Finite Group Databasebeta

← #46847829

Subgroups — #46847829

Order 34688.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774518C211yesmaximal, characteristic, complemented
2#31774187C211yesmaximal, characteristic, complemented
3#31774184C211yesmaximal, characteristic, complemented
4#31774182C211yesmaximal, characteristic, complemented
5#31774179C211yesmaximal, characteristic, complemented
6#31774153C211yesmaximal, characteristic, complemented
7#31774148C211yesmaximal, characteristic
8#31774128C211yesmaximal, characteristic, complemented
9#31774124C211yesmaximal, characteristic, complemented
10#31774119C211yesmaximal, characteristic, complemented
11#31774116C211yesmaximal, characteristic, complemented
12#31774112C211yesmaximal, characteristic, complemented
13#31774109C211yesmaximal, characteristic, complemented
14#31773924C211yesmaximal, characteristic, complemented
15#542×C271C211yesmaximal, characteristic, complemented
16C2×#16864644C2211yescharacteristic
17C2×#16864642C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864639C2211yescharacteristic
20C2×#16864638C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22#22576160C2211yescharacteristic
23#22576155—21no—
24#22576148C2211yescharacteristic
25#22576146—21no—
26#22576142—21no—
27#22576139C2211yescharacteristic
28#22576138—21no—
29#22576134C2211yescharacteristic
30C4×Dic542C2211yescharacteristic
31C4×D1084C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34#22576091C2211yescharacteristic
35#22576091C2211yescharacteristic
36C4⋊(C271⋊C8)C2211yescharacteristic
37C4⋊(C271⋊C8)C2211yescharacteristic
38C4⋊(C271⋊C8)C2211yescharacteristic
39C4⋊(C271⋊C8)C2211yescharacteristic
40#22576084C2211yescharacteristic
41#22576084C2211yescharacteristic
42#22576083C2211yescharacteristic
43#22576083C2211yescharacteristic
44#22576081C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576080C2211yescharacteristic
47#22576080C2211yescharacteristic
48C4⋊Q8×C271C2211yescharacteristic
49#126×C271—21no—
50#124×C271—21no—
51#124×C271C2211yescharacteristic
52#123×C271—21no—
53C22⋊Q8×C271—21no—
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56C4×Q8×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic
58#117×C271C2211yescharacteristic
59C2×Dic542—21no—
60C2×Dic542C2311yescharacteristic
61C2×D1084C2311yescharacteristic
62C2×C22⋊D271—21no—
63C2×C4×D271—21no—
64#16864644—41no—
65#16864642—41no—
66#16864641—41no—
67#16864639—41no—
68#16864638—41no—
69C2×C271⋊C8—41no—
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C4⋊Dic271—21no—
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79C22⋊Dic271—21no—
80(C2×D271)⋊C4—21no—
81(C2×D271)⋊C4—21no—
82(C2×D271)⋊C4—21no—
83(C2×D271)⋊C4—21no—
84C4×Dic271—21no—
85C2×Q8×C271—21no—
86C2×Q8×C271C2311yescharacteristic
87C2×D4×C271—21no—
88C2×D4×C271C2311yescharacteristic
89C22×C4×C271—21no—
90C22×C4×C271C2311yescharacteristic
91C4⋊C4×C271—21no—
92C4⋊C4×C271—21no—
93C4⋊C4×C271—21no—
94C4⋊C4×C271D411yescharacteristic
95C4⋊C4×C271D411yescharacteristic
96C4⋊C4×C271C2311yescharacteristic
97C4⋊C4×C271C2311yescharacteristic
98C22⋊C4×C271—21no—
99C22⋊C4×C271—21no—
100C22⋊C4×C271—21no—
101C22⋊C4×C271—21no—
102C22⋊C4×C271—21no—
103C22⋊C4×C271D411yescharacteristic
104C22⋊C4×C271D411yescharacteristic
105C42×C271—21no—
106C42×C271C2311yescharacteristic
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109Dic542—21no—
110Dic542—21no—
111Dic542—81no—
112D1084—21no—
113D1084—21no—
114C22⋊D271—81no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C4×D271—81no—
119C271⋊C8—21no—
120C271⋊C8—21no—
121C271⋊C8—21no—
122C271⋊C8—21no—
123C23×C271—21no—
124C23×C271C2×D411yescharacteristic
125Q8×C271—21no—
126Q8×C271—21no—
127Q8×C271—81no—
128D4×C271—21no—
129D4×C271—21no—
130D4×C271—81no—
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—81no—
137C2×C4×C271C2×D411yescharacteristic
138C2×C4×C271C2×D411yescharacteristic
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2411yescharacteristic
143C22×C271—21no—
144C22×C271—21no—
145C22×C271—21no—
146C22×C271—41no—
147C22×C271—41no—
148C22×C271—41no—
149C22×C271C22×D411yescharacteristic
150C2×D271—41no—
151C2×D271—41no—
152C2×D271—41no—
153C4×C271—21no—
154C4×C271—21no—
155C4×C271—21no—
156C4×C271—21no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271H(5,𝔽2)11yescharacteristic
162C4×C2712−1+411yescharacteristic
163Dic271—41no—
164Dic271—41no—
165Dic271—41no—
166C2×C271—41no—
167C2×C271—81no—
168C2×C271#53711yescharacteristic
169C2×C271#57711yescharacteristic
170C2×C271#57711yescharacteristic
171D271—81no—
172C271#326311yescharacteristic, complemented, Sylow
173#3263—2711nomaximal, Sylow
174#542—2711no—
175#542D27111yescharacteristic, complemented
176#476—2711no—
177#476—2711no—
178#475—2711no—
179#475—2711no—
180#470—2711no—
181#469—2711no—
182#465—2711no—
183#465—2711no—
184#464—2711no—
185#464—2711no—
186#463—2711no—
187#463—2711no—
188#424—2711no—
189C2×Q16—2711no—
190C2×SD16—2711no—
191C2×SD16—2711no—
192C2×D8—2711no—
193C2×M16—2711no—
194C22×C8—2711no—
195C4⋊Q8—2711no—
196C4⋊Q8C2×D27111yescharacteristic
197#126—21no—
198#126—5421no—
199#124—21no—
200#124—2711no—
201#124—5421no—
202#124C2×D27111yescharacteristic
203#123—21no—
204#123—5421no—
205C22⋊Q8—21no—
206C22⋊Q8—2711no—
207C22⋊Q8—5421no—
208C22⋊Q8C2×D27111yescharacteristic
209#121—2711no—
210#121C2×D27111yescharacteristic
211C4×Q8—2711no—
212C4×Q8C2×D27111yescharacteristic
213C4×D4—2711no—
214C4×D4C2×D27111yescharacteristic
215#117C2×D27111yescharacteristic
216#106—2711no—
217#106—2711no—
218#106—2711no—
219#106—2711no—
220C4⋊C8—2711no—
221C4⋊C8—2711no—
222C4⋊C8—2711no—
223C4⋊C8—2711no—
224#103—2711no—
225#103—2711no—
226#103—2711no—
227#103—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102—2711no—
232C2×Q8—21no—
233C2×Q8—2711no—
234C2×Q8—5421no—
235C2×Q8C22×D27111yescharacteristic
236C2×D4—21no—
237C2×D4—2711no—
238C2×D4—5421no—
239C2×D4C22×D27111yescharacteristic
240C22×C4—21no—
241C22×C4—5421no—
242C22×C4C22×D27111yescharacteristic
243Q16—10841no—
244SD16—10841no—
245SD16—10841no—
246D8—10841no—
247M16—10841no—
248C2×C8—2711no—
249C2×C8—2711no—
250C2×C8—2711no—
251C2×C8—2711no—
252C2×C8—10841no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—2711no—
257C4⋊C4—2711no—
258C4⋊C4—5421no—
259C4⋊C4—5421no—
260C4⋊C4—5421no—
261C4⋊C4C22⋊D27111yescharacteristic
262C4⋊C4C22⋊D27111yescharacteristic
263C4⋊C4C22×D27111yescharacteristic
264C4⋊C4C22×D27111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C22⋊C4C22⋊D27111yescharacteristic
276C22⋊C4C22⋊D27111yescharacteristic
277C42—21no—
278C42—5421no—
279C42C22×D27111yescharacteristic
280C42C22×D27111yescharacteristic
281C23—21no—
282C23—5421no—
283C23C2×C22⋊D27111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—5421no—
288Q8—5421no—
289Q8—21681no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—5421no—
294D4—5421no—
295D4—21681no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—5421no—
303C2×C4—5421no—
304C2×C4—5421no—
305C2×C4—21681no—
306C2×C4C2×C22⋊D27111yescharacteristic
307C2×C4C2×C22⋊D27111yescharacteristic
308C2×C4C2×C22⋊D27111yescharacteristic
309C2×C4C2×C22⋊D27111yescharacteristic
310C2×C4C2×C22⋊D27111yescharacteristic
311C2×C4C23×D27111yescharacteristic
312C8—5421no—
313C8—5421no—
314C8—5421no—
315C8—5421no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—10841no—
323C22—10841no—
324C22—10841no—
325C22C22×C22⋊D27111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—10841no—
335C4—10841no—
336C4—10841no—
337C4#2257620611yescharacteristic
338C4#2257620711yescharacteristic
339C2—41no—
340C2—81no—
341C2—21681no—
342C2#3177447511yescharacteristic
343C2#3177470511yescharacteristic
344C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#542×C271C2
socleC22×C271C22×D4
prime radical#31774148C2
hypercenter#542D271
hypocenterC271#3263
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #31774518C2semidirect
2. #31774187C2semidirect
3. #31774184C2semidirect
4. #31774182C2semidirect
5. #31774179C2semidirect
6. #31774153C2semidirect
8. #31774128C2semidirect
9. #31774124C2semidirect
10. #31774119C2semidirect
11. #31774116C2semidirect
12. #31774112C2semidirect
13. #31774109C2semidirect
14. #31773924C2semidirect
15. #542×C271C2semidirect
172. C271#3263semidirect
175. #542D271semidirect

Maximal subgroups

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

subgroupindex
#317745182
#317741872
#317741842
#317741822
#317741792
#317741532
#317741482
#317741282
#317741242
#317741192
#317741162
#317741122
#317741092
#317739242
#542×C2712
#3263271