Finite Group Databasebeta

← #46839969

Subgroups — #46839969

Order 34688.

914 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16864653C211yesmaximal, characteristic, complemented
2C2×#22576145C211yesmaximal, characteristic, complemented
3C2×#22576133C212yesmaximal, complemented
4#31773817C211yesmaximal, characteristic, complemented
5#31773810C211yesmaximal, characteristic, complemented
6#31773805C211yesmaximal, characteristic, complemented
7#31773774C212yesmaximal, complemented
8#31773773C212yesmaximal, complemented
9#31773769C212yesmaximal, complemented
10#95×D271C211yesmaximal, characteristic, complemented
11#394×C271C211yesmaximal, characteristic, complemented
12C2×#16864653—28no—
13C2×#16864653—24no—
14C22×Dic542C2211yescharacteristic
15C22×D1084C2211yescharacteristic
16C22×C22⋊D271C2212yes—
17C22×C22⋊D271C2212yes—
18C23×Dic271C2212yes—
19C22×C4×D271C2212yes—
20C22×C4×D271C2211yescharacteristic
21#22576145—24no—
22#22576133—28no—
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2212yes—
25C2×C22⋊Dic271C2212yes—
26C2×(C2×D271)⋊C4C2211yescharacteristic
27C2×(C2×D271)⋊C4C2211yescharacteristic
28C2×(C2×D271)⋊C4C2212yes—
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C271⋊#95C2212yes—
31C271⋊#95C2211yescharacteristic
32C271⋊#95C2212yes—
33C271⋊#95C2212yes—
34C22×D4×C271C2211yescharacteristic
35C2×C4⋊C4×C271C2211yescharacteristic
36C2×C22⋊C4×C271C2212yes—
37C2×C22⋊C4×C271C2212yes—
38#95×C271C2211yescharacteristic
39C23×D271C2311yescharacteristic
40C23×D271C2311yescharacteristic
41#16864653—232no—
42C2×Dic542—24no—
43C2×Dic542—22no—
44C2×D1084—24no—
45C2×D1084—22no—
46C2×C22⋊D271—24no—
47C2×C22⋊D271—24no—
48C2×C22⋊D271—24no—
49C2×C22⋊D271—24no—
50C2×C22⋊D271—42no—
51C2×C22⋊D271D418yescomplemented
52C22×Dic271—24no—
53C22×Dic271—24no—
54C22×Dic271—24no—
55C22×Dic271C2311yescharacteristic
56C22×Dic271C2311yescharacteristic
57C22×Dic271C2312yes—
58C22×Dic271C2312yes—
59C2×C4×D271—24no—
60C2×C4×D271—24no—
61C2×C4×D271—24no—
62C2×C4×D271—42no—
63C2×C4×D271D414yescomplemented
64Dic271⋊C4—24no—
65Dic271⋊C4—22no—
66C22⋊Dic271—24no—
67(C2×D271)⋊C4—22no—
68(C2×D271)⋊C4—24no—
69(C2×D271)⋊C4—22no—
70(C2×D271)⋊C4—22no—
71C24×C271C2312yes—
72C2×D4×C271—24no—
73C2×D4×C271—22no—
74C22×C4×C271C2312yes—
75C22×C4×C271C2311yescharacteristic
76C22×C4×C271C2311yescharacteristic
77C22×C4×C271C2311yescharacteristic
78C4⋊C4×C271—22no—
79C22⋊C4×C271—24no—
80C22⋊C4×C271—42no—
81C22×D271—22no—
82C22×D271—21no—
83C22×D271—21no—
84C22×D271—21no—
85C22×D271—22no—
86C22×D271—22no—
87C22×D271—21no—
88C22×D271—21no—
89C22×D271C2×D412yes—
90C22×D271C2×D414yes—
91Dic542—28no—
92D1084—44no—
93C22⋊D271—216no—
94C22⋊D271—48no—
95C2×Dic271—216no—
96C2×Dic271—22no—
97C2×Dic271—22no—
98C2×Dic271—22no—
99C2×Dic271—22no—
100C2×Dic271—22no—
101C2×Dic271—21no—
102C2×Dic271—22no—
103C2×Dic271C2×D412yes—
104C2×Dic271C2×D414yes—
105C2×Dic271#4114yescomplemented
106C4×D271—28no—
107C4×D271—48no—
108C23×C271—22no—
109C23×C271—22no—
110C23×C271—22no—
111C23×C271—22no—
112C23×C271—22no—
113C23×C271—22no—
114C23×C271C2×D414yes—
115C23×C271C2411yescharacteristic
116D4×C271—28no—
117C2×C4×C271—22no—
118C2×C4×C271—22no—
119C2×C4×C271—22no—
120C2×C4×C271—21no—
121C2×C4×C271—21no—
122C2×C4×C271—22no—
123C2×C4×C271—21no—
124C2×C4×C271—22no—
125C2×C4×C271—21no—
126C2×C4×C271C2×D412yes—
127C22×C271—24no—
128C22×C271—24no—
129C22×C271—24no—
130C22×C271—42no—
131C22×C271—42no—
132C22×C271—42no—
133C22×C271—42no—
134C22×C271C22×D411yescharacteristic
135C22×C271C22×D412yes—
136C22×C271C2×#4112yes—
137C22×C271H(5,𝔽2)11yescharacteristic
138C22×C2712−1+411yescharacteristic
139C2×D271—22no—
140C2×D271—22no—
141C2×D271—24no—
142C2×D271—24no—
143C2×D271—41no—
144C2×D271—41no—
145C2×D271—41no—
146C2×D271—42no—
147C2×D271—42no—
148C2×D271#12014yes—
149C4×C271—22no—
150C4×C271—41no—
151C4×C271—41no—
152C4×C271—42no—
153Dic271—22no—
154Dic271—24no—
155Dic271—24no—
156Dic271—41no—
157C2×C271—44no—
158C2×C271C2×#12011yescharacteristic
159C2×C271#53211yescharacteristic
160C2×C271#53711yescharacteristic
161C2×C271#54612yes—
162C2×C271#54712yes—
163D271—24no—
164D271—81no—
165C271#243011yescharacteristic, complemented, Sylow
166#2430—2711nomaximal, Sylow
167C22×#41—2711no—
168C2×#123—2711no—
169C2×#121—2712no—
170#395—2711no—
171#394D27111yescharacteristic, complemented
172#393—2711no—
173#392—2711no—
174#386—2712no—
175#386—2712no—
176#385—2712no—
177C2×#95—2711no—
178C2×#41—5424no—
179C2×#41—5428no—
180C22×Q8—2711no—
181C22×D4—2712no—
182C22×D4—2712no—
183C22×D4—2711no—
184C22×D4C2×D27111yescharacteristic
185C23×C4—2711no—
186C23×C4—2712no—
187C23×C4—2712no—
188#123—5424no—
189#121—5428no—
190C2×C4⋊C4—2712no—
191C2×C4⋊C4—2711no—
192C2×C4⋊C4C2×D27111yescharacteristic
193C2×C22⋊C4—2712no—
194C2×C22⋊C4—2712no—
195C2×C22⋊C4—2711no—
196C2×C22⋊C4—2711no—
197C2×C22⋊C4—2711no—
198C2×C22⋊C4C2×D27112yes—
199C2×C22⋊C4C2×D27112yes—
200#95—2712no—
201#95—2711no—
202#95—2712no—
203#95—2712no—
204#95C2×D27111yescharacteristic
205C24—2711no—
206C24—2711no—
207C24C22×D27112yes—
208#41—54232no—
209C2×Q8—5424no—
210C2×Q8—5422no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—2718no—
214C2×D4—5424no—
215C2×D4—5422no—
216C2×D4—5424no—
217C2×D4—5424no—
218C2×D4—5424no—
219C2×D4—5424no—
220C2×D4—10842no—
221C22×C4—2712no—
222C22×C4—2711no—
223C22×C4—2712no—
224C22×C4—2714no—
225C22×C4—2711no—
226C22×C4—5424no—
227C22×C4—5424no—
228C22×C4—5424no—
229C22×C4—5424no—
230C22×C4—5424no—
231C22×C4—5424no—
232C22×C4—10842no—
233C22×C4C22×D27112yes—
234C22×C4C22×D27111yescharacteristic
235C22×C4C22×D27111yescharacteristic
236C22×C4C22×D27111yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—5424no—
239C4⋊C4—5422no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—5424no—
243C22⋊C4—5422no—
244C22⋊C4—5422no—
245C22⋊C4—5424no—
246C22⋊C4—5422no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—2712no—
254C23—2714no—
255C23—5422no—
256C23—5421no—
257C23—5421no—
258C23—5421no—
259C23—5422no—
260C23—5422no—
261C23—5421no—
262C23—5421no—
263C23D4×D27114yes—
264C23C23×D27111yescharacteristic
265Q8—5428no—
266D4—28no—
267D4—54216no—
268D4—10844no—
269D4—10848no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—2712no—
280C2×C4—2714no—
281C2×C4—2714no—
282C2×C4—5428no—
283C2×C4—54216no—
284C2×C4—5422no—
285C2×C4—5422no—
286C2×C4—5422no—
287C2×C4—5422no—
288C2×C4—5422no—
289C2×C4—5421no—
290C2×C4—5422no—
291C2×C4—10848no—
292C2×C4D4×D27112yes—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—2714no—
301C22—5422no—
302C22—5424no—
303C22—5424no—
304C22—5422no—
305C22—10841no—
306C22—10842no—
307C22—10842no—
308C22—10841no—
309C22—10841no—
310C22C2×D4×D27111yescharacteristic
311C22C2×D4×D27112yes—
312C22#41×D27112yes—
313C22#2257620511yescharacteristic
314C22#2257620711yescharacteristic
315C4—22no—
316C4—41no—
317C4—42no—
318C4—41no—
319C4—5424no—
320C4—5422no—
321C4—5424no—
322C4—10841no—
323C2—44no—
324C2—5424no—
325C2—21681no—
326C2#120×D27111yescharacteristic
327C2#3177444611yescharacteristic
328C2#3177447611yescharacteristic
329C2#3177453612yes—
330C2#3177454512yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#394×C271C2
hypercenter#394D271
hypocenterC271#2430
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#16864653C2semidirect
2. C2×#22576145C2semidirect
3. C2×#22576133C2semidirect
4. #31773817C2semidirect
5. #31773810C2semidirect
6. #31773805C2semidirect
7. #31773774C2semidirect
8. #31773773C2semidirect
9. #31773769C2semidirect
10. #95×D271C2semidirect
11. #394×C271C2semidirect
51. C2×C22⋊D271D4semidirect
63. C2×C4×D271D4semidirect
105. C2×Dic271#41semidirect
165. C271#2430semidirect
171. #394D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646532
C2×#225761452
C2×#225761332
#317738172
#317738102
#317738052
#317737742
#317737732
#317737692
#95×D2712
#394×C2712
#2430271