Finite Group Databasebeta

← #46848307

Subgroups — #46848307

Order 34688.

408 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774611C211yesmaximal, characteristic, complemented
2#31774566C211yesmaximal, characteristic, complemented
3#31774242C211yesmaximal, characteristic, complemented
4#31774222C211yesmaximal, characteristic, complemented
5#31774220C211yesmaximal, characteristic, complemented
6#31774190C211yesmaximal, characteristic, complemented
7#31774131C211yesmaximal, characteristic, complemented
8#31774130C211yesmaximal, characteristic, complemented
9#31774090C211yesmaximal, characteristic, complemented
10#31774041C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12(C22⋊D271)⋊C8C211yesmaximal, characteristic, complemented
13#31773893C211yesmaximal, characteristic, complemented
14#31773892C211yesmaximal, characteristic, complemented
15#449×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864639C2211yescharacteristic
18C22×C271⋊C8C2211yescharacteristic
19C2×C8×D271C2211yescharacteristic
20#22576159C2211yescharacteristic
21#22576156—21no—
22#22576156C2211yescharacteristic
23#22576147—21no—
24#22576146—21no—
25#22576134—21no—
26#22576134C2211yescharacteristic
27#22576132—21no—
28#22576131—21no—
29#22576131C2211yescharacteristic
30#22576131C2211yescharacteristic
31Q8×Dic271C2211yescharacteristic
32D4×Dic271C2211yescharacteristic
33C4×C22⋊D271—21no—
34C4×C22⋊D271C2211yescharacteristic
35#22576119C2211yescharacteristic
36#22576118C2211yescharacteristic
37#22576116—21no—
38#22576094C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576092C2211yescharacteristic
41Dic271⋊C8C2211yescharacteristic
42#22576084C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576082C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49#22576080C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×#41×C271C2211yescharacteristic
53C2×SD16×C271C2211yescharacteristic
54C2×D8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#103×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58C22⋊C8×C271C2211yescharacteristic
59#16864655—22no—
60#16864655—41no—
61C2×Dic542—21no—
62C2×D1084—21no—
63C2×D1084C2311yescharacteristic
64C2×C22⋊D271—21no—
65C2×C22⋊D271—21no—
66C2×C22⋊D271—21no—
67C2×C22⋊D271—21no—
68C2×C22⋊D271—21no—
69C2×C22⋊D271D411yescharacteristic
70C22×Dic271—21no—
71C2×C4×D271—21no—
72C2×C4×D271C2311yescharacteristic
73#16864639—41no—
74C2×C271⋊C8—41no—
75C2×C271⋊C8C2311yescharacteristic
76C2×C271⋊C8C2311yescharacteristic
77C8×D271—41no—
78C4⋊Dic271—21no—
79C4⋊Dic271C2311yescharacteristic
80C4⋊Dic271C2311yescharacteristic
81C4⋊Dic271C2311yescharacteristic
82Dic271⋊C4—21no—
83Dic271⋊C4D411yescharacteristic
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271—21no—
88C22⋊Dic271D411yescharacteristic
89(C2×D271)⋊C4—21no—
90(C2×D271)⋊C4—21no—
91(C2×D271)⋊C4D411yescharacteristic
92C4×Dic271—21no—
93C4×Dic271—21no—
94C4×Dic271—21no—
95C4×Dic271C2311yescharacteristic
96#41×C271—22no—
97#41×C271—41no—
98C2×Q8×C271C2311yescharacteristic
99C2×D4×C271—21no—
100C2×D4×C271—21no—
101C2×D4×C271C2311yescharacteristic
102C2×D4×C271C2311yescharacteristic
103C22×C4×C271—21no—
104C22×C4×C271C2311yescharacteristic
105SD16×C271—22no—
106D8×C271—41no—
107C2×C8×C271C2311yescharacteristic
108C2×C8×C271C2311yescharacteristic
109C4⋊C4×C271C2311yescharacteristic
110C22⋊C4×C271—21no—
111C22×D271—21no—
112C22×D271C2×D411yescharacteristic
113Dic542—81no—
114D1084—21no—
115D1084—21no—
116D1084—42no—
117C22⋊D271—22no—
118C22⋊D271—42no—
119C22⋊D271—81no—
120C22⋊D271—81no—
121C22⋊D271—81no—
122C22⋊D271—81no—
123C2×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—42no—
127C2×Dic271C2×D411yescharacteristic
128C2×Dic271C2×D411yescharacteristic
129C2×Dic271C2×D411yescharacteristic
130C4×D271—21no—
131C4×D271—21no—
132C4×D271—42no—
133C271⋊C8—21no—
134C271⋊C8—21no—
135C23×C271—21no—
136C23×C271—21no—
137C23×C271C2×D411yescharacteristic
138Q8×C271—21no—
139Q8×C271#4112yes—
140D4×C271—21no—
141D4×C271—21no—
142D4×C271—21no—
143D4×C271—21no—
144D4×C271—42no—
145D4×C271—81no—
146C2×C4×C271—21no—
147C2×C4×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—42no—
151C2×C4×C271C2×D411yescharacteristic
152C2×C4×C271C2411yescharacteristic
153C8×C271—21no—
154C8×C271—21no—
155C22×C271—21no—
156C22×C271—21no—
157C22×C271—21no—
158C22×C271—41no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271—41no—
164C22×C271C22×D411yescharacteristic
165C2×D271—21no—
166C2×D271—21no—
167C2×D271—21no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C4×C271—21no—
172C4×C271—22no—
173C4×C271—41no—
174C4×C271C2×#4111yescharacteristic
175C4×C271H(5,𝔽2)11yescharacteristic
176Dic271—21no—
177Dic271—21no—
178Dic271—41no—
179Dic271—41no—
180Dic271—41no—
181Dic271#13512yes—
182C2×C271—41no—
183C2×C271—81no—
184C2×C271—81no—
185C2×C271#54611yescharacteristic
186C2×C271C2×#13511yescharacteristic
187C2×C271#57611yescharacteristic
188D271—41no—
189D271—81no—
190C271#330511yescharacteristic, complemented, Sylow
191#3305—2711nomaximal, Sylow
192#555—2711no—
193#550—2711no—
194#486—2711no—
195#482—2711no—
196#482—2711no—
197#477—2711no—
198#466—2711no—
199#466—2711no—
200#459—2711no—
201#449—2711no—
202#449D27111yescharacteristic, complemented
203C4×SD16—2711no—
204C8×D4—2711no—
205#416—2711no—
206#416—2711no—
207C2×#41—2711no—
208C2×#41C2×D27111yescharacteristic
209C2×SD16C2×D27111yescharacteristic
210C2×D8—2711no—
211C2×D8C2×D27111yescharacteristic
212C22×C8—2711no—
213C22×C8—2711no—
214C4⋊Q8—2711no—
215#127—2711no—
216#127—5421no—
217#124—5421no—
218#124—5421no—
219#121—2711no—
220#121—2711no—
221#121—2711no—
222#121—5421no—
223#121—5421no—
224#121—5421no—
225#121C2×D27111yescharacteristic
226C4×Q8—2711no—
227C4×D4—2711no—
228C4×D4—2711no—
229C4×D4—5421no—
230#117—2711no—
231#117—2711no—
232#117—5421no—
233#107—2711no—
234#107—2711no—
235#106—2711no—
236C4⋊C8—2711no—
237#103—2711no—
238#103—2711no—
239#103C2×D27111yescharacteristic
240#102—2711no—
241#102—2711no—
242#102—2711no—
243#102—2711no—
244#102—2711no—
245#102—2711no—
246#102C2×D27111yescharacteristic
247C22⋊C8—2711no—
248C22⋊C8C2×D27111yescharacteristic
249C4×C8—2711no—
250#41—22no—
251#41—41no—
252#41—5422no—
253#41—10841no—
254C2×Q8—5421no—
255C2×Q8C22×D27111yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—2711no—
259C2×D4—2711no—
260C2×D4—5421no—
261C2×D4—5421no—
262C2×D4—5421no—
263C2×D4—5421no—
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4C22×D27111yescharacteristic
267C2×D4C22×D27111yescharacteristic
268C22×C4—21no—
269C22×C4—2711no—
270C22×C4—5421no—
271C22×C4—5421no—
272C22×C4C22×D27111yescharacteristic
273SD16—22no—
274D8—41no—
275D8—10841no—
276C2×C8—2711no—
277C2×C8—2711no—
278C2×C8—10841no—
279C2×C8—10841no—
280C2×C8C22×D27111yescharacteristic
281C2×C8C22×D27111yescharacteristic
282C4⋊C4—2711no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—2711no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4C22×D27111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—2711no—
291C22⋊C4—2711no—
292C22⋊C4—5421no—
293C22⋊C4—5421no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C42—2711no—
299C42—5421no—
300C42—5421no—
301C42—5421no—
302C23—21no—
303C23—21no—
304C23—2711no—
305C23—5421no—
306C23D4×D27111yescharacteristic
307Q8—21no—
308Q8—21681no—
309Q8#1686465312yes—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—81no—
316D4—5422no—
317D4—5421no—
318D4—5421no—
319D4—10842no—
320D4—10842no—
321D4—21681no—
322D4—21681no—
323D4—21681no—
324D4—21681no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—2711no—
331C2×C4—2711no—
332C2×C4—2711no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—10842no—
339C2×C4—10842no—
340C2×C4D4×D27111yescharacteristic
341C2×C4C23×D27111yescharacteristic
342C8—21no—
343C8—21no—
344C8—5421no—
345C8—5421no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—5421no—
356C22—5421no—
357C22—5421no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22C2×D4×D27111yescharacteristic
362C4—21no—
363C4—22no—
364C4—41no—
365C4—2712no—
366C4—5421no—
367C4—5421no—
368C4—10841no—
369C4—10841no—
370C4—10841no—
371C4C2×#1686465311yescharacteristic
372C4#2257620511yescharacteristic
373C2—41no—
374C2—81no—
375C2—81no—
376C2—10841no—
377C2—21681no—
378C2#3177453411yescharacteristic
379C2#135×D27111yescharacteristic
380C2#3177470111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#449×C271C2
socleC22×C271C22×D4
hypercenter#449D271
hypocenterC271#3305
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31774611C2semidirect
2. #31774566C2semidirect
3. #31774242C2semidirect
4. #31774222C2semidirect
5. #31774220C2semidirect
6. #31774190C2semidirect
7. #31774131C2semidirect
8. #31774130C2semidirect
9. #31774090C2semidirect
10. #31774041C2semidirect
11. SD16×Dic271C2semidirect
12. (C22⋊D271)⋊C8C2semidirect
13. #31773893C2semidirect
14. #31773892C2semidirect
15. #449×C271C2semidirect
190. C271#3305semidirect
202. #449D271semidirect

Maximal subgroups

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

subgroupindex
#317746112
#317745662
#317742422
#317742222
#317742202
#317741902
#317741312
#317741302
#317740902
#317740412
SD16×Dic2712
(C22⋊D271)⋊C82
#317738932
#317738922
#449×C2712
#3305271