Finite Group Databasebeta

← #46846467

Subgroups — #46846467

Order 34688.

392 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
1C4×#16864654C211yesmaximal, characteristic, complemented
2#31774235C211yesmaximal, characteristic, complemented
3#31774218C211yesmaximal, characteristic, complemented
4#31774201C211yesmaximal, characteristic, complemented
5#31774194C211yesmaximal, characteristic, complemented
6#31774116C211yesmaximal, characteristic, complemented
7#31774109C211yesmaximal, characteristic, complemented
8#31774057C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic
10#31774007C211yesmaximal, characteristic, complemented
11#31774002C211yesmaximal, characteristic, complemented
12#31773997C211yesmaximal, characteristic, complemented
13#31773995C211yesmaximal, characteristic, complemented
14#31773954C211yesmaximal, characteristic, complemented
15#529×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C2×#16864644C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C4×Dic542C2211yescharacteristic
22C4×Dic542C2211yescharacteristic
23C4×D1084C2211yescharacteristic
24C4×D1084C2211yescharacteristic
25C4×C22⋊D271—21no—
26C4×C22⋊D271—21no—
27#22576119C2211yescharacteristic
28#22576117—21no—
29C42×D271—21no—
30#22576093C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35C4⋊(C271⋊C8)C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576083C2211yescharacteristic
39#22576083C2211yescharacteristic
40#22576081C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44C22⋊(C271⋊C8)C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46#22576071C2211yescharacteristic
47#22576071C2211yescharacteristic
48#126×C271—21no—
49#125×C271C2211yescharacteristic
50#124×C271C2211yescharacteristic
51#123×C271—21no—
52C22⋊Q8×C271C2211yescharacteristic
53#121×C271C2211yescharacteristic
54C4×Q8×C271C2211yescharacteristic
55C4×D4×C271—21no—
56C4×D4×C271C2211yescharacteristic
57#117×C271—21no—
58C2×C42×C271C2211yescharacteristic
59#16864654—22no—
60#16864654—22no—
61C2×Dic542C2311yescharacteristic
62C2×D1084C2311yescharacteristic
63C2×C22⋊D271—21no—
64C2×C4×D271—21no—
65C2×C4×D271—21no—
66C2×C4×D271—21no—
67#16864644—22no—
68#16864642—22no—
69#16864641—22no—
70#16864639—22no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C2×C271⋊C8C2311yescharacteristic
74C2×C271⋊C8C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79C22⋊Dic271—21no—
80(C2×D271)⋊C4—21no—
81(C2×D271)⋊C4—21no—
82C4×Dic271—21no—
83C4×Dic271—21no—
84C4×Dic271—21no—
85C2×Q8×C271C2311yescharacteristic
86C2×D4×C271—21no—
87C2×D4×C271C2311yescharacteristic
88C22×C4×C271—21no—
89C22×C4×C271D411yescharacteristic
90C22×C4×C271D411yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C4⋊C4×C271—21no—
93C4⋊C4×C271—21no—
94C4⋊C4×C271—21no—
95C4⋊C4×C271—21no—
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—
103C42×C271—21no—
104C42×C271D411yescharacteristic
105C42×C271D411yescharacteristic
106C42×C271C2311yescharacteristic
107C42×C271C2311yescharacteristic
108C22×D271—21no—
109Dic542—21no—
110Dic542#4112yes—
111D1084—21no—
112D1084#4112yes—
113C22⋊D271—24no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C4×D271—24no—
118C4×D271—42no—
119C4×D271—42no—
120C271⋊C8—21no—
121C271⋊C8—21no—
122C271⋊C8—21no—
123C271⋊C8—21no—
124C23×C271—21no—
125C23×C271C2×D411yescharacteristic
126Q8×C271—21no—
127Q8×C271—21no—
128D4×C271—21no—
129D4×C271—21no—
130D4×C271—42no—
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—41no—
137C2×C4×C271—41no—
138C2×C4×C271—42no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2×D411yescharacteristic
142C2×C4×C271C2×D411yescharacteristic
143C2×C4×C271C2×D411yescharacteristic
144C2×C4×C271C2411yescharacteristic
145C22×C271—21no—
146C22×C271—21no—
147C22×C271—21no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271C22×D411yescharacteristic
152C2×D271—22no—
153C2×D271—41no—
154C2×D271—41no—
155C4×C271—21no—
156C4×C271—21no—
157C4×C271—21no—
158C4×C271—21no—
159C4×C271—21no—
160C4×C271—41no—
161C4×C271—41no—
162C4×C271—41no—
163C4×C271C2×#4111yescharacteristic
164C4×C271C2×#4111yescharacteristic
165Dic271—22no—
166Dic271—41no—
167Dic271—41no—
168C2×C271—41no—
169C2×C271—81no—
170C2×C271#52511yescharacteristic
171C2×C271#57511yescharacteristic
172C2×C271#57511yescharacteristic
173D271—42no—
174C271#311111yescharacteristic, complemented, Sylow
175#3111—2711nomaximal, Sylow
176#529D27111yescharacteristic, complemented
177C4×#41—2711no—
178#485—2711no—
179#482—2711no—
180#479—2711no—
181#478—2711no—
182#464—2711no—
183#463—2711no—
184#452—2711no—
185#449—2711no—
186#442—2711no—
187#441—2711no—
188#440—2711no—
189#440—2711no—
190#432—2711no—
191C2×#41—2711no—
192C2×Q16—2711no—
193C2×SD16—2711no—
194C2×SD16—2711no—
195C2×D8—2711no—
196#126—21no—
197#125C2×D27111yescharacteristic
198#124C2×D27111yescharacteristic
199#123—21no—
200C22⋊Q8C2×D27111yescharacteristic
201#121C2×D27111yescharacteristic
202C4×Q8—2711no—
203C4×Q8—2711no—
204C4×Q8C2×D27111yescharacteristic
205C4×D4—21no—
206C4×D4—2711no—
207C4×D4—2711no—
208C4×D4—5421no—
209C4×D4—5421no—
210C4×D4C2×D27111yescharacteristic
211#117—21no—
212#117—2711no—
213#117—5421no—
214C2×C42—5421no—
215C2×C42C2×D27111yescharacteristic
216#107—2711no—
217#107—2711no—
218#106—2711no—
219#106—2711no—
220C4⋊C8—2711no—
221C4⋊C8—2711no—
222#103—2711no—
223#103—2711no—
224#103—2711no—
225#103—2711no—
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230C22⋊C8—2711no—
231C22⋊C8—2711no—
232#97—2711no—
233#97—2711no—
234#41—5422no—
235#41—5422no—
236C2×Q8—2711no—
237C2×Q8C22×D27111yescharacteristic
238C2×D4—21no—
239C2×D4—2711no—
240C2×D4—5421no—
241C2×D4C22×D27111yescharacteristic
242C22×C4—21no—
243C22×C4—5421no—
244C22×C4—5421no—
245C22×C4—5421no—
246C22×C4C22⋊D27111yescharacteristic
247C22×C4C22⋊D27111yescharacteristic
248C22×C4C22×D27111yescharacteristic
249Q16—5422no—
250SD16—5422no—
251SD16—5422no—
252D8—5422no—
253C2×C8—2711no—
254C2×C8—2711no—
255C2×C8—2711no—
256C2×C8—2711no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—5421no—
264C4⋊C4—5421no—
265C4⋊C4C22×D27111yescharacteristic
266C4⋊C4C22×D27111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C42—21no—
276C42—5421no—
277C42—5421no—
278C42—5421no—
279C42C22⋊D27111yescharacteristic
280C42C22⋊D27111yescharacteristic
281C42C22×D27111yescharacteristic
282C42C22×D27111yescharacteristic
283C23—21no—
284C23—5421no—
285C23C2×C22⋊D27111yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—2712no—
289Q8—5421no—
290D4—21no—
291D4—21no—
292D4—42no—
293D4—2712no—
294D4—5424no—
295D4—5421no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—5424no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—10842no—
309C2×C4—10842no—
310C2×C4C2×C22⋊D27111yescharacteristic
311C2×C4C2×C22⋊D27111yescharacteristic
312C2×C4C2×C22⋊D27111yescharacteristic
313C2×C4C2×C22⋊D27111yescharacteristic
314C2×C4C2×C22⋊D27111yescharacteristic
315C2×C4C23×D27111yescharacteristic
316C8—5421no—
317C8—5421no—
318C8—5421no—
319C8—5421no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5422no—
327C22—10841no—
328C22—10841no—
329C22C22×C22⋊D27111yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—5422no—
339C4—10841no—
340C4—10841no—
341C4#41×D27111yescharacteristic
342C4#41×D27111yescharacteristic
343C2—41no—
344C2—81no—
345C2—10842no—
346C2#3177440411yescharacteristic
347C2#3177469511yescharacteristic
348C2#3177469511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#529×C271C2
socleC22×C271C22×D4
prime radical#31774040C2
hypercenter#529D271
hypocenterC271#3111
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C4×#16864654C2semidirect
2. #31774235C2semidirect
3. #31774218C2semidirect
4. #31774201C2semidirect
5. #31774194C2semidirect
6. #31774116C2semidirect
7. #31774109C2semidirect
8. #31774057C2semidirect
10. #31774007C2semidirect
11. #31774002C2semidirect
12. #31773997C2semidirect
13. #31773995C2semidirect
14. #31773954C2semidirect
15. #529×C271C2semidirect
174. C271#3111semidirect
176. #529D271semidirect

Maximal subgroups

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

subgroupindex
C4×#168646542
#317742352
#317742182
#317742012
#317741942
#317741162
#317741092
#317740572
#317740402
#317740072
#317740022
#317739972
#317739952
#317739542
#529×C2712
#3111271