Finite Group Databasebeta

← #27174344

Subgroups — #27174344

Order 12928.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 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×#13108073C211yesmaximal, characteristic, complemented
2C2×D4×Dic101C212yesmaximal, complemented
3C2×#15692215C211yesmaximal, characteristic, complemented
4#20265098C211yesmaximal, characteristic, complemented
5#20265093C212yesmaximal, complemented
6#20265091C211yesmaximal, characteristic, complemented
7#20265089C211yesmaximal, characteristic, complemented
8#20265085C212yesmaximal, complemented
9C22⋊C4×Dic101C212yesmaximal, complemented
10#20265024C211yesmaximal, characteristic, complemented
11#392×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—28no—
13C2×#13108073—24no—
14C22×Dic202C2211yescharacteristic
15C22×C22⋊D101C2212yes—
16C22×C22⋊D101C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18C23×Dic101C2212yes—
19C22×C4×D101C2211yescharacteristic
20D4×Dic101—28no—
21#15692215—24no—
22C2×C4⋊Dic101C2212yes—
23C2×#13108053C2211yescharacteristic
24C2×C22⋊Dic101C2212yes—
25C2×C22⋊Dic101C2212yes—
26C2×C22⋊Dic101C2212yes—
27C2×#13108051C2212yes—
28C2×#13108051C2211yescharacteristic
29C2×C4×Dic101C2212yes—
30C2×C4×Dic101C2211yescharacteristic
31#15692162C2211yescharacteristic
32#15692162C2212yes—
33#15692162C2212yes—
34C22×D4×C101C2211yescharacteristic
35C22×D4×C101C2212yes—
36C2×C22⋊C4×C101C2212yes—
37C2×C22⋊C4×C101C2211yescharacteristic
38#95×C101C2211yescharacteristic
39C23×D101C2311yescharacteristic
40#13108073—232no—
41C2×Dic202—22no—
42C2×Dic202—24no—
43C2×C22⋊D101—24no—
44C2×C22⋊D101—22no—
45C2×C22⋊D101—24no—
46C2×C22⋊D101—22no—
47C2×C22⋊D101—24no—
48C2×C22⋊D101—22no—
49C22×Dic101—24no—
50C22×Dic101—24no—
51C22×Dic101—22no—
52C22×Dic101—24no—
53C22×Dic101D418yescomplemented
54C22×Dic101C2311yescharacteristic
55C22×Dic101C2312yes—
56C22×Dic101C2312yes—
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C2×C4×D101—24no—
60C2×C4×D101D414yescomplemented
61C4⋊Dic101—24no—
62#13108053—22no—
63C22⋊Dic101—24no—
64C22⋊Dic101—24no—
65C22⋊Dic101—24no—
66#13108051—22no—
67#13108051—24no—
68C4×Dic101—22no—
69C4×Dic101—24no—
70C24×C101C2312yes—
71C24×C101C2311yescharacteristic
72C2×D4×C101—24no—
73C2×D4×C101—24no—
74C2×D4×C101—24no—
75C2×D4×C101—22no—
76C2×D4×C101—24no—
77C22×C4×C101C2311yescharacteristic
78C22×C4×C101C2311yescharacteristic
79C22×C4×C101C2312yes—
80C22⋊C4×C101—22no—
81C22⋊C4×C101—24no—
82C23×C101—21no—
83C23×C101—22no—
84C23×C101—22no—
85C23×C101—22no—
86C23×C101—22no—
87C23×C101—22no—
88C23×C101—22no—
89C23×C101—22no—
90C23×C101—22no—
91C23×C101—21no—
92C23×C101C2×D414yes—
93C23×C101#4112yescomplemented
94C23×C101C2411yescharacteristic
95C22×D101—22no—
96C22×D101—22no—
97C22×D101—21no—
98C22×D101—21no—
99C22×D101C2×D412yes—
100D4×C101—216no—
101D4×C101—28no—
102C2×C4×C101—22no—
103C2×C4×C101—22no—
104C2×C4×C101—22no—
105C2×C4×C101—21no—
106C2×C4×C101—22no—
107C2×C4×C101C2×D412yes—
108C2×C4×C101#4114yescomplemented
109C22⋊D101—28no—
110C22⋊D101—216no—
111C2×Dic101—21no—
112C2×Dic101—22no—
113C2×Dic101—22no—
114C2×Dic101—21no—
115C2×Dic101—216no—
116C2×Dic101—22no—
117C2×Dic101—21no—
118C2×Dic101—22no—
119C2×Dic101—22no—
120C2×Dic101—22no—
121C2×Dic101—44no—
122C2×Dic101C2×D414yes—
123C2×Dic101C2×D414yes—
124C2×Dic101C2×D412yes—
125C2×Dic101#4112yescomplemented
126C4×D101—28no—
127Dic202—28no—
128C22×C101—24no—
129C22×C101—22no—
130C22×C101—24no—
131C22×C101—24no—
132C22×C101—24no—
133C22×C101—41no—
134C22×C101—42no—
135C22×C101—42no—
136C22×C101—42no—
137C22×C101—42no—
138C22×C101—41no—
139C22×C101—42no—
140C22×C101C22×D411yescharacteristic
141C22×C101C22×D412yes—
142C22×C101C2×#4111yescharacteristic
143C22×C101C2×#4111yescharacteristic
144C22×C101C2×#4112yes—
145C2×D101—22no—
146C2×D101—22no—
147C2×D101—22no—
148C2×D101—42no—
149C2×D101—42no—
150C4×C101—22no—
151C4×C101—24no—
152C4×C101—41no—
153Dic101—24no—
154Dic101—22no—
155Dic101—41no—
156Dic101—42no—
157Dic101#12014yes—
158C2×C101—44no—
159C2×C101—42no—
160C2×C101C2×#12011yescharacteristic
161C2×C101#52512yes—
162C2×C101#52511yescharacteristic
163C2×C101#53212yes—
164C2×C101#53211yescharacteristic
165D101—42no—
166#2388—1011nomaximal, Sylow
167C101#238811yescharacteristic, complemented, Sylow
168C22×#41—1011no—
169C2×C4×D4—1012no—
170C2×C4×D4—1011no—
171#395—1011no—
172#394—1011no—
173#394—1012no—
174#393—1011no—
175#392—1012no—
176#392D10111yescharacteristic, complemented
177C4×C22⋊C4—1012no—
178C4×C22⋊C4—1011no—
179C2×#41—2028no—
180C2×#41—2024no—
181C22×Q8—1011no—
182C22×D4—1011no—
183C22×D4—1012no—
184C22×D4C2×D10112yes—
185C22×D4C2×D10111yescharacteristic
186C23×C4—1012no—
187C23×C4—1011no—
188C23×C4—1011no—
189C4×D4—2024no—
190C4×D4—2028no—
191C2×C4⋊C4—1011no—
192C2×C4⋊C4—1012no—
193C2×C22⋊C4—1012no—
194C2×C22⋊C4—1012no—
195C2×C22⋊C4—1012no—
196C2×C22⋊C4—1011no—
197C2×C22⋊C4—1012no—
198C2×C22⋊C4C2×D10111yescharacteristic
199C2×C22⋊C4C2×D10112yes—
200C2×C42—1011no—
201C2×C42—1012no—
202#95—1012no—
203#95—1012no—
204#95—1011no—
205#95C2×D10111yescharacteristic
206C24—1011no—
207C24C22×D10112yes—
208C24C22×D10111yescharacteristic
209#41—20232no—
210C2×Q8—2024no—
211C2×Q8—2022no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—2022no—
218C2×D4—2024no—
219C2×D4—2022no—
220C2×D4—2024no—
221C2×D4—2024no—
222C2×D4—2022no—
223C22×C4—1011no—
224C22×C4—1018no—
225C22×C4—1011no—
226C22×C4—1012no—
227C22×C4—1014no—
228C22×C4—1012no—
229C22×C4—1011no—
230C22×C4—2024no—
231C22×C4—2022no—
232C22×C4—2024no—
233C22×C4—2024no—
234C22×C4—2024no—
235C22×C4C22×D10111yescharacteristic
236C22×C4C22×D10111yescharacteristic
237C22×C4C22×D10112yes—
238C4⋊C4—2024no—
239C4⋊C4—2022no—
240C22⋊C4—22no—
241C22⋊C4—24no—
242C22⋊C4—2022no—
243C22⋊C4—2024no—
244C22⋊C4—2024no—
245C22⋊C4—2024no—
246C22⋊C4—2024no—
247C42—2024no—
248C42—2022no—
249C23—21no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—1012no—
260C23—2022no—
261C23—2021no—
262C23—2022no—
263C23—2021no—
264C23D4×D10114yes—
265C23#1310807312yescomplemented
266C23C23×D10111yescharacteristic
267Q8—2028no—
268D4—28no—
269D4—216no—
270D4—20216no—
271D4—2028no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—1014no—
278C2×C4—1012no—
279C2×C4—1014no—
280C2×C4—1012no—
281C2×C4—4044no—
282C2×C4—2028no—
283C2×C4—20216no—
284C2×C4—2022no—
285C2×C4—2022no—
286C2×C4—2021no—
287C2×C4—2022no—
288C2×C4—2022no—
289C2×C4—2021no—
290C2×C4—2022no—
291C2×C4—2022no—
292C2×C4—2021no—
293C2×C4D4×D10112yes—
294C2×C4#1310807314yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—42no—
307C22—4042no—
308C22—4042no—
309C22—2022no—
310C22—2022no—
311C22—2022no—
312C22C2×D4×D10112yes—
313C22C2×D4×D10111yescharacteristic
314C22#41×D10111yescharacteristic
315C22C2×#1310807311yescharacteristic
316C22C2×#1310807312yes—
317C4—22no—
318C4—24no—
319C4—41no—
320C4—1014no—
321C4—4041no—
322C4—4042no—
323C4—2024no—
324C4—2022no—
325C2—44no—
326C2—42no—
327C2—4042no—
328C2#120×D10111yescharacteristic
329C2#2026568312yes—
330C2#2026568711yescharacteristic
331C2#2026572212yes—
332C2#2026572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#392×C101C2
hypercenter#392D101
hypocenterC101#2388
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#13108073C2semidirect
2. C2×D4×Dic101C2semidirect
3. C2×#15692215C2semidirect
4. #20265098C2semidirect
5. #20265093C2semidirect
6. #20265091C2semidirect
7. #20265089C2semidirect
8. #20265085C2semidirect
9. C22⋊C4×Dic101C2semidirect
10. #20265024C2semidirect
11. #392×C101C2semidirect
53. C22×Dic101D4semidirect
60. C2×C4×D101D4semidirect
93. C23×C101#41semidirect
108. C2×C4×C101#41semidirect
125. C2×Dic101#41semidirect
167. C101#2388semidirect
176. #392D101semidirect
265. C23#13108073semidirect
294. C2×C4#13108073semidirect

Maximal subgroups

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

subgroupindex
C22×#131080732
C2×D4×Dic1012
C2×#156922152
#202650982
#202650932
#202650912
#202650892
#202650852
C22⋊C4×Dic1012
#202650242
#392×C1012
#2388101