Finite Group Databasebeta

← #46848260

Subgroups — #46848260

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#31774556C211yesmaximal, characteristic, complemented
2#31774254C211yesmaximal, characteristic, complemented
3#31774238C211yesmaximal, characteristic, complemented
4#31774215C211yesmaximal, characteristic, complemented
5#31774201C211yesmaximal, characteristic, complemented
6#31774131C211yesmaximal, characteristic, complemented
7#31774124C211yesmaximal, characteristic, complemented
8#31774112C211yesmaximal, characteristic, complemented
9#31774040C211yesmaximal, characteristic, complemented
10#31774035C211yesmaximal, characteristic, complemented
11C4×#16864642C211yesmaximal, characteristic, complemented
12D4×C271⋊C8C211yesmaximal, characteristic, complemented
13#31773891C211yesmaximal, characteristic, complemented
14C2×#22576083C211yesmaximal, characteristic, complemented
15#553×C271C211yesmaximal, characteristic, complemented
16C2×#16864654C2211yescharacteristic
17C22×Dic542C2211yescharacteristic
18C2×#16864642C2211yescharacteristic
19C2×#16864642C2211yescharacteristic
20C2×#16864639C2211yescharacteristic
21C22×C271⋊C8C2211yescharacteristic
22C22×C271⋊C8C2211yescharacteristic
23#22576148C2211yescharacteristic
24#22576146—21no—
25#22576139C2211yescharacteristic
26#22576138—21no—
27#22576134C2211yescharacteristic
28#22576132—21no—
29C4×Dic542C2211yescharacteristic
30(C22⋊D271)⋊C4—21no—
31#22576093C2211yescharacteristic
32#22576091C2211yescharacteristic
33#22576091C2211yescharacteristic
34C4⋊(C271⋊C8)C2211yescharacteristic
35#22576083C2211yescharacteristic
36#22576083C2211yescharacteristic
37#22576083C2211yescharacteristic
38#22576083C2211yescharacteristic
39#22576083C2211yescharacteristic
40#22576083C2211yescharacteristic
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
59#16864654—22no—
60#16864654—41no—
61C2×Dic542—22no—
62C2×Dic542—21no—
63C2×Dic542—41no—
64C2×Dic542C2311yescharacteristic
65C2×Dic542C2311yescharacteristic
66C2×D1084C2311yescharacteristic
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C22×Dic271—21no—
70C2×C4×D271—21no—
71#16864642—22no—
72#16864642—41no—
73#16864639—41no—
74C2×C271⋊C8—41no—
75C2×C271⋊C8—41no—
76C2×C271⋊C8C2311yescharacteristic
77C2×C271⋊C8C2311yescharacteristic
78C2×C271⋊C8C2311yescharacteristic
79C2×C271⋊C8C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81Dic271⋊C4—21no—
82Dic271⋊C4—21no—
83C22⋊Dic271—21no—
84(C2×D271)⋊C4—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87C4×Dic271—21no—
88C2×D4×C271—21no—
89C2×D4×C271—21no—
90C2×D4×C271D411yescharacteristic
91C2×D4×C271C2311yescharacteristic
92C22×C4×C271—21no—
93C22×C4×C271—21no—
94C22×C4×C271C2311yescharacteristic
95C22×C4×C271C2311yescharacteristic
96C4⋊C4×C271—21no—
97C4⋊C4×C271—21no—
98C4⋊C4×C271—21no—
99C4⋊C4×C271D411yescharacteristic
100C4⋊C4×C271C2311yescharacteristic
101C4⋊C4×C271C2311yescharacteristic
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—
112Dic542—21no—
113Dic542—21no—
114Dic542—21no—
115Dic542—42no—
116Dic542#4112yes—
117D1084—21no—
118D1084—21no—
119C22⋊D271—42no—
120C22⋊D271—42no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271—21no—
124C2×Dic271—42no—
125C4×D271—42no—
126C271⋊C8—21no—
127C271⋊C8—21no—
128C271⋊C8—21no—
129C271⋊C8—21no—
130C23×C271—21no—
131C23×C271C2×D411yescharacteristic
132C23×C271C2×D411yescharacteristic
133D4×C271—22no—
134D4×C271—21no—
135D4×C271—21no—
136D4×C271—42no—
137D4×C271—81no—
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—81no—
146C2×C4×C271—81no—
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—21no—
168C4×C271—21no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271C2×#4111yescharacteristic
173C4×C271H(5,𝔽2)11yescharacteristic
174Dic271—22no—
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#553D27111yescharacteristic, complemented
187#548—2711no—
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×#41—2711no—
202C22×Q8—2711no—
203C2×SD16—2711no—
204C2×SD16—2711no—
205C2×D8—2711no—
206C22×C8—2711no—
207C22×C8—2711no—
208#126—21no—
209#125C2×D27111yescharacteristic
210#124—2711no—
211#124—5421no—
212#123—21no—
213C22⋊Q8—2711no—
214C22⋊Q8—5421no—
215#121—21no—
216#121—2711no—
217#121—5421no—
218#121C2×D27111yescharacteristic
219#121C2×D27111yescharacteristic
220C4×Q8—2711no—
221C4×D4—21no—
222C4×D4—5421no—
223C4×D4C2×D27111yescharacteristic
224C4×D4C2×D27111yescharacteristic
225#117C2×D27111yescharacteristic
226C2×C4⋊C4C2×D27111yescharacteristic
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—5422no—
245#41—10841no—
246C2×Q8—2711no—
247C2×Q8—2711no—
248C2×Q8—5422no—
249C2×Q8—5421no—
250C2×Q8—10841no—
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—21no—
276C4⋊C4—2711no—
277C4⋊C4—5421no—
278C4⋊C4—5421no—
279C4⋊C4C22⋊D27111yescharacteristic
280C4⋊C4C22×D27111yescharacteristic
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—2712no—
301Q8—5421no—
302Q8—5421no—
303Q8—5421no—
304Q8—10842no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—5421no—
311D4—5421no—
312D4—10842no—
313D4—10842no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—10842no—
327C2×C4—10842no—
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—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—5422no—
358C4—10841no—
359C4—10841no—
360C4#41×D27111yescharacteristic
361C4#2257620611yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—21681no—
366C2#3177453511yescharacteristic
367C2C2×#2257618011yescharacteristic
368C2#3177470811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#553×C271C2
socleC22×C271C22×D4
hypercenter#553D271
hypocenterC271#3301
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. #31774556C2semidirect
2. #31774254C2semidirect
3. #31774238C2semidirect
4. #31774215C2semidirect
5. #31774201C2semidirect
6. #31774131C2semidirect
7. #31774124C2semidirect
8. #31774112C2semidirect
9. #31774040C2semidirect
10. #31774035C2semidirect
11. C4×#16864642C2semidirect
12. D4×C271⋊C8C2semidirect
13. #31773891C2semidirect
14. C2×#22576083C2semidirect
15. #553×C271C2semidirect
184. C271#3301semidirect
186. #553D271semidirect

Maximal subgroups

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

subgroupindex
#317745562
#317742542
#317742382
#317742152
#317742012
#317741312
#317741242
#317741122
#317740402
#317740352
C4×#168646422
D4×C271⋊C82
#317738912
C2×#225760832
#553×C2712
#3301271