Finite Group Databasebeta

← #46848339

Subgroups — #46848339

Order 34688.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774600C211yesmaximal, characteristic, complemented
2#31774550C211yesmaximal, characteristic, complemented
3#31774236C211yesmaximal, characteristic, complemented
4#31774226C211yesmaximal, characteristic, complemented
5#31774194C211yesmaximal, characteristic, complemented
6#31774146C211yesmaximal, characteristic, complemented
7#31774129C211yesmaximal, characteristic, complemented
8#31774099C211yesmaximal, characteristic, complemented
9#31774045C211yesmaximal, characteristic, complemented
10#31774045C211yesmaximal, characteristic, complemented
11#31774010C211yesmaximal, characteristic, complemented
12#31773970C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#31773892C211yesmaximal, characteristic, complemented
15#491×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×#16864641C2211yescharacteristic
19C2×#16864641C2211yescharacteristic
20C2×D2168C2211yescharacteristic
21C2×#16864637C2211yescharacteristic
22C2×C8×D271C2211yescharacteristic
23#22576155—21no—
24#22576145—21no—
25#22576145—21no—
26#22576145—21no—
27#22576136C2211yescharacteristic
28#22576136C2211yescharacteristic
29#22576135—21no—
30#22576135—21no—
31#22576135C2211yescharacteristic
32#22576135C2211yescharacteristic
33C4×D1084C2211yescharacteristic
34C4×D1084C2211yescharacteristic
35C4×D1084C2211yescharacteristic
36#22576122—21no—
37#22576118C2211yescharacteristic
38C4⋊C4×D271—21no—
39C4⋊C4×D271C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576092C2211yescharacteristic
42#22576091C2211yescharacteristic
43C4⋊(C271⋊C8)C2211yescharacteristic
44#22576084C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576082C2211yescharacteristic
47#22576082C2211yescharacteristic
48#22576080C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52C4⋊Q8×C271C2211yescharacteristic
53#125×C271C2211yescharacteristic
54#103×C271C2211yescharacteristic
55#103×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#97×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—22no—
61#16864655—41no—
62#16864655—41no—
63C2×D1084—21no—
64C2×D1084—21no—
65C2×D1084—21no—
66C2×D1084—21no—
67C2×D1084D411yescharacteristic
68C2×D1084C2311yescharacteristic
69C2×D1084C2311yescharacteristic
70C2×C4×D271—21no—
71C2×C4×D271—21no—
72C2×C4×D271—21no—
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271C2311yescharacteristic
76C2×C4×D271C2311yescharacteristic
77#16864641—41no—
78#16864641—41no—
79D2168—22no—
80#16864637—41no—
81C2×C271⋊C8C2311yescharacteristic
82C2×C271⋊C8C2311yescharacteristic
83C8×D271—41no—
84C4⋊Dic271D411yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86C4⋊Dic271C2311yescharacteristic
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊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)⋊C4—21no—
97(C2×D271)⋊C4D411yescharacteristic
98(C2×D271)⋊C4D411yescharacteristic
99C4×Dic271—21no—
100C2×Q8×C271C2311yescharacteristic
101C2×Q8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C2×C8×C271C2311yescharacteristic
104C4⋊C4×C271—21no—
105C4⋊C4×C271—21no—
106C4⋊C4×C271—21no—
107C4⋊C4×C271—21no—
108C4⋊C4×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112C22×D271—21no—
113C22×D271C2×D411yescharacteristic
114C22×D271C2×D411yescharacteristic
115D1084—21no—
116D1084—21no—
117D1084—21no—
118D1084—41no—
119D1084—42no—
120D1084—42no—
121D1084—42no—
122D1084—81no—
123D1084#4112yes—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271C2×D411yescharacteristic
127C2×Dic271C2×D411yescharacteristic
128C4×D271—21no—
129C4×D271—21no—
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C4×D271—42no—
134C4×D271—42no—
135C4×D271—81no—
136C4×D271—81no—
137C271⋊C8—21no—
138C271⋊C8—21no—
139Q8×C271—21no—
140Q8×C271—21no—
141Q8×C271—21no—
142Q8×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2411yescharacteristic
150C8×C271—21no—
151C8×C271—21no—
152C22×C271C22×D411yescharacteristic
153C2×D271—21no—
154C2×D271—21no—
155C2×D271—22no—
156C2×D271—21no—
157C2×D271—21no—
158C2×D271—21no—
159C2×D271—21no—
160C2×D271—41no—
161C2×D271—41no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—41no—
168C4×C271—41no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271C2×#4111yescharacteristic
172C4×C271H(5,𝔽2)11yescharacteristic
173Dic271—21no—
174Dic271—21no—
175Dic271—41no—
176Dic271—41no—
177C2×C271#54611yescharacteristic
178C2×C271#57511yescharacteristic
179C2×C271#57711yescharacteristic
180D271—41no—
181D271—41no—
182D271—42no—
183D271—81no—
184C271#330711yescharacteristic, complemented, Sylow
185#3307—2711nomaximal, Sylow
186#553—2711no—
187#547—2711no—
188#491D27111yescharacteristic, complemented
189#485—2711no—
190#483—2711no—
191#478—2711no—
192#468—2711no—
193#465—2711no—
194#461—2711no—
195#449—2711no—
196#449—2711no—
197#442—2711no—
198#435—2711no—
199#417—2711no—
200#416—2711no—
201C2×#41—2711no—
202C2×#41—2711no—
203C2×SD16—2711no—
204C2×SD16—2711no—
205C2×D8—2711no—
206C2×M16—2711no—
207C22×C8—2711no—
208C4⋊Q8C2×D27111yescharacteristic
209#126—5421no—
210#125C2×D27111yescharacteristic
211#123—5421no—
212#123—5421no—
213#123—5421no—
214C22⋊Q8—2711no—
215C22⋊Q8—2711no—
216#121—2711no—
217#121—2711no—
218#121—5421no—
219#121—5421no—
220C4×D4—2711no—
221C4×D4—2711no—
222C4×D4—2711no—
223C4×D4—5421no—
224#117—2711no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—5421no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103C2×D27111yescharacteristic
234#103C2×D27111yescharacteristic
235#103C2×D27111yescharacteristic
236#103C2×D27111yescharacteristic
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241C22⋊C8—2711no—
242C22⋊C8—2711no—
243#97C2×D27111yescharacteristic
244#41—5422no—
245#41—5422no—
246#41—10841no—
247#41—10841no—
248C2×Q8C22×D27111yescharacteristic
249C2×Q8C22×D27111yescharacteristic
250C2×D4—2711no—
251C2×D4—2711no—
252C2×D4—2711no—
253C2×D4—5421no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4—5421no—
257C22×C4—2711no—
258C22×C4—2711no—
259C22×C4—5421no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4—5421no—
263C22×C4—5421no—
264SD16—10841no—
265SD16—10841no—
266D8—5422no—
267M16—10841no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—10841no—
271C2×C8C22×D27111yescharacteristic
272C2×C8C22×D27111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—2711no—
278C4⋊C4—2711no—
279C4⋊C4—2711no—
280C4⋊C4—5421no—
281C4⋊C4—5421no—
282C4⋊C4—5421no—
283C4⋊C4C22×D27111yescharacteristic
284C4⋊C4C22×D27111yescharacteristic
285C22⋊C4—2711no—
286C22⋊C4—2711no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4—5421no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C42—5421no—
295C42C22×D27111yescharacteristic
296C23—2711no—
297C23—2711no—
298C23—5421no—
299C23—5421no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304D4—2712no—
305D4—5421no—
306D4—5421no—
307D4—5421no—
308D4—10841no—
309D4—10842no—
310D4—10842no—
311D4—10842no—
312D4—21681no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—2711no—
318C2×C4—2711no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—10842no—
326C2×C4—10842no—
327C2×C4—10842no—
328C2×C4—21681no—
329C2×C4—21681no—
330C2×C4D4×D27111yescharacteristic
331C2×C4D4×D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—21no—
334C8—21no—
335C8—5421no—
336C8—5421no—
337C22—5421no—
338C22—5421no—
339C22—5422no—
340C22—5421no—
341C22—5421no—
342C22—5421no—
343C22—5421no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22—10841no—
349C22C2×D4×D27111yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—5421no—
357C4—5421no—
358C4—10841no—
359C4—10841no—
360C4#41×D27111yescharacteristic
361C4#2257620611yescharacteristic
362C2—10841no—
363C2—10841no—
364C2—10842no—
365C2—21681no—
366C2#3177454011yescharacteristic
367C2#3177469611yescharacteristic
368C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#491×C271C2
socleC22×C271C22×D4
hypercenter#491D271
hypocenterC271#3307
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #31774600C2semidirect
2. #31774550C2semidirect
3. #31774236C2semidirect
4. #31774226C2semidirect
5. #31774194C2semidirect
6. #31774146C2semidirect
7. #31774129C2semidirect
8. #31774099C2semidirect
9. #31774045C2semidirect
10. #31774045C2semidirect
11. #31774010C2semidirect
12. #31773970C2semidirect
13. #31773897C2semidirect
14. #31773892C2semidirect
15. #491×C271C2semidirect
184. C271#3307semidirect
188. #491D271semidirect

Maximal subgroups

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

subgroupindex
#317746002
#317745502
#317742362
#317742262
#317741942
#317741462
#317741292
#317740992
#317740452
#317740102
#317739702
#317738972
#317738922
#491×C2712
#3307271