Finite Group Databasebeta

← #27174346

Subgroups — #27174346

Order 12928.

944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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×#15692217C211yesmaximal, characteristic, complemented
3C2×#15692215C212yesmaximal, complemented
4#20265097C211yesmaximal, characteristic, complemented
5#20265092C212yesmaximal, complemented
6#20265090C211yesmaximal, characteristic, complemented
7#20265086C212yesmaximal, complemented
8#20265085C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic101C212yesmaximal, complemented
10#20265024C211yesmaximal, characteristic, complemented
11#394×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—28no—
13C2×#13108073—24no—
14C22×Dic202C2211yescharacteristic
15C22×D404C2211yescharacteristic
16C22×C22⋊D101C2212yes—
17C22×C22⋊D101C2212yes—
18C23×Dic101C2212yes—
19C22×C4×D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21#15692217—24no—
22#15692215—28no—
23C2×#13108053C2212yes—
24C2×C22⋊Dic101C2212yes—
25C2×#13108051C2212yes—
26C2×#13108051C2211yescharacteristic
27C2×#13108051C2212yes—
28C2×#13108051C2211yescharacteristic
29C2×C4×Dic101C2212yes—
30C2×C4×Dic101C2211yescharacteristic
31#15692162C2211yescharacteristic
32#15692162C2212yes—
33#15692162C2212yes—
34C22×D4×C101C2211yescharacteristic
35C2×C4⋊C4×C101C2211yescharacteristic
36C2×C22⋊C4×C101C2212yes—
37C2×C22⋊C4×C101C2212yes—
38#95×C101C2211yescharacteristic
39C23×D101C2311yescharacteristic
40C23×D101C2311yescharacteristic
41#13108073—232no—
42C2×Dic202—22no—
43C2×Dic202—24no—
44C2×D404—22no—
45C2×D404—22no—
46C2×D404—22no—
47C2×C22⋊D101—24no—
48C2×C22⋊D101—24no—
49C2×C22⋊D101—24no—
50C2×C22⋊D101—24no—
51C2×C22⋊D101—24no—
52C2×C22⋊D101—24no—
53C22×Dic101—24no—
54C22×Dic101—24no—
55C22×Dic101D418yescomplemented
56C22×Dic101C2311yescharacteristic
57C22×Dic101C2312yes—
58C22×Dic101C2312yes—
59C22×Dic101C2311yescharacteristic
60C2×C4×D101—24no—
61C2×C4×D101—24no—
62C2×C4×D101—22no—
63C2×C4×D101D414yescomplemented
64#13108053—24no—
65C22⋊Dic101—24no—
66#13108051—22no—
67#13108051—22no—
68#13108051—24no—
69#13108051—24no—
70C4×Dic101—24no—
71C4×Dic101—22no—
72C24×C101C2312yes—
73C2×D4×C101—24no—
74C2×D4×C101—22no—
75C22×C4×C101C2311yescharacteristic
76C22×C4×C101C2312yes—
77C22×C4×C101C2311yescharacteristic
78C22×C4×C101C2311yescharacteristic
79C4⋊C4×C101—22no—
80C22⋊C4×C101—24no—
81C22⋊C4×C101—24no—
82C23×C101—22no—
83C23×C101—22no—
84C23×C101—22no—
85C23×C101—22no—
86C23×C101—22no—
87C23×C101—22no—
88C23×C101C2×D414yes—
89C23×C101C2411yescharacteristic
90C22×D101—22no—
91C22×D101—21no—
92C22×D101—22no—
93C22×D101—22no—
94C22×D101—22no—
95C22×D101—21no—
96C22×D101—21no—
97C22×D101—21no—
98C22×D101C2×D412yes—
99C22×D101#4112yescomplemented
100D4×C101—28no—
101C2×C4×C101—21no—
102C2×C4×C101—22no—
103C2×C4×C101—21no—
104C2×C4×C101—22no—
105C2×C4×C101—22no—
106C2×C4×C101—21no—
107C2×C4×C101—22no—
108C2×C4×C101—22no—
109C2×C4×C101C2×D412yes—
110C2×C4×C101#4112yescomplemented
111C22⋊D101—216no—
112C22⋊D101—216no—
113C2×Dic101—22no—
114C2×Dic101—22no—
115C2×Dic101—22no—
116C2×Dic101—22no—
117C2×Dic101—216no—
118C2×Dic101—21no—
119C2×Dic101—22no—
120C2×Dic101C2×D414yes—
121C2×Dic101C2×D414yes—
122C2×Dic101C2×D412yes—
123C2×Dic101#4114yescomplemented
124D404—28no—
125C4×D101—28no—
126C4×D101—44no—
127Dic202—28no—
128C22×C101—24no—
129C22×C101—24no—
130C22×C101—24no—
131C22×C101—42no—
132C22×C101—42no—
133C22×C101—42no—
134C22×C101—42no—
135C22×C101C22×D411yescharacteristic
136C22×C101C22×D412yes—
137C22×C101C2×#4111yescharacteristic
138C22×C101C2×#4112yes—
139C22×C101C2×#4111yescharacteristic
140C2×D101—22no—
141C2×D101—22no—
142C2×D101—22no—
143C2×D101—24no—
144C2×D101—22no—
145C2×D101—41no—
146C2×D101—42no—
147C2×D101—42no—
148C2×D101—42no—
149C2×D101—41no—
150C4×C101—22no—
151C4×C101—22no—
152C4×C101—41no—
153C4×C101—42no—
154Dic101—24no—
155Dic101—24no—
156Dic101—41no—
157Dic101#12014yes—
158C2×C101—44no—
159C2×C101C2×#12011yescharacteristic
160C2×C101#52512yes—
161C2×C101#52511yescharacteristic
162C2×C101#53212yes—
163C2×C101#53211yescharacteristic
164D101—42no—
165D101—42no—
166#2388—1011nomaximal, Sylow
167C101#238811yescharacteristic, complemented, Sylow
168C22×#41—1011no—
169C2×C4×D4—1011no—
170C2×C4×D4—1012no—
171#395—1011no—
172#394—1012no—
173#394D10111yescharacteristic, complemented
174#393—1011no—
175#392—1011no—
176#392—1012no—
177C4×C22⋊C4—1012no—
178C4×C22⋊C4—1011no—
179C2×#41—2028no—
180C2×#41—2024no—
181C22×Q8—1011no—
182C22×D4—1012no—
183C22×D4—1012no—
184C22×D4—1011no—
185C22×D4C2×D10111yescharacteristic
186C23×C4—1012no—
187C23×C4—1011no—
188C23×C4—1011no—
189C4×D4—2028no—
190C4×D4—2024no—
191C2×C4⋊C4—1012no—
192C2×C4⋊C4C2×D10111yescharacteristic
193C2×C22⋊C4—1012no—
194C2×C22⋊C4—1012no—
195C2×C22⋊C4—1011no—
196C2×C22⋊C4—1011no—
197C2×C22⋊C4—1012no—
198C2×C22⋊C4C2×D10112yes—
199C2×C22⋊C4C2×D10112yes—
200C2×C42—1012no—
201C2×C42—1011no—
202#95—1011no—
203#95—1012no—
204#95—1012no—
205#95C2×D10111yescharacteristic
206C24—1011no—
207C24—1011no—
208C24C22×D10112yes—
209#41—20232no—
210C2×Q8—2024no—
211C2×Q8—2022no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—2022no—
215C2×D4—2022no—
216C2×D4—2024no—
217C2×D4—2024no—
218C2×D4—2024no—
219C2×D4—2024no—
220C2×D4—2024no—
221C2×D4—2024no—
222C2×D4—2022no—
223C22×C4—1011no—
224C22×C4—1018no—
225C22×C4—1012no—
226C22×C4—1012no—
227C22×C4—1014no—
228C22×C4—1011no—
229C22×C4—2024no—
230C22×C4—2022no—
231C22×C4—2024no—
232C22×C4—2024no—
233C22×C4—2024no—
234C22×C4C22×D10112yes—
235C22×C4C22×D10111yescharacteristic
236C22×C4C22×D10111yescharacteristic
237C22×C4C22×D10111yescharacteristic
238C4⋊C4—22no—
239C4⋊C4—2024no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—2024no—
243C22⋊C4—2024no—
244C22⋊C4—2022no—
245C22⋊C4—2024no—
246C22⋊C4—2022no—
247C42—2022no—
248C42—2024no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—1012no—
256C23—1012no—
257C23—2022no—
258C23—2021no—
259C23—2022no—
260C23—2022no—
261C23—2021no—
262C23—2022no—
263C23—2021no—
264C23—2021no—
265C23D4×D10114yes—
266C23C23×D10111yescharacteristic
267Q8—2028no—
268D4—28no—
269D4—2028no—
270D4—20216no—
271D4—20216no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—1014no—
281C2×C4—1012no—
282C2×C4—1014no—
283C2×C4—1014no—
284C2×C4—4044no—
285C2×C4—2028no—
286C2×C4—20216no—
287C2×C4—2021no—
288C2×C4—2022no—
289C2×C4—2022no—
290C2×C4—2022no—
291C2×C4—2022no—
292C2×C4—2022no—
293C2×C4D4×D10112yes—
294C2×C4#1310807312yescomplemented
295C22—24no—
296C22—24no—
297C22—24no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—4041no—
303C22—4042no—
304C22—4042no—
305C22—4042no—
306C22—4041no—
307C22—2022no—
308C22—2024no—
309C22—2022no—
310C22—2022no—
311C22—2022no—
312C22C2×D4×D10112yes—
313C22C2×D4×D10111yescharacteristic
314C22#41×D10111yescharacteristic
315C22#41×D10112yes—
316C22C2×#1310807311yescharacteristic
317C4—22no—
318C4—22no—
319C4—41no—
320C4—42no—
321C4—1014no—
322C4—4041no—
323C4—2024no—
324C4—2024no—
325C2—44no—
326C2—4042no—
327C2—4042no—
328C2#120×D10111yescharacteristic
329C2#2026568512yes—
330C2#2026568711yescharacteristic
331C2#2026572212yes—
332C2#2026572611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#394×C101C2
hypercenter#394D101
hypocenterC101#2388
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#13108073C2semidirect
2. C2×#15692217C2semidirect
3. C2×#15692215C2semidirect
4. #20265097C2semidirect
5. #20265092C2semidirect
6. #20265090C2semidirect
7. #20265086C2semidirect
8. #20265085C2semidirect
9. C22⋊C4×Dic101C2semidirect
10. #20265024C2semidirect
11. #394×C101C2semidirect
55. C22×Dic101D4semidirect
63. C2×C4×D101D4semidirect
99. C22×D101#41semidirect
110. C2×C4×C101#41semidirect
123. C2×Dic101#41semidirect
167. C101#2388semidirect
173. #394D101semidirect
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×#156922172
C2×#156922152
#202650972
#202650922
#202650902
#202650862
#202650852
C22⋊C4×Dic1012
#202650242
#394×C1012
#2388101