311yescharacteristic
84Dic271⋊C4—21no—
85Dic271⋊C4—21no—
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4—21no—
92(C2×D271)⋊C4—21no—
93(C2×D271)⋊C4—21no—
94(C2×D271)⋊C4—21no—
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4D411yescharacteristic
97(C2×D271)⋊C4D411yescharacteristic
98C4×Dic271—21no—
99C2×Q8×C271C2311yescharacteristic
100C2×Q8×C271C2311yescharacteristic
101Q16×C271—22no—
102C2×C8×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C4⋊C4×C271—21no—
105C4⋊C4×C271—21no—
106C4⋊C4×C271C2311yescharacteristic
107C4⋊C4×C271C2311yescharacteristic
108C42×C271—21no—
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113C22×D271C2×D411yescharacteristic
114Dic542—21no—
115Dic542—21no—
116Dic542—42no—
117D1084—22no—
118D1084—21no—
119D1084—21no—
120D1084—42no—
121D1084—42no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271C2×D411yescharacteristic
126C2×Dic271C2×D411yescharacteristic
127C4×D271—21no—
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—42no—
132C4×D271—42no—
133C4×D271—42no—
134C271⋊C8—21no—
135C271⋊C8—21no—
136Q8×C271—21no—
137Q8×C271—21no—
138Q8×C271—21no—
139Q8×C271#4112yes—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271C2×D411yescharacteristic
145C2×C4×C271C2×D411yescharacteristic
146C2×C4×C271C2411yescharacteristic
147C8×C271—21no—
148C8×C271—21no—
149C22×C271C22×D411yescharacteristic
150C2×D271—21no—
151C2×D271—21no—
152C2×D271—21no—
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—41no—
156C2×D271—41no—
157C2×D271—41no—
158C2×D271#13512yes—
159C4×C271—22no—
160C4×C271—21no—
161C4×C271—21no—
162C4×C271—41no—
163C4×C271—41no—
164C4×C271—41no—
165C4×C271C2×#4111yescharacteristic
166C4×C271H(5,𝔽2)11yescharacteristic
167Dic271—21no—
168Dic271—21no—
169Dic271—41no—
170Dic271—41no—
171Dic271—41no—
172C2×C271#54611yescharacteristic
173C2×C271C2×#13511yescharacteristic
174C2×C271#57811yescharacteristic
175D271—22no—
176D271—41no—
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#467—2711no—
187#465—2711no—
188#464—2711no—
189#452—2711no—
190#451—2711no—
191C4×Q16D27111yescharacteristic, complemented
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×C8—2711no—
202#126—5421no—
203#125—2711no—
204#125—5421no—
205#124—2711no—
206#124—5421no—
207#123—5421no—
208C22⋊Q8—2711no—
209C22⋊Q8—2711no—
210C22⋊Q8—2711no—
211C22⋊Q8—5421no—
212#121—2711no—
213#121—5421no—
214C4×Q8C2×D27111yescharacteristic
215C4×Q8C2×D27111yescharacteristic
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×C8C2×D27111yescharacteristic
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—2711no—
252C22×C4—5421no—
253C22×C4—5421no—
254C22×C4—5421no—
255Q16—22no—
256Q16—10841no—
257SD16—10841no—
258C2×C8—2711no—
259C2×C8—2711no—
260C2×C8—10841no—
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—2711no—
278C22⋊C4—2711no—
279C22⋊C4—5421no—
280C22⋊C4—5421no—
281C22⋊C4—5421no—
282C22⋊C4—5421no—
283C22⋊C4—5421no—
284C22⋊C4—5421no—
285C22⋊C4—5421no—
286C42—21no—
287C42—21no—
288C42—5421no—
289C42C22×D27111yescharacteristic
290C23—2711no—
291C23—2711no—
292C23—5421no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—5421no—
297Q8—5421no—
298Q8—10842no—
299Q8#1686465412yes—
300D4—5422no—
301D4—5421no—
302D4—5421no—
303D4—10842no—
304D4—10842no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—2711no—
310C2×C4—2711no—
311C2×C4—5421no—
312C2×C4—5421no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—10842no—
319C2×C4—10842no—
320C2×C4—10842no—
321C2×C4D4×D27111yescharacteristic
322C2×C4D4×D27111yescharacteristic
323C2×C4C23×D27111yescharacteristic
324C8—21no—
325C8—21no—
326C8—5421no—
327C8—5421no—
328C22—2712no—
329C22—5421no—
330C22—5421no—
331C22—5421no—
332C22—5421no—
333C22—5421no—
334C22—10841no—
335C22—10841no—
336C22—10841no—
337C22C2×D4×D27111yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—5421no—
345C4—5421no—
346C4—10841no—
347C4—10841no—
348C4—10841no—
349C4C2×#1686465411yescharacteristic
350C4#2257620611yescharacteristic
351C2—5422no—
352C2—10841no—
353C2—21681no—
354C2#3177454311yescharacteristic
355C2#135×D27111yescharacteristic
356C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC4×Q16×C271C2
socleC22×C271C22×D4
prime radical#31774559C2
hypercenterC4×Q16D271
hypocenterC271#3300
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
1. #31774588C2semidirect
3. #31774252C2semidirect
4. #31774240C2semidirect
5. #31774212C2semidirect
6. #31774204C2semidirect
7. #31774137C2semidirect
8. #31774129C2semidirect
9. #31774119C2semidirect
10. #31774058C2semidirect
11. #31774051C2semidirect
12. C8×D1084C2semidirect
13. #31773892C2semidirect
14. #103×D271C2semidirect
15. C4×Q16×C271C2semidirect
178. C271#3300semidirect
191. C4×Q16D271semidirect

Maximal subgroups

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

subgroupindex
#317745882
#317745592
#317742522
#317742402
#317742122
#317742042
#317741372
#317741292
#317741192
#317740582
#317740512
C8×D10842
#317738922
#103×D2712
C4×Q16×C2712
#3300271