Finite Group Databasebeta

← #46839749

Subgroups — #46839749

Order 34688.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576143C211yesmaximal, characteristic, complemented
2C2×#22576129C211yesmaximal, characteristic
3C2×#22576119C211yesmaximal, characteristic, complemented
4#31773820C212yesmaximal, complemented
5#31773813C212yesmaximal, complemented
6#31773797C212yesmaximal, complemented
7#31773791C212yesmaximal, complemented
8#31773776C211yesmaximal, characteristic, complemented
9#31773770C211yesmaximal, characteristic, complemented
10#31773753C211yesmaximal, characteristic, complemented
11C22×C22⋊C4×C271C211yesmaximal, characteristic, complemented
12C22×C22⋊D271C2211yescharacteristic
13C22×C22⋊D271C2212yes—
14#22576143—24no—
15#22576129—24no—
16#22576119—24no—
17C2×C4⋊Dic271C2212yescomplemented
18C2×Dic271⋊C4C2211yescharacteristic, complemented
19C2×Dic271⋊C4C2211yescharacteristic, complemented
20C2×Dic271⋊C4C2212yescomplemented
21C2×C22⋊Dic271C2211yescharacteristic
22C2×C22⋊Dic271C2211yescharacteristic
23C2×C22⋊Dic271C2212yes—
24C2×C22⋊Dic271C2212yes—
25C2×C22⋊Dic271C2212yes—
26C2×C22⋊Dic271C2211yescharacteristic
27C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
28C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
29C2×(C2×D271)⋊C4C2212yescomplemented
30C2×C4×Dic271C2212yescomplemented
31C271⋊#95C2212yescomplemented
32C271⋊#95C2212yescomplemented
33C25×C271C2211yescharacteristic
34C23×C4×C271C2211yescharacteristic
35C23×C4×C271C2211yescharacteristic
36C2×C22⋊C4×C271—24no—
37C2×C22⋊C4×C271C2212yes—
38C2×C22⋊C4×C271C2212yes—
39C23×D271C2311yescharacteristic
40C2×C22⋊D271—24no—
41C2×C22⋊D271—22no—
42C2×C22⋊D271—22no—
43C2×C22⋊D271—22no—
44C2×C22⋊D271—24no—
45C2×C22⋊D271—42no—
46C22×Dic271C2311yescharacteristic
47C22×Dic271C2311yescharacteristic
48C22×Dic271C2312yes—
49C22×Dic271C2311yescharacteristic
50C22×Dic271C2312yes—
51C4⋊Dic271—24no—
52Dic271⋊C4—22no—
53Dic271⋊C4—22no—
54Dic271⋊C4—42no—
55C22⋊Dic271—24no—
56C22⋊Dic271—22no—
57C22⋊Dic271—22no—
58C22⋊Dic271—24no—
59C22⋊Dic271—42no—
60C22⋊Dic271D414yes—
61(C2×D271)⋊C4—22no—
62(C2×D271)⋊C4—22no—
63(C2×D271)⋊C4—42no—
64C4×Dic271—24no—
65C24×C271—22no—
66C24×C271—24no—
67C24×C271—22no—
68C24×C271—22no—
69C24×C271—42no—
70C24×C271C2312yes—
71C24×C271C2311yescharacteristic
72C22×C4×C271—22no—
73C22×C4×C271—22no—
74C22×C4×C271—22no—
75C22×C4×C271—42no—
76C22×C4×C271D414yescomplemented
77C22×C4×C271C2312yes—
78C22×C4×C271C2311yescharacteristic
79C22×C4×C271C2311yescharacteristic
80C22⋊C4×C271—24no—
81C22⋊C4×C271—24no—
82C22×D271—21no—
83C22×D271—21no—
84C22×D271—21no—
85C22×D271—22no—
86C22×D271—22no—
87C22⋊D271—48no—
88C22⋊D271—44no—
89C2×Dic271—22no—
90C2×Dic271—22no—
91C2×Dic271—22no—
92C2×Dic271—21no—
93C2×Dic271—21no—
94C2×Dic271—21no—
95C2×Dic271—21no—
96C2×Dic271—22no—
97C2×Dic271—22no—
98C2×Dic271—21no—
99C2×Dic271—22no—
100C2×Dic271—22no—
101C2×Dic271C2×D414yes—
102C23×C271—22no—
103C23×C271—22no—
104C23×C271—28no—
105C23×C271—22no—
106C23×C271—22no—
107C23×C271—22no—
108C23×C271—22no—
109C23×C271—22no—
110C23×C271—22no—
111C23×C271—48no—
112C23×C271—48no—
113C23×C271—44no—
114C23×C271—44no—
115C23×C271C2×D412yes—
116C23×C271C2×D412yes—
117C23×C271#4112yes—
118C23×C271#4114yes—
119C23×C271C2411yescharacteristic
120C2×C4×C271—28no—
121C2×C4×C271—21no—
122C2×C4×C271—21no—
123C2×C4×C271—21no—
124C2×C4×C271—21no—
125C2×C4×C271—22no—
126C2×C4×C271—22no—
127C2×C4×C271—22no—
128C2×C4×C271—44no—
129C2×C4×C271C2×D412yes—
130C2×C4×C271C2×D412yes—
131C22×C271—22no—
132C22×C271—22no—
133C22×C271—24no—
134C22×C271—24no—
135C22×C271—24no—
136C22×C271—24no—
137C22×C271—24no—
138C22×C271—416no—
139C22×C271—42no—
140C22×C271—42no—
141C22×C271—42no—
142C22×C271—42no—
143C22×C271#12114yes—
144C22×C271C22×D411yescharacteristic
145C22×C271C22×D411yescharacteristic
146C22×C271C2×#4111yescharacteristic
147C22×C271C2×#4112yes—
148C22×C271H(5,𝔽2)12yes—
149C2×D271—41no—
150C2×D271—41no—
151C2×D271—41no—
152C2×D271—42no—
153C2×D271—42no—
154C4×C271—22no—
155C4×C271—22no—
156C4×C271—42no—
157Dic271—24no—
158Dic271—41no—
159Dic271—41no—
160Dic271—41no—
161Dic271—42no—
162C2×C271—24no—
163C2×C271—44no—
164C2×C271C2×#12111yescharacteristic
165C2×C271#52511yescharacteristic
166C2×C271#53511yescharacteristic
167C2×C271#54612yes—
168C2×C271#55112yes—
169D271—81no—
170C271#241111yescharacteristic, complemented, Sylow
171#2411—2711nomaximal, Sylow
172C2×#123—2711no—
173C2×#120—2711no—
174C2×#117—2711no—
175C22×C22⋊C4D27111yescharacteristic, complemented
176#396—2712no—
177#394—2712no—
178#390—2712no—
179#388—2712no—
180#386—2711no—
181#385—2711no—
182#379—2711no—
183C25C2×D27111yescharacteristic
184C22×D4—2711no—
185C22×D4—2712no—
186C23×C4C2×D27111yescharacteristic
187C23×C4C2×D27111yescharacteristic
188#123—5424no—
189#120—5424no—
190#117—5424no—
191C2×C4⋊C4—2712no—
192C2×C4⋊C4—2711no—
193C2×C4⋊C4—2711no—
194C2×C4⋊C4—2712no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—2711no—
197C2×C22⋊C4—2711no—
198C2×C22⋊C4—2712no—
199C2×C22⋊C4—2711no—
200C2×C22⋊C4—2712no—
201C2×C22⋊C4—2711no—
202C2×C22⋊C4—2711no—
203C2×C22⋊C4—2712no—
204C2×C22⋊C4—2712no—
205C2×C22⋊C4C2×D27112yes—
206C2×C22⋊C4C2×D27112yes—
207C2×C42—2712no—
208#95—2712no—
209#95—2712no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—2711no—
216C24C22×D27112yes—
217C24C22×D27111yescharacteristic
218C2×D4—5424no—
219C2×D4—5422no—
220C2×D4—5422no—
221C2×D4—5422no—
222C2×D4—5424no—
223C2×D4—10842no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—2712no—
229C22×C4—2711no—
230C22×C4—2711no—
231C22×C4—2712no—
232C22×C4—2711no—
233C22×C4C22⋊D27114yescomplemented
234C22×C4C22×D27112yes—
235C22×C4C22×D27111yescharacteristic
236C22×C4C22×D27111yescharacteristic
237C4⋊C4—5424no—
238C4⋊C4—5422no—
239C4⋊C4—5422no—
240C4⋊C4—10842no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—2714no—
244C22⋊C4—5424no—
245C22⋊C4—5422no—
246C22⋊C4—5422no—
247C22⋊C4—5424no—
248C22⋊C4—5422no—
249C22⋊C4—5422no—
250C22⋊C4—10842no—
251C22⋊C4—10842no—
252C42—5424no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—48no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—5421no—
267C23—5421no—
268C23—5421no—
269C23—5422no—
270C23—5422no—
271C23D4×D27112yes—
272C23C2×C22⋊D27112yes—
273C23#1686465312yes—
274C23#1686465414yes—
275C23C23×D27111yescharacteristic
276D4—10848no—
277D4—10844no—
278C2×C4—28no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—2714no—
288C2×C4—5422no—
289C2×C4—5422no—
290C2×C4—5422no—
291C2×C4—5421no—
292C2×C4—5421no—
293C2×C4—5421no—
294C2×C4—5421no—
295C2×C4—5421no—
296C2×C4—5422no—
297C2×C4—5422no—
298C2×C4—5422no—
299C2×C4—5422no—
300C2×C4C2×C22⋊D27112yes—
301C2×C4C2×C22⋊D27112yes—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—10841no—
315C22—10842no—
316C22—10842no—
317C22—10841no—
318C22—10841no—
319C22#2257613214yes—
320C22C2×D4×D27111yescharacteristic
321C22C22×C22⋊D27111yescharacteristic
322C22C2×#1686465311yescharacteristic
323C22C2×#1686465412yes—
324C22#2257620512yes—
325C4—22no—
326C4—22no—
327C4—42no—
328C4—5424no—
329C4—10841no—
330C4—10842no—
331C4—10841no—
332C4—10841no—
333C2—24no—
334C2—44no—
335C2—21681no—
336C2C2×#2257613211yescharacteristic
337C2#3177440811yescharacteristic
338C2#3177446411yescharacteristic
339C2#3177453712yes—
340C2#3177457312yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC22×C22⋊C4×C271C2
prime radicalC2×#22576129C2
hypercenterC22×C22⋊C4D271
hypocenterC271#2411
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#22576143C2semidirect
3. C2×#22576119C2semidirect
4. #31773820C2semidirect
5. #31773813C2semidirect
6. #31773797C2semidirect
7. #31773791C2semidirect
8. #31773776C2semidirect
9. #31773770C2semidirect
10. #31773753C2semidirect
11. C22×C22⋊C4×C271C2semidirect
17. C2×C4⋊Dic271C22semidirect
18. C2×Dic271⋊C4C22semidirect
19. C2×Dic271⋊C4C22semidirect
20. C2×Dic271⋊C4C22semidirect
27. C2×(C2×D271)⋊C4C22semidirect
28. C2×(C2×D271)⋊C4C22semidirect
29. C2×(C2×D271)⋊C4C22semidirect
30. C2×C4×Dic271C22semidirect
31. C271⋊#95C22semidirect
32. C271⋊#95C22semidirect
76. C22×C4×C271D4semidirect
170. C271#2411semidirect
175. C22×C22⋊C4D271semidirect
233. C22×C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761432
C2×#225761292
C2×#225761192
#317738202
#317738132
#317737972
#317737912
#317737762
#317737702
#317737532
C22×C22⋊C4×C2712
#2411271