Finite Group Databasebeta

← #27179460

Subgroups — #27179460

Order 12928.

688 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 688 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×#15692239C211yesmaximal, characteristic, complemented
2C2×#15692236C211yesmaximal, characteristic, complemented
3C2×#15692227C211yesmaximal, characteristic
4C22×C4×Dic101C211yesmaximal, characteristic, complemented
5#20265077C212yesmaximal, complemented
6#20265074C212yesmaximal, complemented
7#20265070C212yesmaximal, complemented
8#20265066C212yesmaximal, complemented
9#20265056C211yesmaximal, characteristic, complemented
10#20265035C211yesmaximal, characteristic, complemented
11#386×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101C2211yescharacteristic
13C22×C22⋊D101C2211yescharacteristic
14C23×Dic101C2211yescharacteristic
15C23×Dic101C2211yescharacteristic
16#15692239—24no—
17#15692236—24no—
18#15692227—24no—
19C2×C4⋊Dic101C2212yes—
20C2×#13108053C2212yes—
21C2×#13108053C2212yes—
22C2×#13108053C2211yescharacteristic
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C22⋊Dic101C2211yescharacteristic
25C2×C22⋊Dic101C2212yes—
26C2×#13108051C2212yes—
27C2×#13108051C2211yescharacteristic
28C2×#13108051C2212yes—
29C2×C4×Dic101—24no—
30C2×C4×Dic101C2212yes—
31C2×C4×Dic101C2212yes—
32#15692162C2212yes—
33#15692162C2212yes—
34C22×D4×C101C2211yescharacteristic
35C23×C4×C101C2211yescharacteristic
36C2×C22⋊C4×C101C2211yescharacteristic
37C2×C22⋊C4×C101C2212yes—
38#95×C101C2212yes—
39C23×D101C2311yescharacteristic
40C2×C22⋊D101—22no—
41C2×C22⋊D101—22no—
42C2×C22⋊D101—22no—
43C2×C22⋊D101—22no—
44C2×C22⋊D101—24no—
45C22×Dic101—22no—
46C22×Dic101—22no—
47C22×Dic101—24no—
48C22×Dic101D414yes—
49C22×Dic101D414yescomplemented
50C22×Dic101C2311yescharacteristic
51C22×Dic101C2312yes—
52C22×Dic101C2311yescharacteristic
53C22×Dic101C2312yes—
54C22×Dic101C2311yescharacteristic
55C4⋊Dic101—24no—
56#13108053—22no—
57#13108053—24no—
58#13108053—24no—
59C22⋊Dic101—22no—
60C22⋊Dic101—22no—
61C22⋊Dic101—24no—
62#13108051—24no—
63#13108051—22no—
64#13108051—24no—
65C4×Dic101—24no—
66C4×Dic101—24no—
67C24×C101C2311yescharacteristic
68C24×C101C2311yescharacteristic
69C2×D4×C101—22no—
70C2×D4×C101—22no—
71C2×D4×C101—22no—
72C22×C4×C101—22no—
73C22×C4×C101—24no—
74C22×C4×C101C2311yescharacteristic
75C22×C4×C101C2312yes—
76C22×C4×C101C2312yes—
77C22⋊C4×C101—22no—
78C22⋊C4×C101—24no—
79C23×C101—22no—
80C23×C101—22no—
81C23×C101—21no—
82C23×C101—21no—
83C23×C101—22no—
84C23×C101—21no—
85C23×C101—22no—
86C23×C101C2×D412yes—
87C23×C101C2×D412yes—
88C23×C101#4112yes—
89C23×C101C2411yescharacteristic
90C22×D101—21no—
91C22×D101—21no—
92C22×D101—22no—
93C22×D101—22no—
94C22×D101—21no—
95D4×C101—44no—
96C2×C4×C101—28no—
97C2×C4×C101—22no—
98C2×C4×C101—22no—
99C2×C4×C101—22no—
100C2×C4×C101—21no—
101C2×C4×C101—21no—
102C2×C4×C101—22no—
103C2×C4×C101—21no—
104C2×C4×C101—22no—
105C2×C4×C101#4114yes—
106C22⋊D101—44no—
107C22⋊D101—44no—
108C2×Dic101—28no—
109C2×Dic101—22no—
110C2×Dic101—22no—
111C2×Dic101—22no—
112C2×Dic101—28no—
113C2×Dic101—22no—
114C2×Dic101—21no—
115C2×Dic101—21no—
116C2×Dic101—22no—
117C2×Dic101—22no—
118C2×Dic101—21no—
119C2×Dic101C2×D412yes—
120C2×Dic101C2×D414yes—
121C2×Dic101C2×D412yes—
122C2×Dic101#4114yes—
123C22×C101—22no—
124C22×C101—22no—
125C22×C101—24no—
126C22×C101—24no—
127C22×C101—41no—
128C22×C101—41no—
129C22×C101—41no—
130C22×C101—42no—
131C22×C101—42no—
132C22×C101#12114yes—
133C22×C101C22×D411yescharacteristic
134C22×C101C22×D411yescharacteristic
135C22×C101C2×#4112yes—
136C22×C101C2×#4112yes—
137C22×C101C2×#4111yescharacteristic
138C2×D101—41no—
139C2×D101—41no—
140C2×D101—41no—
141C2×D101—42no—
142C2×D101—42no—
143C4×C101—24no—
144C4×C101—41no—
145C4×C101—42no—
146Dic101—24no—
147Dic101—22no—
148Dic101—41no—
149Dic101—42no—
150Dic101#12314yescomplemented
151C2×C101—24no—
152C2×C101—81no—
153C2×C101C2×#12111yescharacteristic
154C2×C101C2×#12311yescharacteristic
155C2×C101#52511yescharacteristic
156C2×C101#52912yes—
157C2×C101#53212yes—
158D101—81no—
159#2854—1011nomaximal, Sylow
160C101#285411yescharacteristic, complemented, Sylow
161C2×#123—1011no—
162C2×#123—1011no—
163C2×#121—1011no—
164C22×C42—1011no—
165#390—1012no—
166#389—1012no—
167#388—1012no—
168#388—1012no—
169#386—1011no—
170#386D10111yescharacteristic, complemented
171#380—1011no—
172C22×D4—1011no—
173C22×D4—1011no—
174C22×D4C2×D10111yescharacteristic
175C23×C4—1011no—
176C23×C4—1011no—
177C23×C4C2×D10111yescharacteristic
178#123—2024no—
179#123—2024no—
180#121—2024no—
181C2×C4⋊C4—1011no—
182C2×C4⋊C4—1012no—
183C2×C4⋊C4—1012no—
184C2×C4⋊C4—1012no—
185C2×C22⋊C4—1011no—
186C2×C22⋊C4—1011no—
187C2×C22⋊C4—1012no—
188C2×C22⋊C4—1011no—
189C2×C22⋊C4—1012no—
190C2×C22⋊C4—1012no—
191C2×C22⋊C4C2×D10111yescharacteristic
192C2×C22⋊C4C2×D10112yes—
193C2×C42—1012no—
194C2×C42—1012no—
195C2×C42—2024no—
196#95—1012no—
197#95—1012no—
198#95C2×D10112yes—
199C24—1011no—
200C24C22×D10111yescharacteristic
201C24C22×D10111yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—2022no—
206C2×D4—2024no—
207C2×D4—2022no—
208C2×D4—2022no—
209C2×D4—2022no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—1011no—
213C22×C4—1014no—
214C22×C4—1014no—
215C22×C4—1011no—
216C22×C4—1012no—
217C22×C4—1012no—
218C22×C4—1011no—
219C22×C4—2024no—
220C22×C4—2022no—
221C22×C4—2022no—
222C22×C4C22×D10112yes—
223C22×C4C22×D10111yescharacteristic
224C22×C4C22×D10112yes—
225C4⋊C4—2024no—
226C4⋊C4—2022no—
227C4⋊C4—2024no—
228C4⋊C4—2024no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—2022no—
232C22⋊C4—2024no—
233C22⋊C4—2022no—
234C22⋊C4—2024no—
235C22⋊C4—2022no—
236C22⋊C4—2024no—
237C42—2024no—
238C42—2024no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—2021no—
247C23—2021no—
248C23—2022no—
249C23—2022no—
250C23—2021no—
251C23D4×D10112yes—
252C23D4×D10112yes—
253C23#1310807412yes—
254C23C23×D10111yescharacteristic
255D4—44no—
256D4—4044no—
257D4—4044no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—1012no—
268C2×C4—1014no—
269C2×C4—1012no—
270C2×C4—1014no—
271C2×C4—2028no—
272C2×C4—2022no—
273C2×C4—2021no—
274C2×C4—2021no—
275C2×C4—2022no—
276C2×C4—2022no—
277C2×C4—2021no—
278C2×C4—2022no—
279C2×C4—2022no—
280C2×C4—2022no—
281C2×C4—2028no—
282C2×C4#1310807314yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—4041no—
293C22—4041no—
294C22—4041no—
295C22—4042no—
296C22—4042no—
297C22#1569222814yes—
298C22C2×D4×D10111yescharacteristic
299C22C2×D4×D10111yescharacteristic
300C22#41×D10112yes—
301C22C2×#1310807312yes—
302C22C2×#1310807411yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—1014no—
307C4—4041no—
308C4—4042no—
309C4—2024no—
310C4—2022no—
311C2—24no—
312C2—81no—
313C2—8081no—
314C2C2×#1569222811yescharacteristic
315C2#123×D10111yescharacteristic
316C2#2026568311yescharacteristic
317C2#2026570712yes—
318C2#2026572212yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#386×C101C2
prime radicalC2×#15692227C2
hypercenter#386D101
hypocenterC101#2854
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#15692239C2semidirect
2. C2×#15692236C2semidirect
4. C22×C4×Dic101C2semidirect
5. #20265077C2semidirect
6. #20265074C2semidirect
7. #20265070C2semidirect
8. #20265066C2semidirect
9. #20265056C2semidirect
10. #20265035C2semidirect
11. #386×C101C2semidirect
49. C22×Dic101D4semidirect
150. Dic101#123semidirect
160. C101#2854semidirect
170. #386D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922392
C2×#156922362
C2×#156922272
C22×C4×Dic1012
#202650772
#202650742
#202650702
#202650662
#202650562
#202650352
#386×C1012
#2854101