Finite Group Databasebeta

← #46839745

Subgroups — #46839745

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×#22576141C211yesmaximal, characteristic, complemented
2C2×#22576116C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic271C211yesmaximal, characteristic, complemented
4#31773820C212yesmaximal, complemented
5#31773811C212yesmaximal, complemented
6#31773800C212yesmaximal, complemented
7#31773785C212yesmaximal, complemented
8#31773777C211yesmaximal, characteristic, complemented
9#31773772C211yesmaximal, characteristic, complemented
10#31773753C211yesmaximal, characteristic, complemented
11C2×#120×C271C211yesmaximal, characteristic
12C23×Dic271C2211yescharacteristic
13C23×Dic271C2211yescharacteristic
14#22576141—24no—
15#22576116—24no—
16C2×C4⋊Dic271C2212yescomplemented
17C2×Dic271⋊C4C2211yescharacteristic, complemented
18C2×Dic271⋊C4C2212yescomplemented
19C2×Dic271⋊C4C2211yescharacteristic, complemented
20C2×C22⋊Dic271—24no—
21C2×C22⋊Dic271C2211yescharacteristic
22C2×C22⋊Dic271C2211yescharacteristic
23C2×C22⋊Dic271C2212yes—
24C2×C22⋊Dic271C2212yes—
25C2×C22⋊Dic271C2212yes—
26C2×C22⋊Dic271C2212yes—
27C2×C22⋊Dic271C2211yescharacteristic, complemented
28C2×C22⋊Dic271C2211yescharacteristic, complemented
29C2×C22⋊Dic271C2212yescomplemented
30C2×C4×Dic271C2212yescomplemented
31C271⋊#95C2212yescomplemented
32C271⋊#95C2212yescomplemented
33C25×C271C2211yescharacteristic
34C22×D4×C271C2211yescharacteristic
35C22×D4×C271C2212yes—
36#120×C271—24no—
37C2×C22⋊C4×C271C2212yes—
38C2×C22⋊C4×C271C2211yescharacteristic
39C22×Dic271—22no—
40C22×Dic271—22no—
41C22×Dic271—22no—
42C22×Dic271—42no—
43C22×Dic271D414yescomplemented
44C22×Dic271C2312yes—
45C22×Dic271C2311yescharacteristic
46C22×Dic271C2312yes—
47C22×Dic271C2311yescharacteristic
48C22×Dic271C2311yescharacteristic
49C22×Dic271C2311yescharacteristic
50C4⋊Dic271—42no—
51Dic271⋊C4—22no—
52Dic271⋊C4—22no—
53Dic271⋊C4—24no—
54C22⋊Dic271—22no—
55C22⋊Dic271—22no—
56C22⋊Dic271—24no—
57C22⋊Dic271—24no—
58C22⋊Dic271—22no—
59C22⋊Dic271—22no—
60C22⋊Dic271—24no—
61C22⋊Dic271—42no—
62C22⋊Dic271—42no—
63C4×Dic271—24no—
64C24×C271—22no—
65C24×C271—24no—
66C24×C271—22no—
67C24×C271—22no—
68C24×C271—42no—
69C24×C271C2311yescharacteristic
70C24×C271C2312yes—
71C24×C271C2311yescharacteristic
72C2×D4×C271—22no—
73C2×D4×C271—22no—
74C2×D4×C271—24no—
75C2×D4×C271—22no—
76C2×D4×C271—24no—
77C2×D4×C271—42no—
78C22×C4×C271C2311yescharacteristic
79C22×C4×C271C2312yes—
80C22⋊C4×C271—24no—
81C22⋊C4×C271D414yes—
82C2×Dic271—28no—
83C2×Dic271—21no—
84C2×Dic271—21no—
85C2×Dic271—21no—
86C2×Dic271—21no—
87C2×Dic271—22no—
88C2×Dic271—21no—
89C2×Dic271—21no—
90C2×Dic271—22no—
91C2×Dic271—22no—
92C2×Dic271—22no—
93C2×Dic271—22no—
94C2×Dic271—22no—
95C2×Dic271—22no—
96C2×Dic271—22no—
97C2×Dic271—44no—
98C2×Dic271C2×D412yes—
99C2×Dic271C2×D412yes—
100C23×C271—22no—
101C23×C271—22no—
102C23×C271—28no—
103C23×C271—22no—
104C23×C271—22no—
105C23×C271—22no—
106C23×C271—22no—
107C23×C271—22no—
108C23×C271—22no—
109C23×C271—21no—
110C23×C271—21no—
111C23×C271—21no—
112C23×C271—22no—
113C23×C271—22no—
114C23×C271—48no—
115C23×C271—48no—
116C23×C271—44no—
117C23×C271—44no—
118C23×C271C2×D412yes—
119C23×C271C2×D412yes—
120C23×C271#4114yes—
121C23×C271#4112yes—
122C23×C271C2411yescharacteristic
123D4×C271—48no—
124D4×C271—44no—
125C2×C4×C271—22no—
126C2×C4×C271—22no—
127C2×C4×C271—21no—
128C2×C4×C271—21no—
129C2×C4×C271—21no—
130C2×C4×C271C2×D414yes—
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—41no—
144C22×C271—41no—
145C22×C271—41no—
146C22×C271—42no—
147C22×C271—42no—
148C22×C271#12114yes—
149C22×C271C22×D411yescharacteristic
150C22×C271C22×D411yescharacteristic
151C22×C271C2×#4111yescharacteristic
152C22×C271C2×#4112yes—
153C22×C271H(5,𝔽2)12yes—
154C4×C271—24no—
155C4×C271—41no—
156Dic271—22no—
157Dic271—22no—
158Dic271—41no—
159Dic271—41no—
160Dic271—42no—
161Dic271—42no—
162C2×C271—24no—
163C2×C271—44no—
164C2×C271—81no—
165C2×C271C2×#12111yescharacteristic
166C2×C271#52511yescharacteristic
167C2×C271#53511yescharacteristic
168C2×C271#54612yes—
169C2×C271#55112yes—
170C271#241111yescharacteristic, complemented, Sylow
171#2411—2711nomaximal, Sylow
172C2×#123—2711no—
173C2×#120D27111yescharacteristic
174C2×#117—2711no—
175C22×C22⋊C4—2711no—
176#396—2712no—
177#394—2712no—
178#390—2712no—
179#388—2712no—
180#386—2711no—
181#385—2711no—
182#379—2711no—
183C25C2×D27111yescharacteristic
184C22×D4C2×D27111yescharacteristic
185C22×D4C2×D27112yes—
186C23×C4—2711no—
187C23×C4—2711no—
188#123—5424no—
189#120—24no—
190#117—5424no—
191C2×C4⋊C4—2712no—
192C2×C4⋊C4—2711no—
193C2×C4⋊C4—2711no—
194C2×C4⋊C4—2712no—
195C2×C22⋊C4—2712no—
196C2×C22⋊C4—2712no—
197C2×C22⋊C4—2711no—
198C2×C22⋊C4—2711no—
199C2×C22⋊C4—2711no—
200C2×C22⋊C4—2711no—
201C2×C22⋊C4—2712no—
202C2×C22⋊C4—2712no—
203C2×C22⋊C4—2712no—
204C2×C22⋊C4—5424no—
205C2×C22⋊C4C2×D27112yes—
206C2×C22⋊C4C2×D27111yescharacteristic
207C2×C42—2712no—
208#95—2712no—
209#95—2712no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C22×D27111yescharacteristic
216C24C22×D27112yes—
217C24C22×D27111yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C22×C4—2714no—
225C22×C4—2711no—
226C22×C4—2711no—
227C22×C4—2711no—
228C22×C4—2712no—
229C22×C4—2712no—
230C22×C4—2711no—
231C22×C4—5422no—
232C22×C4—5422no—
233C22×C4—5422no—
234C22×C4—10842no—
235C22×C4C22×D27111yescharacteristic
236C22×C4C22×D27112yes—
237C4⋊C4—5422no—
238C4⋊C4—5422no—
239C4⋊C4—5424no—
240C4⋊C4—10842no—
241C22⋊C4—24no—
242C22⋊C4—5424no—
243C22⋊C4—5424no—
244C22⋊C4—5422no—
245C22⋊C4—5422no—
246C22⋊C4—5422no—
247C22⋊C4—5422no—
248C22⋊C4—5424no—
249C22⋊C4—10842no—
250C22⋊C4—10842no—
251C22⋊C4C22⋊D27114yes—
252C42—5424no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23D4×D27112yes—
272C23C2×C22⋊D27112yes—
273C23#1686465314yes—
274C23#1686465312yes—
275C23C23×D27111yescharacteristic
276D4—48no—
277D4—44no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—2712no—
284C2×C4—2712no—
285C2×C4—5428no—
286C2×C4—5421no—
287C2×C4—5421no—
288C2×C4—5421no—
289C2×C4—5421no—
290C2×C4—5421no—
291C2×C4—5421no—
292C2×C4—5422no—
293C2×C4—5422no—
294C2×C4—5422no—
295C2×C4—5422no—
296C2×C4—5422no—
297C2×C4—5422no—
298C2×C4—5422no—
299C2×C4—5422no—
300C2×C4—10844no—
301C2×C4C2×C22⋊D27114yes—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22#2257613114yes—
320C22C2×D4×D27111yescharacteristic
321C22C22×C22⋊D27111yescharacteristic
322C22C2×#1686465312yes—
323C22C2×#1686465311yescharacteristic
324C22#2257620512yes—
325C4—24no—
326C4—41no—
327C4—5422no—
328C4—5422no—
329C4—10842no—
330C4—10841no—
331C4—10842no—
332C4—10841no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2C2×#2257613111yescharacteristic
337C2#3177440311yescharacteristic
338C2#3177446211yescharacteristic
339C2#3177454112yes—
340C2#3177457812yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting, prime radicalC2×#120×C271C2
hypercenterC2×#120D271
hypocenterC271#2411
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#22576141C2semidirect
2. C2×#22576116C2semidirect
3. C22×C22⋊Dic271C2semidirect
4. #31773820C2semidirect
5. #31773811C2semidirect
6. #31773800C2semidirect
7. #31773785C2semidirect
8. #31773777C2semidirect
9. #31773772C2semidirect
10. #31773753C2semidirect
16. C2×C4⋊Dic271C22semidirect
17. C2×Dic271⋊C4C22semidirect
18. C2×Dic271⋊C4C22semidirect
19. C2×Dic271⋊C4C22semidirect
27. C2×C22⋊Dic271C22semidirect
28. C2×C22⋊Dic271C22semidirect
29. C2×C22⋊Dic271C22semidirect
30. C2×C4×Dic271C22semidirect
31. C271⋊#95C22semidirect
32. C271⋊#95C22semidirect
43. C22×Dic271D4semidirect
170. C271#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#225761412
C2×#225761162
C22×C22⋊Dic2712
#317738202
#317738112
#317738002
#317737852
#317737772
#317737722
#317737532
C2×#120×C2712
#2411271