Finite Group Databasebeta

← #46848266

Subgroups — #46848266

Order 34688.

398 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#31774593C211yesmaximal, characteristic, complemented
2#31774251C211yesmaximal, characteristic, complemented
3#31774235C211yesmaximal, characteristic, complemented
4#31774213C211yesmaximal, characteristic, complemented
5#31774204C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774128C211yesmaximal, characteristic, complemented
8#31774109C211yesmaximal, characteristic, complemented
9#31774042C211yesmaximal, characteristic, complemented
10#31774036C211yesmaximal, characteristic, complemented
11C4×#16864641C211yesmaximal, characteristic, complemented
12D4×C271⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14C2×#22576084C211yesmaximal, characteristic, complemented
15#548×C271C211yesmaximal, characteristic, complemented
16C2×#16864641C2211yescharacteristic
17C2×#16864641C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C22×C271⋊C8C2211yescharacteristic
20C22×C271⋊C8C2211yescharacteristic
21#22576155—21no—
22#22576152C2211yescharacteristic
23#22576142—21no—
24#22576134C2211yescharacteristic
25#22576134C2211yescharacteristic
26#22576132—21no—
27C4×D1084C2211yescharacteristic
28(C22⋊D271)⋊C4—21no—
29#22576119C2211yescharacteristic
30C2×C4⋊Dic271C2211yescharacteristic
31#22576093C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576084C2211yescharacteristic
36#22576084C2211yescharacteristic
37#22576084C2211yescharacteristic
38#22576084C2211yescharacteristic
39#22576084C2211yescharacteristic
40#22576084C2211yescharacteristic
41#22576081C2211yescharacteristic
42#22576081C2211yescharacteristic
43#22576081C2211yescharacteristic
44#22576080C2211yescharacteristic
45C22⋊(C271⋊C8)C2211yescharacteristic
46C22⋊(C271⋊C8)C2211yescharacteristic
47C4×C271⋊C8C2211yescharacteristic
48C2×#41×C271C2211yescharacteristic
49C22×Q8×C271C2211yescharacteristic
50#124×C271—21no—
51#124×C271C2211yescharacteristic
52C22⋊Q8×C271—21no—
53C22⋊Q8×C271C2211yescharacteristic
54#121×C271—21no—
55#121×C271C2211yescharacteristic
56C4×Q8×C271C2211yescharacteristic
57C4×D4×C271—21no—
58C4×D4×C271C2211yescharacteristic
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C2×C22⋊D271—21no—
62C22×Dic271—21no—
63C2×C4×D271—21no—
64#16864641—22no—
65#16864641—41no—
66#16864639—41no—
67C2×C271⋊C8—41no—
68C2×C271⋊C8—41no—
69C2×C271⋊C8C2311yescharacteristic
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271—21no—
74C4⋊Dic271C2311yescharacteristic
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—
83C4×Dic271—21no—
84#41×C271—22no—
85#41×C271—41no—
86C2×Q8×C271—21no—
87C2×Q8×C271—22no—
88C2×Q8×C271—41no—
89C2×Q8×C271C2311yescharacteristic
90C2×Q8×C271C2311yescharacteristic
91C2×D4×C271—21no—
92C2×D4×C271—21no—
93C2×D4×C271D411yescharacteristic
94C2×D4×C271C2311yescharacteristic
95C22×C4×C271—21no—
96C22×C4×C271—21no—
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C4⋊C4×C271—21no—
100C4⋊C4×C271—21no—
101C4⋊C4×C271D411yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C22⋊C4×C271—21no—
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22⋊C4×C271—21no—
107C22⋊C4×C271D411yescharacteristic
108C22⋊C4×C271D411yescharacteristic
109C42×C271—21no—
110C42×C271C2311yescharacteristic
111C22×D271—21no—
112D1084—21no—
113D1084—21no—
114C22⋊D271—42no—
115C22⋊D271—81no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—81no—
120C4×D271—81no—
121C271⋊C8—21no—
122C271⋊C8—21no—
123C271⋊C8—21no—
124C271⋊C8—21no—
125C23×C271—21no—
126C23×C271C2×D411yescharacteristic
127C23×C271C2×D411yescharacteristic
128Q8×C271—21no—
129Q8×C271—21no—
130Q8×C271—21no—
131Q8×C271—42no—
132Q8×C271#4112yes—
133D4×C271—22no—
134D4×C271—21no—
135D4×C271—21no—
136D4×C271—42no—
137D4×C271—42no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—42no—
146C2×C4×C271—42no—
147C2×C4×C271C2×D411yescharacteristic
148C2×C4×C271C2×D411yescharacteristic
149C2×C4×C271C2×D411yescharacteristic
150C2×C4×C271C2×D411yescharacteristic
151C2×C4×C271C2411yescharacteristic
152C22×C271—21no—
153C22×C271—21no—
154C22×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—41no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271#13512yes—
161C22×C271C22×D411yescharacteristic
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C4×C271—21no—
166C4×C271—21no—
167C4×C271—22no—
168C4×C271—21no—
169C4×C271—21no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271C2×#4111yescharacteristic
173C4×C271H(5,𝔽2)11yescharacteristic
174Dic271—41no—
175Dic271—41no—
176Dic271—41no—
177C2×C271—22no—
178C2×C271—41no—
179C2×C271—81no—
180C2×C271#54611yescharacteristic
181C2×C271C2×#13511yescharacteristic
182C2×C271#57711yescharacteristic
183D271—81no—
184C271#330111yescharacteristic, complemented, Sylow
185#3301—2711nomaximal, Sylow
186#553—2711no—
187#548D27111yescharacteristic, complemented
188#488—2711no—
189#485—2711no—
190#481—2711no—
191#479—2711no—
192#466—2711no—
193#465—2711no—
194#463—2711no—
195#449—2711no—
196#448—2711no—
197C4×SD16—2711no—
198C8×D4—2711no—
199#416—2711no—
200C2×#103—2711no—
201C2×#41C2×D27111yescharacteristic
202C22×Q8C2×D27111yescharacteristic
203C2×SD16—2711no—
204C2×SD16—2711no—
205C2×D8—2711no—
206C22×C8—2711no—
207C22×C8—2711no—
208#126—5421no—
209#125—2711no—
210#124—21no—
211#124C2×D27111yescharacteristic
212#123—5421no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D27111yescharacteristic
215#121—21no—
216#121—2711no—
217#121—2711no—
218#121—5421no—
219#121C2×D27111yescharacteristic
220C4×Q8C2×D27111yescharacteristic
221C4×D4—21no—
222C4×D4—2711no—
223C4×D4—5421no—
224C4×D4C2×D27111yescharacteristic
225#117—2711no—
226C2×C4⋊C4—2711no—
227#107—2711no—
228#106—2711no—
229#106—2711no—
230C4⋊C8—2711no—
231#103—2711no—
232#103—2711no—
233#103—2711no—
234#103—2711no—
235#103—2711no—
236#103—2711no—
237#102—2711no—
238#102—2711no—
239#102—2711no—
240#102—2711no—
241C22⋊C8—2711no—
242C22⋊C8—2711no—
243C4×C8—2711no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D27111yescharacteristic
250C2×Q8C22×D27111yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2711no—
254C2×D4—5421no—
255C2×D4—5421no—
256C2×D4C22⋊D27111yescharacteristic
257C2×D4C22×D27111yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—5421no—
261C22×C4—5421no—
262C22×C4C22×D27111yescharacteristic
263C22×C4C22×D27111yescharacteristic
264SD16—5422no—
265SD16—10841no—
266D8—10841no—
267C2×C8—2711no—
268C2×C8—2711no—
269C2×C8—2711no—
270C2×C8—2711no—
271C2×C8—10841no—
272C2×C8—10841no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—2711no—
276C4⋊C4—2711no—
277C4⋊C4—2711no—
278C4⋊C4—5421no—
279C4⋊C4—5421no—
280C4⋊C4—5421no—
281C4⋊C4C22⋊D27111yescharacteristic
282C4⋊C4C22×D27111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5421no—
288C22⋊C4—5421no—
289C22⋊C4—5421no—
290C22⋊C4—5421no—
291C22⋊C4C22⋊D27111yescharacteristic
292C22⋊C4C22⋊D27111yescharacteristic
293C42—21no—
294C42—5421no—
295C42C22×D27111yescharacteristic
296C23—21no—
297C23—5421no—
298C23C2×C22⋊D27111yescharacteristic
299C23C2×C22⋊D27111yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1686465312yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—5421no—
311D4—5421no—
312D4—10842no—
313D4—21681no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—21681no—
327C2×C4—21681no—
328C2×C4C2×C22⋊D27111yescharacteristic
329C2×C4C2×C22⋊D27111yescharacteristic
330C2×C4C2×C22⋊D27111yescharacteristic
331C2×C4C2×C22⋊D27111yescharacteristic
332C2×C4C23×D27111yescharacteristic
333C8—5421no—
334C8—5421no—
335C8—5421no—
336C8—5421no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22#2257618012yes—
349C22C22×C22⋊D27111yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—10841no—
358C4—10841no—
359C4—10841no—
360C4C2×#1686465311yescharacteristic
361C4#2257620511yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—21681no—
366C2#3177454111yescharacteristic
367C2C2×#2257618011yescharacteristic
368C2#3177470511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#548×C271C2
socleC22×C271C22×D4
hypercenter#548D271
hypocenterC271#3301
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #31774593C2semidirect
2. #31774251C2semidirect
3. #31774235C2semidirect
4. #31774213C2semidirect
5. #31774204C2semidirect
6. #31774131C2semidirect
7. #31774128C2semidirect
8. #31774109C2semidirect
9. #31774042C2semidirect
10. #31774036C2semidirect
11. C4×#16864641C2semidirect
12. D4×C271⋊C8C2semidirect
13. #31773893C2semidirect
14. C2×#22576084C2semidirect
15. #548×C271C2semidirect
184. C271#3301semidirect
187. #548D271semidirect

Maximal subgroups

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

subgroupindex
#317745932
#317742512
#317742352
#317742132
#317742042
#317741312
#317741282
#317741092
#317740422
#317740362
C4×#168646412
D4×C271⋊C82
#317738932
C2×#225760842
#548×C2712
#3301271