Finite Group Databasebeta

← #46846927

Subgroups — #46846927

Order 34688.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31774433C211yesmaximal, characteristic, complemented
2#31774428C211yesmaximal, characteristic, complemented
3#31774297C211yesmaximal, characteristic, complemented
4#31774278C211yesmaximal, characteristic, complemented
5#31774253C211yesmaximal, characteristic, complemented
6#31774250C211yesmaximal, characteristic, complemented
7#31774136C211yesmaximal, characteristic, complemented
8#31774131C211yesmaximal, characteristic
9#31774126C211yesmaximal, characteristic, complemented
10#31773989C211yesmaximal, characteristic, complemented
11SD16×Dic271C211yesmaximal, characteristic, complemented
12#31773984C211yesmaximal, characteristic, complemented
13#31773977C211yesmaximal, characteristic, complemented
14C2×C8×Dic271C211yesmaximal, characteristic, complemented
15#465×C271C211yesmaximal, characteristic, complemented
16C2×#16864644C2211yescharacteristic
17C2×#16864643C2211yescharacteristic
18C2×#16864639C2211yescharacteristic
19C22×C271⋊C8C2211yescharacteristic
20#22576155—21no—
21#22576153—21no—
22#22576151C2211yescharacteristic
23#22576151C2211yescharacteristic
24#22576149C2211yescharacteristic
25#22576147C2211yescharacteristic
26#22576144—21no—
27#22576141—21no—
28#22576139C2211yescharacteristic
29#22576134C2211yescharacteristic
30Q8×Dic271C2211yescharacteristic
31#22576125C2211yescharacteristic
32D4×Dic271—21no—
33D4×Dic271C2211yescharacteristic
34#22576122C2211yescharacteristic
35(C22⋊D271)⋊C4—21no—
36#22576118—21no—
37#22576116—21no—
38C2×C4×Dic271C2211yescharacteristic
39#22576093C2211yescharacteristic
40#22576093C2211yescharacteristic
41#22576092C2211yescharacteristic
42#22576085C2211yescharacteristic
43#22576084C2211yescharacteristic
44#22576083C2211yescharacteristic
45#22576082C2211yescharacteristic
46#22576081C2211yescharacteristic
47#22576080C2211yescharacteristic
48C8×Dic271C2211yescharacteristic
49C8×Dic271C2211yescharacteristic
50C8×Dic271C2211yescharacteristic
51C8×Dic271C2211yescharacteristic
52C2×SD16×C271C2211yescharacteristic
53C22×C8×C271C2211yescharacteristic
54C22⋊Q8×C271C2211yescharacteristic
55#121×C271C2211yescharacteristic
56#106×C271C2211yescharacteristic
57#103×C271C2211yescharacteristic
58#102×C271C2211yescharacteristic
59C2×Dic542C2311yescharacteristic
60C2×D1084C2311yescharacteristic
61C2×C22⋊D271—21no—
62C22×Dic271—21no—
63C22×Dic271D411yescharacteristic
64C22×Dic271D411yescharacteristic
65C2×C4×D271—21no—
66#16864644—22no—
67#16864643—22no—
68#16864639—22no—
69C2×C271⋊C8—22no—
70C2×C271⋊C8C2311yescharacteristic
71C2×C271⋊C8C2311yescharacteristic
72C4⋊Dic271—21no—
73C4⋊Dic271—21no—
74C4⋊Dic271—21no—
75C4⋊Dic271C2311yescharacteristic
76C4⋊Dic271C2311yescharacteristic
77Dic271⋊C4—21no—
78Dic271⋊C4—21no—
79Dic271⋊C4—21no—
80Dic271⋊C4—21no—
81C22⋊Dic271—21no—
82C22⋊Dic271—21no—
83C22⋊Dic271—21no—
84C22⋊Dic271—21no—
85(C2×D271)⋊C4—21no—
86(C2×D271)⋊C4—21no—
87(C2×D271)⋊C4—21no—
88(C2×D271)⋊C4—21no—
89C4×Dic271—21no—
90C4×Dic271—21no—
91C4×Dic271D411yescharacteristic
92C4×Dic271D411yescharacteristic
93C4×Dic271C2311yescharacteristic
94C4×Dic271C2311yescharacteristic
95C2×Q8×C271C2311yescharacteristic
96C2×D4×C271—21no—
97C2×D4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99SD16×C271—22no—
100C2×C8×C271—22no—
101C2×C8×C271C2311yescharacteristic
102C2×C8×C271C2311yescharacteristic
103C4⋊C4×C271—21no—
104C4⋊C4×C271C2311yescharacteristic
105C4⋊C4×C271C2311yescharacteristic
106C22⋊C4×C271—21no—
107C22⋊C4×C271—21no—
108C22×D271—21no—
109Dic542—21no—
110Dic542—21no—
111D1084—21no—
112D1084—21no—
113C22⋊D271—42no—
114C2×Dic271—22no—
115C2×Dic271—22no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—42no—
120C2×Dic271C2×D411yescharacteristic
121C2×Dic271C2×D411yescharacteristic
122C2×Dic271C2×D411yescharacteristic
123C2×Dic271C2×D411yescharacteristic
124C4×D271—42no—
125C271⋊C8—21no—
126C271⋊C8#4112yes—
127C23×C271—21no—
128C23×C271C2×D411yescharacteristic
129Q8×C271—21no—
130Q8×C271—21no—
131D4×C271—21no—
132D4×C271—21no—
133D4×C271—42no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271C2×D411yescharacteristic
140C2×C4×C271C2411yescharacteristic
141C8×C271—21no—
142C8×C271#4112yes—
143C22×C271—21no—
144C22×C271—21no—
145C22×C271—41no—
146C22×C271—41no—
147C22×C271—41no—
148C22×C271#13512yes—
149C22×C271C22×D411yescharacteristic
150C2×D271—41no—
151C2×D271—41no—
152C2×D271—41no—
153C4×C271—21no—
154C4×C271—41no—
155C4×C271—41no—
156C4×C271—41no—
157C4×C271C2×#4111yescharacteristic
158C4×C271C2×#4111yescharacteristic
159Dic271—21no—
160Dic271—21no—
161Dic271—21no—
162Dic271—41no—
163Dic271—41no—
164Dic271—41no—
165Dic271#13512yes—
166C2×C271—22no—
167C2×C271—81no—
168C2×C271#53211yescharacteristic
169C2×C271C2×#13511yescharacteristic
170C2×C271C2×#13511yescharacteristic
171D271—81no—
172C271#316811yescharacteristic, complemented, Sylow
173#3168—2711nomaximal, Sylow
174#529—2711no—
175#529—2711no—
176#499—2711no—
177#495—2711no—
178#488—2711no—
179#488—2711no—
180#467—2711no—
181#466—2711no—
182#465—2711no—
183#465D27111yescharacteristic, complemented
184C4×Q16—2711no—
185C4×SD16—2711no—
186C4×SD16—2711no—
187C4×D8—2711no—
188C2×C4×C8—2711no—
189C2×Q16—2711no—
190C2×SD16—2711no—
191C2×SD16C2×D27111yescharacteristic
192C2×D8—2711no—
193C22×C8—2711no—
194C22×C8C2×D27111yescharacteristic
195#126—5421no—
196#126—5421no—
197#125—2711no—
198#125—2711no—
199#124—2711no—
200#124—2711no—
201#123—5421no—
202#123—5421no—
203C22⋊Q8—2711no—
204C22⋊Q8C2×D27111yescharacteristic
205#121—2711no—
206#121C2×D27111yescharacteristic
207C4×Q8—2711no—
208C4×Q8—2711no—
209C4×D4—2711no—
210C4×D4—2711no—
211C4×D4—5421no—
212C4×D4—5421no—
213#117—5421no—
214#117—5421no—
215C2×C42—2711no—
216#107—2711no—
217#107—2711no—
218#106—2711no—
219#106C2×D27111yescharacteristic
220#103—2711no—
221#103—2711no—
222#103—2711no—
223#103C2×D27111yescharacteristic
224#102—2711no—
225#102—2711no—
226#102—2711no—
227#102C2×D27111yescharacteristic
228C4×C8—2711no—
229C4×C8—2711no—
230C4×C8—2711no—
231C4×C8—2711no—
232C2×Q8—2711no—
233C2×Q8C22×D27111yescharacteristic
234C2×D4—21no—
235C2×D4—2711no—
236C2×D4—5421no—
237C2×D4C22×D27111yescharacteristic
238C22×C4—2711no—
239C22×C4—2711no—
240C22×C4—5421no—
241C22×C4—5421no—
242C22×C4C22×D27111yescharacteristic
243Q16—5422no—
244SD16—22no—
245SD16—5422no—
246D8—5422no—
247C2×C8—22no—
248C2×C8—2711no—
249C2×C8—2711no—
250C2×C8—5422no—
251C2×C8C22×D27111yescharacteristic
252C2×C8C22×D27111yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—2711no—
255C4⋊C4—2711no—
256C4⋊C4—5421no—
257C4⋊C4—5421no—
258C4⋊C4—5421no—
259C4⋊C4—5421no—
260C4⋊C4—5421no—
261C4⋊C4—5421no—
262C4⋊C4—5421no—
263C4⋊C4C22×D27111yescharacteristic
264C4⋊C4C22×D27111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—5421no—
268C22⋊C4—5421no—
269C22⋊C4—5421no—
270C22⋊C4—5421no—
271C22⋊C4—5421no—
272C22⋊C4—5421no—
273C22⋊C4—5421no—
274C22⋊C4—5421no—
275C42—2711no—
276C42—2711no—
277C42—2711no—
278C42—2711no—
279C42—5421no—
280C42—5421no—
281C23—21no—
282C23—5421no—
283C23D4×D27111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—5421no—
287Q8—5421no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—5421no—
292D4—5421no—
293D4—10842no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—2711no—
300C2×C4—2711no—
301C2×C4—2711no—
302C2×C4—2711no—
303C2×C4—5422no—
304C2×C4—5422no—
305C2×C4—5421no—
306C2×C4—5421no—
307C2×C4—5421no—
308C2×C4—10842no—
309C2×C4—10842no—
310C2×C4D4×D27111yescharacteristic
311C2×C4C23×D27111yescharacteristic
312C8—21no—
313C8—2712no—
314C8—5421no—
315C8#1686465312yes—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—10841no—
322C22—10841no—
323C22—10841no—
324C22#2257618112yes—
325C22C2×D4×D27111yescharacteristic
326C4—21no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—2712no—
331C4—5421no—
332C4—5421no—
333C4—5421no—
334C4—10841no—
335C4—10841no—
336C4—10841no—
337C4#41×D27111yescharacteristic
338C4C2×#1686465311yescharacteristic
339C2—22no—
340C2—81no—
341C2—21681no—
342C2#3177444211yescharacteristic
343C2#135×D27111yescharacteristic
344C2C2×#2257618111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#465×C271C2
socleC22×C271C22×D4
prime radical#31774131C2
hypercenter#465D271
hypocenterC271#3168
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31774433C2semidirect
2. #31774428C2semidirect
3. #31774297C2semidirect
4. #31774278C2semidirect
5. #31774253C2semidirect
6. #31774250C2semidirect
7. #31774136C2semidirect
9. #31774126C2semidirect
10. #31773989C2semidirect
11. SD16×Dic271C2semidirect
12. #31773984C2semidirect
13. #31773977C2semidirect
14. C2×C8×Dic271C2semidirect
15. #465×C271C2semidirect
172. C271#3168semidirect
183. #465D271semidirect

Maximal subgroups

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

subgroupindex
#317744332
#317744282
#317742972
#317742782
#317742532
#317742502
#317741362
#317741312
#317741262
#317739892
SD16×Dic2712
#317739842
#317739772
C2×C8×Dic2712
#465×C2712
#3168271