Finite Group Databasebeta

← #46848964

Subgroups — #46848964

Order 34688.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774593C211yesmaximal, characteristic, complemented
2#31774296C211yesmaximal, characteristic, complemented
3#31774221C211yesmaximal, characteristic, complemented
4#31774218C211yesmaximal, characteristic, complemented
5#31774210C211yesmaximal, characteristic, complemented
6#31774208C211yesmaximal, characteristic, complemented
7#31774197C211yesmaximal, characteristic, complemented
8#31774194C211yesmaximal, characteristic, complemented
9#31774148C211yesmaximal, characteristic, complemented
10#31774148C211yesmaximal, characteristic, complemented
11C4×#16864639C211yesmaximal, characteristic, complemented
12#31773975C211yesmaximal, characteristic, complemented
13#31773940C211yesmaximal, characteristic, complemented
14#31773937C211yesmaximal, characteristic, complemented
15#553×C271C211yesmaximal, characteristic, complemented
16C2×#16864639C2211yescharacteristic
17C2×#16864638C2211yescharacteristic
18C2×#16864638C2211yescharacteristic
19#22576155—21no—
20#22576152C2211yescharacteristic
21#22576142—21no—
22#22576134C2211yescharacteristic
23#22576134C2211yescharacteristic
24#22576132—21no—
25C4×D1084C2211yescharacteristic
26(C22⋊D271)⋊C4—21no—
27#22576119C2211yescharacteristic
28C2×C4⋊Dic271C2211yescharacteristic
29#22576093C2211yescharacteristic
30#22576093C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576093C2211yescharacteristic
33#22576093C2211yescharacteristic
34#22576091C2211yescharacteristic
35#22576091C2211yescharacteristic
36#22576091C2211yescharacteristic
37#22576091C2211yescharacteristic
38C4⋊(C271⋊C8)C2211yescharacteristic
39#22576081C2211yescharacteristic
40#22576081C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576080C2211yescharacteristic
43#22576080C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47C4×C271⋊C8C2211yescharacteristic
48#126×C271—21no—
49#125×C271C2211yescharacteristic
50#123×C271—21no—
51#121×C271—21no—
52#121×C271C2211yescharacteristic
53#121×C271C2211yescharacteristic
54C4×D4×C271—21no—
55C4×D4×C271C2211yescharacteristic
56C4×D4×C271C2211yescharacteristic
57#117×C271C2211yescharacteristic
58C2×C4⋊C4×C271C2211yescharacteristic
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C22×Dic271—21no—
63C2×C4×D271—21no—
64#16864639—22no—
65#16864638—22no—
66#16864638—22no—
67C2×C271⋊C8C2311yescharacteristic
68C2×C271⋊C8C2311yescharacteristic
69C2×C271⋊C8C2311yescharacteristic
70C2×C271⋊C8C2311yescharacteristic
71C4⋊Dic271—21no—
72C4⋊Dic271C2311yescharacteristic
73C4⋊Dic271C2311yescharacteristic
74C4⋊Dic271C2311yescharacteristic
75Dic271⋊C4—21no—
76Dic271⋊C4—21no—
77C22⋊Dic271—21no—
78(C2×D271)⋊C4—21no—
79(C2×D271)⋊C4—21no—
80(C2×D271)⋊C4—21no—
81C4×Dic271—21no—
82C2×D4×C271—21no—
83C2×D4×C271—21no—
84C2×D4×C271D411yescharacteristic
85C2×D4×C271C2311yescharacteristic
86C22×C4×C271—21no—
87C22×C4×C271—21no—
88C22×C4×C271C2311yescharacteristic
89C22×C4×C271C2311yescharacteristic
90C4⋊C4×C271—21no—
91C4⋊C4×C271—21no—
92C4⋊C4×C271—21no—
93C4⋊C4×C271D411yescharacteristic
94C4⋊C4×C271C2311yescharacteristic
95C4⋊C4×C271C2311yescharacteristic
96C4⋊C4×C271C2311yescharacteristic
97C22⋊C4×C271—21no—
98C22⋊C4×C271—21no—
99C22⋊C4×C271—21no—
100C22⋊C4×C271—21no—
101C22⋊C4×C271D411yescharacteristic
102C22⋊C4×C271D411yescharacteristic
103C42×C271—21no—
104C42×C271C2311yescharacteristic
105C22×D271—21no—
106D1084—21no—
107D1084—21no—
108C22⋊D271—42no—
109C22⋊D271—81no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—21no—
113C2×Dic271—81no—
114C4×D271—81no—
115C271⋊C8—21no—
116C271⋊C8—21no—
117C271⋊C8—21no—
118C271⋊C8#4112yes—
119C23×C271—21no—
120C23×C271C2×D411yescharacteristic
121C23×C271C2×D411yescharacteristic
122D4×C271—21no—
123D4×C271—21no—
124D4×C271—41no—
125D4×C271—42no—
126D4×C271—81no—
127C2×C4×C271—21no—
128C2×C4×C271—21no—
129C2×C4×C271—21no—
130C2×C4×C271—21no—
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—21no—
134C2×C4×C271—81no—
135C2×C4×C271—81no—
136C2×C4×C271C2×D411yescharacteristic
137C2×C4×C271C2×D411yescharacteristic
138C2×C4×C271C2×D411yescharacteristic
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2411yescharacteristic
141C22×C271—21no—
142C22×C271—21no—
143C22×C271—21no—
144C22×C271—21no—
145C22×C271—21no—
146C22×C271—21no—
147C22×C271—41no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271C22×D411yescharacteristic
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C4×C271—21no—
155C4×C271—21no—
156C4×C271—21no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271—41no—
161C4×C271C2×#4111yescharacteristic
162C4×C2712−1+411yescharacteristic
163Dic271—41no—
164Dic271—41no—
165Dic271—41no—
166C2×C271—41no—
167C2×C271—41no—
168C2×C271—81no—
169C2×C271#54711yescharacteristic
170C2×C271#57511yescharacteristic
171C2×C271#57611yescharacteristic
172D271—81no—
173C271#335511yescharacteristic, complemented, Sylow
174#3355—2711nomaximal, Sylow
175#553—2711no—
176#553D27111yescharacteristic, complemented
177#499—2711no—
178#482—2711no—
179#482—2711no—
180#480—2711no—
181#480—2711no—
182#478—2711no—
183#478—2711no—
184#469—2711no—
185#469—2711no—
186C4×D8—2711no—
187#436—2711no—
188#428—2711no—
189#428—2711no—
190C2×D8—2711no—
191C2×M16—2711no—
192C2×M16—2711no—
193#126—21no—
194#126—5421no—
195#125—2711no—
196#125C2×D27111yescharacteristic
197#123—21no—
198#123—5421no—
199#121—21no—
200#121—2711no—
201#121—2711no—
202#121—5421no—
203#121C2×D27111yescharacteristic
204#121C2×D27111yescharacteristic
205C4×D4—21no—
206C4×D4—2711no—
207C4×D4—5421no—
208C4×D4C2×D27111yescharacteristic
209C4×D4C2×D27111yescharacteristic
210#117—2711no—
211#117C2×D27111yescharacteristic
212C2×C4⋊C4—2711no—
213C2×C4⋊C4C2×D27111yescharacteristic
214#107—2711no—
215#107—2711no—
216#107—2711no—
217#107—2711no—
218#107—2711no—
219#106—2711no—
220#106—2711no—
221#106—2711no—
222#106—2711no—
223C4⋊C8—2711no—
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230C22⋊C8—2711no—
231C22⋊C8—2711no—
232C4×C8—2711no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—2711no—
236C2×D4—5421no—
237C2×D4—5421no—
238C2×D4C22⋊D27111yescharacteristic
239C2×D4C22×D27111yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—5421no—
243C22×C4—5421no—
244C22×C4C22×D27111yescharacteristic
245C22×C4C22×D27111yescharacteristic
246D8—5422no—
247M16—5422no—
248M16—5422no—
249C2×C8—2711no—
250C2×C8—2711no—
251C2×C8—2711no—
252C2×C8—2711no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—2711no—
257C4⋊C4—2711no—
258C4⋊C4—2711no—
259C4⋊C4—5421no—
260C4⋊C4—5421no—
261C4⋊C4—5421no—
262C4⋊C4C22⋊D27111yescharacteristic
263C4⋊C4C22×D27111yescharacteristic
264C4⋊C4C22×D27111yescharacteristic
265C4⋊C4C22×D27111yescharacteristic
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⋊C4C22⋊D27111yescharacteristic
275C22⋊C4C22⋊D27111yescharacteristic
276C42—21no—
277C42—5421no—
278C42C22×D27111yescharacteristic
279C23—21no—
280C23—5421no—
281C23C2×C22⋊D27111yescharacteristic
282C23C2×C22⋊D27111yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—5421no—
289D4—5421no—
290D4—10842no—
291D4—21681no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—5421no—
302C2×C4—5421no—
303C2×C4—5421no—
304C2×C4—21681no—
305C2×C4—21681no—
306C2×C4C2×C22⋊D27111yescharacteristic
307C2×C4C2×C22⋊D27111yescharacteristic
308C2×C4C2×C22⋊D27111yescharacteristic
309C2×C4C2×C22⋊D27111yescharacteristic
310C2×C4C23×D27111yescharacteristic
311C8—2712no—
312C8—5421no—
313C8—5421no—
314C8—5421no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—10841no—
325C22—10841no—
326C22—10841no—
327C22C22×C22⋊D27111yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—10841no—
336C4—10841no—
337C4—10841no—
338C4#41×D27111yescharacteristic
339C4#2257620711yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—21681no—
344C2#3177454611yescharacteristic
345C2#3177469511yescharacteristic
346C2#3177470011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#553×C271C2
socleC22×C271C22×D4
hypercenter#553D271
hypocenterC271#3355
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #31774593C2semidirect
2. #31774296C2semidirect
3. #31774221C2semidirect
4. #31774218C2semidirect
5. #31774210C2semidirect
6. #31774208C2semidirect
7. #31774197C2semidirect
8. #31774194C2semidirect
9. #31774148C2semidirect
10. #31774148C2semidirect
11. C4×#16864639C2semidirect
12. #31773975C2semidirect
13. #31773940C2semidirect
14. #31773937C2semidirect
15. #553×C271C2semidirect
173. C271#3355semidirect
176. #553D271semidirect

Maximal subgroups

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

subgroupindex
#317745932
#317742962
#317742212
#317742182
#317742102
#317742082
#317741972
#317741942
#317741482
C4×#168646392
#317739752
#317739402
#317739372
#553×C2712
#3355271