Finite Group Databasebeta

← #27171537

Subgroups — #27171537

Order 12928.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15692224C211yesmaximal, characteristic
2#20265650C211yesmaximal, characteristic, complemented
3C2×#15692169C211yesmaximal, characteristic, complemented
4#20265033C211yesmaximal, characteristic, complemented
5#20264944C211yesmaximal, characteristic, complemented
6#20264943C211yesmaximal, characteristic, complemented
7#379×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101—21no—
9C22×C22⋊D101C2211yescharacteristic
10#15692224—22no—
11#15692224C414yescomplemented
12C4×C22⋊D101—22no—
13#15692215—22no—
14#15692211—21no—
15C2×C22⋊Dic101—21no—
16C2×C22⋊Dic101—22no—
17C2×C22⋊Dic101C2211yescharacteristic
18C2×C22⋊Dic101C2211yescharacteristic
19C2×C22⋊Dic101C2211yescharacteristic
20C22⋊C4×D101—21no—
21#15692169—22no—
22C25×C101C2211yescharacteristic
23C2×C22⋊C4×C101—22no—
24C2×C22⋊C4×C101C2211yescharacteristic
25C2×C22⋊C4×C101C2211yescharacteristic
26C23×D101D411yescharacteristic
27C2×C22⋊D101—21no—
28C2×C22⋊D101—21no—
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—24no—
32C2×C22⋊D101—44no—
33C2×C22⋊D101C2×C412yes—
34C2×C22⋊D101D412yes—
35C22×Dic101—21no—
36C22×Dic101—21no—
37C22×Dic101D411yescharacteristic
38C22×Dic101D412yescomplemented
39C2×C4×D101—22no—
40#13108053—22no—
41C22⋊Dic101—21no—
42C22⋊Dic101—21no—
43C22⋊Dic101—21no—
44C22⋊Dic101—24no—
45C22⋊Dic101—21no—
46C22⋊Dic101—21no—
47C22⋊Dic101—44no—
48C22⋊Dic101C2×C412yescomplemented
49#13108051—22no—
50C4×Dic101—22no—
51C24×C101—21no—
52C24×C101—21no—
53C24×C101—21no—
54C24×C101—21no—
55C24×C101—21no—
56C24×C101—21no—
57C24×C101—44no—
58C24×C101C2×C412yescomplemented
59C24×C101C2311yescharacteristic
60C22×C4×C101—21no—
61C22×C4×C101D412yescomplemented
62C22⋊C4×C101—21no—
63C22⋊C4×C101—21no—
64C22⋊C4×C101—21no—
65C22⋊C4×C101—21no—
66C22⋊C4×C101—44no—
67C23×C101—24no—
68C23×C101—24no—
69C23×C101—24no—
70C23×C101—22no—
71C23×C101—22no—
72C23×C101—22no—
73C23×C101—44no—
74C23×C101—42no—
75C23×C101—42no—
76C23×C101—42no—
77C23×C101—42no—
78C23×C101—44no—
79C23×C101—44no—
80C23×C101—44no—
81C23×C101—41no—
82C23×C101—41no—
83C23×C101—41no—
84C23×C101—41no—
85C23×C101C22×C411yescharacteristic
86C23×C101C2×D411yescharacteristic
87C23×C101C2×D411yescharacteristic
88C23×C101C2×D411yescharacteristic
89C23×C101C2×D411yescharacteristic
90C23×C101#4111yescharacteristic
91C23×C101#4111yescharacteristic
92C22×D101—21no—
93C22×D101—21no—
94C22×D101—42no—
95C22×D101C22⋊C412yescomplemented
96C2×C4×C101—22no—
97C2×C4×C101—42no—
98C2×C4×C101—42no—
99C2×C4×C101—41no—
100C22⋊D101—22no—
101C22⋊D101—22no—
102C22⋊D101—22no—
103C22⋊D101—22no—
104C22⋊D101—48no—
105C2×Dic101—21no—
106C2×Dic101—21no—
107C2×Dic101—22no—
108C2×Dic101—22no—
109C2×Dic101—42no—
110C2×Dic101—42no—
111C2×Dic101—42no—
112C2×Dic101—41no—
113C2×Dic101C22⋊C412yescomplemented
114C4×D101—82no—
115C22×C101—22no—
116C22×C101—22no—
117C22×C101—22no—
118C22×C101—24no—
119C22×C101—24no—
120C22×C101—24no—
121C22×C101—41no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101—41no—
125C22×C101—42no—
126C22×C101—44no—
127C22×C101—44no—
128C22×C101—44no—
129C22×C101—42no—
130C22×C101—42no—
131C22×C101—42no—
132C22×C101—44no—
133C22×C101C2×C22⋊C411yescharacteristic
134C22×C101C4×D411yescharacteristic
135C22×C101C4×D411yescharacteristic
136C22×C101#12011yescharacteristic
137C22×C101#12111yescharacteristic
138C22×C101#12111yescharacteristic
139C22×C101#12311yescharacteristic
140C2×D101—22no—
141C2×D101—42no—
142C2×D101—42no—
143C2×D101—81no—
144C4×C101—42no—
145C4×C101—81no—
146Dic101—22no—
147Dic101—42no—
148Dic101—42no—
149Dic101—81no—
150C2×C101—21no—
151C2×C101—21no—
152C2×C101—21no—
153C2×C101—21no—
154C2×C101—21no—
155C2×C101—21no—
156C2×C101—44no—
157C2×C101#38611yescharacteristic
158C2×C101B(4,𝔽2)12yes—
159D101—42no—
160#1889—1011nomaximal, Sylow
161C101#188911yescharacteristic, complemented, Sylow
162C2×#120—1011no—
163#518—1011no—
164C2×#99—1011no—
165#379—1011no—
166#379D10111yescharacteristic, complemented
167#342—1011no—
168#342—1011no—
169C25C2×D10111yescharacteristic
170C22×D4—1011no—
171C22×D4—2021no—
172#120—1014no—
173#120—2022no—
174C4×D4—2022no—
175C4×D4—2022no—
176#117—2021no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1011no—
179C2×C22⋊C4—1011no—
180C2×C22⋊C4—1011no—
181C2×C22⋊C4—2021no—
182C2×C22⋊C4—2022no—
183C2×C22⋊C4—2021no—
184C2×C22⋊C4C2×D10111yescharacteristic
185C2×C22⋊C4C2×D10111yescharacteristic
186#99—2022no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—44no—
194C24—1011no—
195C24C4×D10112yescomplemented
196C24C22×D10111yescharacteristic
197C2×D4—1012no—
198C2×D4—1012no—
199C2×D4—4044no—
200C2×D4—2024no—
201C2×D4—2021no—
202C2×D4—2021no—
203C2×D4—2021no—
204C2×D4—2021no—
205C22×C4—21no—
206C22×C4—1011no—
207C22×C4—1012no—
208C22×C4—2022no—
209C22×C4—2021no—
210C22×C4—2021no—
211C22×C4C22⋊D10112yescomplemented
212C4⋊C4—2022no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—44no—
218C22⋊C4—1012no—
219C22⋊C4—4044no—
220C22⋊C4—2022no—
221C22⋊C4—2024no—
222C22⋊C4—2021no—
223C22⋊C4—2021no—
224C22⋊C4—2021no—
225C22⋊C4—2021no—
226C22⋊C4—2021no—
227C42—2022no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—44no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—1012no—
247C23—4042no—
248C23—2021no—
249C23—2021no—
250C23C2×C4×D10111yescharacteristic
251C23D4×D10111yescharacteristic
252C23D4×D10111yescharacteristic
253C23D4×D10111yescharacteristic
254C23C2×C22⋊D10111yescharacteristic
255C23#1310807311yescharacteristic
256C23#1310807411yescharacteristic
257D4—4048no—
258D4—2022no—
259D4—2022no—
260D4—2022no—
261D4—2022no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—8082no—
267C2×C4—1012no—
268C2×C4—4042no—
269C2×C4—4041no—
270C2×C4—4042no—
271C2×C4—4042no—
272C2×C4—2022no—
273C2×C4—2022no—
274C2×C4—2021no—
275C2×C4—2021no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—24no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—44no—
292C22—42no—
293C22—44no—
294C22—8081no—
295C22—4042no—
296C22—4042no—
297C22—2022no—
298C22C22⋊C4×D10111yescharacteristic
299C22#1569221511yescharacteristic
300C22C4×C22⋊D10111yescharacteristic
301C22#1569222311yescharacteristic
302C22#1569222611yescharacteristic
303C22#1569222811yescharacteristic
304C22#1569223711yescharacteristic
305C4—42no—
306C4—81no—
307C4—8081no—
308C4—4042no—
309C4—4042no—
310C4—2022no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—44no—
318C2—4042no—
319C2#2026505311yescharacteristic
320C2#2026536312yes—
subgroupG/H
centerC22#15692223
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#379×C101C2
socleC22×C101#120
prime radicalC2×#15692224C2
cosocleC24×C101C23
hypercenter#379D101
hypocenterC101#1889
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #20265650C2semidirect
3. C2×#15692169C2semidirect
4. #20265033C2semidirect
5. #20264944C2semidirect
6. #20264943C2semidirect
7. #379×C101C2semidirect
11. #15692224C4semidirect
38. C22×Dic101D4semidirect
48. C22⋊Dic101C2×C4semidirect
58. C24×C101C2×C4semidirect
61. C22×C4×C101D4semidirect
95. C22×D101C22⋊C4semidirect
113. C2×Dic101C22⋊C4semidirect
161. C101#1889semidirect
166. #379D101semidirect
195. C24C4×D101semidirect
211. C22×C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922242
#202656502
C2×#156921692
#202650332
#202649442
#202649432
#379×C1012
#1889101