Finite Group Databasebeta

← #46848954

Subgroups — #46848954

Order 34688.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774611C211yesmaximal, characteristic, complemented
2#31774547C211yesmaximal, characteristic, complemented
3#31774294C211yesmaximal, characteristic, complemented
4#31774219C211yesmaximal, characteristic, complemented
5#31774218C211yesmaximal, characteristic, complemented
6#31774214C211yesmaximal, characteristic, complemented
7#31774213C211yesmaximal, characteristic, complemented
8#31774185C211yesmaximal, characteristic, complemented
9#31774175C211yesmaximal, characteristic, complemented
10#31774147C211yesmaximal, characteristic, complemented
11D8×Dic271C211yesmaximal, characteristic, complemented
12#31773973C211yesmaximal, characteristic, complemented
13#31773939C211yesmaximal, characteristic, complemented
14#31773936C211yesmaximal, characteristic, complemented
15#469×C271C211yesmaximal, characteristic, complemented
16C2×#16864653—21no—
17C2×#16864637C2211yescharacteristic
18#22576159C2211yescharacteristic
19#22576159C2211yescharacteristic
20#22576147—21no—
21#22576146—21no—
22#22576142—21no—
23#22576141—21no—
24#22576137—21no—
25#22576132—21no—
26#22576131C2211yescharacteristic
27#22576131C2211yescharacteristic
28D4×Dic271C2211yescharacteristic
29D4×Dic271C2211yescharacteristic
30C4×C22⋊D271C2211yescharacteristic
31#22576119C2211yescharacteristic
32#22576118C2211yescharacteristic
33#22576116—21no—
34C2×C4⋊Dic271C2211yescharacteristic
35C4⋊C4×D271C2211yescharacteristic
36#22576094C2211yescharacteristic
37#22576094C2211yescharacteristic
38#22576093C2211yescharacteristic
39#22576092C2211yescharacteristic
40#22576092C2211yescharacteristic
41#22576091C2211yescharacteristic
42#22576091C2211yescharacteristic
43#22576091C2211yescharacteristic
44Dic271⋊C8C2211yescharacteristic
45#22576081C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576081C2211yescharacteristic
48#22576081C2211yescharacteristic
49C22⋊(C271⋊C8)C2211yescharacteristic
50(C2×D271)⋊C8C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×D8×C271C2211yescharacteristic
53C2×M16×C271C2211yescharacteristic
54#121×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic
57#102×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59#16864653—44no—
60C2×Dic542—21no—
61C2×Dic542—21no—
62C2×C22⋊D271—21no—
63C2×C22⋊D271—21no—
64C2×C22⋊D271D411yescharacteristic
65C22×Dic271—21no—
66C22×Dic271—21no—
67C22×Dic271—21no—
68C2×C4×D271—21no—
69C2×C4×D271C2311yescharacteristic
70#16864637—22no—
71C2×C271⋊C8C2311yescharacteristic
72C2×C271⋊C8C2311yescharacteristic
73C4⋊Dic271—21no—
74C4⋊Dic271C2311yescharacteristic
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77C4⋊Dic271C2311yescharacteristic
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81Dic271⋊C4D411yescharacteristic
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85C22⋊Dic271—21no—
86C22⋊Dic271—21no—
87C22⋊Dic271D411yescharacteristic
88(C2×D271)⋊C4—21no—
89(C2×D271)⋊C4D411yescharacteristic
90C4×Dic271—21no—
91C4×Dic271—21no—
92C4×Dic271C2311yescharacteristic
93C2×D4×C271—21no—
94C2×D4×C271—21no—
95C2×D4×C271C2311yescharacteristic
96C2×D4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98D8×C271—22no—
99M16×C271—22no—
100C2×C8×C271C2311yescharacteristic
101C2×C8×C271C2311yescharacteristic
102C4⋊C4×C271C2311yescharacteristic
103C4⋊C4×C271C2311yescharacteristic
104C22⋊C4×C271—21no—
105C22⋊C4×C271—21no—
106C22×D271C2×D411yescharacteristic
107Dic542—42no—
108Dic542—81no—
109C22⋊D271—22no—
110C22⋊D271—42no—
111C22⋊D271—81no—
112C2×Dic271—21no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—42no—
117C2×Dic271—42no—
118C2×Dic271—42no—
119C2×Dic271C2×D411yescharacteristic
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C4×D271—21no—
123C4×D271—21no—
124C4×D271—42no—
125C271⋊C8—21no—
126C271⋊C8—21no—
127C23×C271—21no—
128C23×C271—21no—
129C23×C271C2×D411yescharacteristic
130D4×C271—21no—
131D4×C271—21no—
132D4×C271—21no—
133D4×C271—21no—
134D4×C271—42no—
135D4×C271—81no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271C2×D411yescharacteristic
141C2×C4×C271C2411yescharacteristic
142C8×C271—21no—
143C8×C271#4112yescomplemented
144C22×C271—21no—
145C22×C271—21no—
146C22×C271—21no—
147C22×C271—41no—
148C22×C271—41no—
149C22×C271—41no—
150C22×C271—41no—
151C22×C271—41no—
152C22×C271—41no—
153C22×C271C22×D411yescharacteristic
154C2×D271—21no—
155C2×D271—21no—
156C2×D271—21no—
157C4×C271—21no—
158C4×C271—41no—
159C4×C271—41no—
160C4×C271C2×#4111yescharacteristic
161C4×C2712−1+411yescharacteristic
162Dic271—21no—
163Dic271—21no—
164Dic271—41no—
165Dic271—41no—
166Dic271—41no—
167Dic271—41no—
168Dic271AGL(1,ℤ/8)12yes—
169C2×C271—41no—
170C2×C271—81no—
171C2×C271—81no—
172C2×C271#54711yescharacteristic
173C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C271#57711yescharacteristic
175D271—41no—
176C271#335411yescharacteristic, complemented, Sylow
177#3354—2711nomaximal, Sylow
178#555—2711no—
179#547—2711no—
180#498—2711no—
181#482—2711no—
182#482—2711no—
183#481—2711no—
184#481—2711no—
185#476—2711no—
186#474—2711no—
187#469—2711no—
188#469D27111yescharacteristic, complemented
189C4×D8—2711no—
190#436—2711no—
191#428—2711no—
192#428—2711no—
193C2×#41—5421no—
194C2×D8C2×D27111yescharacteristic
195C2×M16—2711no—
196C2×M16C2×D27111yescharacteristic
197C4⋊Q8—2711no—
198C4⋊Q8—2711no—
199#124—5421no—
200#124—5421no—
201#123—5421no—
202#123—5421no—
203C22⋊Q8—5421no—
204#121—2711no—
205#121—2711no—
206#121—5421no—
207#121C2×D27111yescharacteristic
208#121C2×D27111yescharacteristic
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—2711no—
212#117—2711no—
213#117—2711no—
214#117—5421no—
215C2×C4⋊C4—2711no—
216C2×C4⋊C4—2711no—
217#107—2711no—
218#107—2711no—
219#107—2711no—
220#106—2711no—
221#106—2711no—
222#106—2711no—
223#106—2711no—
224#106—2711no—
225#106C2×D27111yescharacteristic
226C4⋊C8—2711no—
227#102—2711no—
228#102—2711no—
229#102—2711no—
230#102—2711no—
231#102C2×D27111yescharacteristic
232#102C2×D27111yescharacteristic
233C22⋊C8—2711no—
234C22⋊C8—2711no—
235C4×C8—2711no—
236#41—10844no—
237C2×Q8—5421no—
238C2×Q8—5421no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—2711no—
242C2×D4—5421no—
243C2×D4—5421no—
244C2×D4C22×D27111yescharacteristic
245C2×D4C22×D27111yescharacteristic
246C22×C4—2711no—
247C22×C4—5421no—
248C22×C4—5421no—
249C22×C4—5421no—
250C22×C4—5421no—
251C22×C4C22×D27111yescharacteristic
252D8—22no—
253M16—22no—
254M16—5422no—
255C2×C8—2711no—
256C2×C8—2711no—
257C2×C8C22×D27111yescharacteristic
258C2×C8C22×D27111yescharacteristic
259C4⋊C4—2711no—
260C4⋊C4—2711no—
261C4⋊C4—2711no—
262C4⋊C4—2711no—
263C4⋊C4—2711no—
264C4⋊C4—5421no—
265C4⋊C4—5421no—
266C4⋊C4—5421no—
267C4⋊C4—5421no—
268C4⋊C4C22×D27111yescharacteristic
269C4⋊C4C22×D27111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2711no—
273C22⋊C4—2711no—
274C22⋊C4—5421no—
275C22⋊C4—5421no—
276C22⋊C4—5421no—
277C22⋊C4—5421no—
278C22⋊C4—5421no—
279C22⋊C4—5421no—
280C42—2711no—
281C42—5421no—
282C42—5421no—
283C23—21no—
284C23—21no—
285C23—2711no—
286C23D4×D27111yescharacteristic
287Q8—10842no—
288Q8—21681no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—5422no—
296D4—10842no—
297D4—21681no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2711no—
303C2×C4—2711no—
304C2×C4—2711no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—5421no—
309C2×C4—5421no—
310C2×C4—5421no—
311C2×C4—10842no—
312C2×C4—10842no—
313C2×C4—10842no—
314C2×C4—10842no—
315C2×C4D4×D27111yescharacteristic
316C2×C4C23×D27111yescharacteristic
317C8—21no—
318C8—5421no—
319C8—5421no—
320C8#1686465312yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—5421no—
331C22—5421no—
332C22—5421no—
333C22C2×D4×D27111yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—2712no—
338C4—5421no—
339C4—5421no—
340C4—10841no—
341C4—10841no—
342C4—10841no—
343C4—10841no—
344C4C2×#1686465311yescharacteristic
345C4#2257620711yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—10841no—
350C2#3177454711yescharacteristic
351C2AGL(1,ℤ/8)×D27111yescharacteristic
352C2#3177470711yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#469×C271C2
socleC22×C271C22×D4
hypercenter#469D271
hypocenterC271#3354
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #31774611C2semidirect
2. #31774547C2semidirect
3. #31774294C2semidirect
4. #31774219C2semidirect
5. #31774218C2semidirect
6. #31774214C2semidirect
7. #31774213C2semidirect
8. #31774185C2semidirect
9. #31774175C2semidirect
10. #31774147C2semidirect
11. D8×Dic271C2semidirect
12. #31773973C2semidirect
13. #31773939C2semidirect
14. #31773936C2semidirect
15. #469×C271C2semidirect
143. C8×C271#41semidirect
176. C271#3354semidirect
188. #469D271semidirect
320. C8#16864653semidirect

Maximal subgroups

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

subgroupindex
#317746112
#317745472
#317742942
#317742192
#317742182
#317742142
#317742132
#317741852
#317741752
#317741472
D8×Dic2712
#317739732
#317739392
#317739362
#469×C2712
#3354271