Finite Group Databasebeta

← #46839700

Subgroups — #46839700

Order 34688.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#16864655C211yesmaximal, characteristic
2C2×#22576145C212yesmaximal, complemented
3C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
4#31773816C211yesmaximal, characteristic, complemented
5#31773815C211yesmaximal, characteristic, complemented
6#31773814C211yesmaximal, characteristic, complemented
7#31773775C212yesmaximal, complemented
8#31773771C212yesmaximal, complemented
9#31773768C212yesmaximal, complemented
10#31773759C211yesmaximal, characteristic, complemented
11#395×C271C211yesmaximal, characteristic, complemented
12C2×#16864655—28no—
13C2×#16864655—24no—
14C22×D1084C2211yescharacteristic
15C22×D1084C2212yes—
16C22×C4×D271C2212yes—
17C22×C4×D271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19C22×C4×D271C2211yescharacteristic
20#22576145—28no—
21C2×C4⋊Dic271C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2212yes—
24C2×Dic271⋊C4C2211yescharacteristic
25C4⋊C4×D271—24no—
26C2×(C2×D271)⋊C4C2212yes—
27C2×(C2×D271)⋊C4C2212yes—
28C2×(C2×D271)⋊C4C2212yes—
29C2×(C2×D271)⋊C4C2212yes—
30C2×(C2×D271)⋊C4C2212yes—
31C271⋊#95C2212yes—
32C271⋊#95C2211yescharacteristic
33C271⋊#95C2212yes—
34C22×Q8×C271C2211yescharacteristic
35C2×C4⋊C4×C271C2212yes—
36C2×C4⋊C4×C271C2211yescharacteristic
37#95×C271C2211yescharacteristic
38#95×C271C2212yes—
39C23×D271C2312yes—
40C23×D271C2311yescharacteristic
41#16864655—232no—
42C2×D1084—24no—
43C2×D1084—24no—
44C2×D1084—24no—
45C2×D1084—22no—
46C2×D1084D418yes—
47C22×Dic271C2311yescharacteristic
48C22×Dic271C2312yes—
49C22×Dic271C2311yescharacteristic
50C22×Dic271C2311yescharacteristic
51C2×C4×D271—24no—
52C2×C4×D271—24no—
53C2×C4×D271—22no—
54C2×C4×D271—24no—
55C2×C4×D271—22no—
56C2×C4×D271—24no—
57C2×C4×D271—22no—
58C2×C4×D271—24no—
59C2×C4×D271—22no—
60C2×C4×D271D414yes—
61C4⋊Dic271—22no—
62Dic271⋊C4—22no—
63Dic271⋊C4—24no—
64Dic271⋊C4—22no—
65(C2×D271)⋊C4—24no—
66(C2×D271)⋊C4—24no—
67(C2×D271)⋊C4—24no—
68(C2×D271)⋊C4—24no—
69(C2×D271)⋊C4—24no—
70C2×Q8×C271—24no—
71C2×Q8×C271—22no—
72C22×C4×C271C2311yescharacteristic
73C22×C4×C271C2312yes—
74C22×C4×C271C2312yes—
75C22×C4×C271C2311yescharacteristic
76C22×C4×C271C2311yescharacteristic
77C4⋊C4×C271—24no—
78C4⋊C4×C271—22no—
79C22×D271—21no—
80C22×D271—21no—
81C22×D271—22no—
82C22×D271—22no—
83C22×D271—22no—
84C22×D271—22no—
85C22×D271—22no—
86C22×D271—22no—
87C22×D271—22no—
88C22×D271C2×D414yes—
89C22×D271C2×D414yes—
90C22×D271C2×D412yes—
91D1084—28no—
92D1084—216no—
93C2×Dic271—21no—
94C2×Dic271—22no—
95C2×Dic271—21no—
96C2×Dic271—22no—
97C2×Dic271—22no—
98C2×Dic271—21no—
99C2×Dic271—22no—
100C2×Dic271—22no—
101C2×Dic271C2×D412yes—
102C2×Dic271#4112yescomplemented
103C4×D271—28no—
104C4×D271—216no—
105C4×D271—28no—
106C4×D271—44no—
107C23×C271C2411yescharacteristic
108Q8×C271—28no—
109C2×C4×C271—21no—
110C2×C4×C271—22no—
111C2×C4×C271—22no—
112C2×C4×C271—22no—
113C2×C4×C271—21no—
114C2×C4×C271—22no—
115C2×C4×C271—22no—
116C2×C4×C271—21no—
117C2×C4×C271—22no—
118C2×C4×C271—22no—
119C2×C4×C271C2×D414yes—
120C2×C4×C271C2×D412yes—
121C2×C4×C271#4112yescomplemented
122C22×C271C22×D411yescharacteristic
123C22×C271C22×D412yes—
124C22×C271C2×#4111yescharacteristic
125C22×C271C2×#4111yescharacteristic
126C22×C2712−1+412yes—
127C2×D271—22no—
128C2×D271—22no—
129C2×D271—24no—
130C2×D271—24no—
131C2×D271—24no—
132C2×D271—24no—
133C2×D271—24no—
134C2×D271—42no—
135C2×D271—42no—
136C2×D271—42no—
137C2×D271—42no—
138C2×D271#12014yes—
139C4×C271—24no—
140C4×C271—22no—
141C4×C271—22no—
142C4×C271—41no—
143C4×C271—42no—
144Dic271—22no—
145Dic271—22no—
146Dic271—41no—
147Dic271—42no—
148C2×C271C2×#12011yescharacteristic
149C2×C271#52511yescharacteristic
150C2×C271#53611yescharacteristic
151C2×C271#54712yes—
152C2×C271#54712yes—
153D271—24no—
154D271—44no—
155C271#240611yescharacteristic, complemented, Sylow
156#2406—2711nomaximal, Sylow
157C22×#41—2711no—
158C2×#123—2712no—
159C22×C4⋊C4—2711no—
160#395—2711no—
161#395D27111yescharacteristic, complemented
162#394—2711no—
163#394—2711no—
164#386—2712no—
165#385—2712no—
166#385—2712no—
167#381—2711no—
168C2×#41—5424no—
169C2×#41—5428no—
170C22×Q8C2×D27111yescharacteristic
171C22×D4—2712no—
172C22×D4—2711no—
173C23×C4—2711no—
174C23×C4—2711no—
175C23×C4—2711no—
176C23×C4—2712no—
177#123—5428no—
178C2×C4⋊C4—2711no—
179C2×C4⋊C4—2711no—
180C2×C4⋊C4—2711no—
181C2×C4⋊C4—2712no—
182C2×C4⋊C4—5424no—
183C2×C4⋊C4C2×D27111yescharacteristic
184C2×C4⋊C4C2×D27112yes—
185C2×C22⋊C4—2712no—
186C2×C22⋊C4—2712no—
187C2×C22⋊C4—2712no—
188C2×C22⋊C4—2712no—
189C2×C22⋊C4—2712no—
190#95—2712no—
191#95—2711no—
192#95—2712no—
193#95C2×D27111yescharacteristic
194#95C2×D27112yes—
195C24—2711no—
196C24—2712no—
197#41—54232no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—2718no—
201C2×D4—5424no—
202C2×D4—5424no—
203C2×D4—5424no—
204C2×D4—5422no—
205C22×C4—2711no—
206C22×C4—2711no—
207C22×C4—2714no—
208C22×C4—2712no—
209C22×C4—2711no—
210C22×C4—5422no—
211C22×C4—5424no—
212C22×C4—5422no—
213C22×C4—5424no—
214C22×C4—5424no—
215C22×C4—5422no—
216C22×C4—5424no—
217C22×C4—5424no—
218C22×C4—5422no—
219C22×C4C22×D27112yes—
220C22×C4C22×D27111yescharacteristic
221C22×C4C22×D27111yescharacteristic
222C22×C4C22×D27112yes—
223C22×C4C22×D27111yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—5422no—
227C4⋊C4—5422no—
228C4⋊C4—5422no—
229C4⋊C4—5424no—
230C22⋊C4—5424no—
231C22⋊C4—5424no—
232C22⋊C4—5424no—
233C22⋊C4—5424no—
234C22⋊C4—5424no—
235C23—2714no—
236C23—2712no—
237C23—2714no—
238C23—5421no—
239C23—5421no—
240C23—5422no—
241C23—5422no—
242C23—5422no—
243C23—5422no—
244C23—5422no—
245C23—5422no—
246C23—5422no—
247C23C23×D27111yescharacteristic
248Q8—28no—
249D4—54216no—
250D4—5428no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—2712no—
262C2×C4—2712no—
263C2×C4—5428no—
264C2×C4—54216no—
265C2×C4—5428no—
266C2×C4—5421no—
267C2×C4—5422no—
268C2×C4—5421no—
269C2×C4—5422no—
270C2×C4—5422no—
271C2×C4—5421no—
272C2×C4—5422no—
273C2×C4—5422no—
274C2×C4—10844no—
275C2×C4D4×D27112yes—
276C2×C4D4×D27114yes—
277C2×C4#1686465512yescomplemented
278C22—2714no—
279C22—5422no—
280C22—5424no—
281C22—5424no—
282C22—5422no—
283C22—5424no—
284C22—5424no—
285C22—5424no—
286C22—10842no—
287C22—10842no—
288C22—10842no—
289C22—10842no—
290C22C2×D4×D27111yescharacteristic
291C22C2×D4×D27112yes—
292C22#41×D27111yescharacteristic
293C22C2×#1686465511yescharacteristic
294C22#2257620812yes—
295C4—22no—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—5422no—
301C4—5422no—
302C4—10842no—
303C4—10841no—
304C2—5424no—
305C2—10844no—
306C2#120×D27111yescharacteristic
307C2#3177441011yescharacteristic
308C2#3177447211yescharacteristic
309C2#3177454912yes—
310C2#3177455012yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#395×C271C2
prime radicalC22×#16864655C2
hypercenter#395D271
hypocenterC271#2406
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#22576145C2semidirect
3. C2×C4⋊C4×D271C2semidirect
4. #31773816C2semidirect
5. #31773815C2semidirect
6. #31773814C2semidirect
7. #31773775C2semidirect
8. #31773771C2semidirect
9. #31773768C2semidirect
10. #31773759C2semidirect
11. #395×C271C2semidirect
102. C2×Dic271#41semidirect
121. C2×C4×C271#41semidirect
155. C271#2406semidirect
161. #395D271semidirect
277. C2×C4#16864655semidirect

Maximal subgroups

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

subgroupindex
C22×#168646552
C2×#225761452
C2×C4⋊C4×D2712
#317738162
#317738152
#317738142
#317737752
#317737712
#317737682
#317737592
#395×C2712
#2406271