Finite Group Databasebeta

← #27171544

Subgroups — #27171544

Order 12928.

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

normal subgroups only

This group has 386 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×#15692239C211yesmaximal, characteristic
2#20265650C211yesmaximal, characteristic
3C2×#15692169C211yesmaximal, characteristic
4#20265038C211yesmaximal, characteristic, complemented
5#20265038C211yesmaximal, characteristic, complemented
6#20264943C211yesmaximal, characteristic, complemented
7#342×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101C2211yescharacteristic
9C23×Dic101C2211yescharacteristic
10#15692239—22no—
11#15692239C414yescomplemented
12C4×C22⋊D101—22no—
13#15692215—22no—
14#15692211—21no—
15C2×C4⋊Dic101—21no—
16C2×C22⋊Dic101C2211yescharacteristic
17C2×C22⋊Dic101C2211yescharacteristic
18C2×#13108051—21no—
19C22⋊C4×D101—21no—
20#15692169—22no—
21#15692162—22no—
22#15692162—22no—
23C2×C22⋊C4×C101C2211yescharacteristic
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⋊D101C2×C412yes—
32C2×C22⋊D101D412yes—
33C22×Dic101—21no—
34C22×Dic101—21no—
35C22×Dic101—21no—
36C22×Dic101—21no—
37C22×Dic101—21no—
38C22×Dic101—21no—
39C22×Dic101—21no—
40C22×Dic101C2×C412yescomplemented
41C22×Dic101D411yescharacteristic
42C22×Dic101D412yes—
43C2×C4×D101—22no—
44C4⋊Dic101—24no—
45#13108053—22no—
46C22⋊Dic101—21no—
47C22⋊Dic101—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50#13108051—22no—
51#13108051—24no—
52C4×Dic101—22no—
53C24×C101C2311yescharacteristic
54C22×C4×C101—21no—
55C22×C4×C101—21no—
56C22×C4×C101D412yes—
57C22⋊C4×C101—21no—
58C22⋊C4×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101—21no—
62C22⋊C4×C101C2×C412yescomplemented
63C23×C101—42no—
64C23×C101C22×C411yescharacteristic
65C23×C101C2×D411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C23×C101C2×D411yescharacteristic
68C23×C101C2×D411yescharacteristic
69C23×C101#4111yescharacteristic
70C23×C101#4111yescharacteristic
71C22×D101—21no—
72C22×D101—21no—
73C22×D101—42no—
74C22×D101C22⋊C412yes—
75C2×C4×C101—22no—
76C2×C4×C101—22no—
77C2×C4×C101—42no—
78C2×C4×C101—42no—
79C2×C4×C101—42no—
80C2×C4×C101—41no—
81C22⋊D101—22no—
82C22⋊D101—22no—
83C22⋊D101—22no—
84C22⋊D101—22no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—22no—
90C2×Dic101—22no—
91C2×Dic101—41no—
92C2×Dic101—41no—
93C2×Dic101—41no—
94C2×Dic101—41no—
95C2×Dic101—42no—
96C2×Dic101—42no—
97C2×Dic101—41no—
98C2×Dic101C22⋊C412yes—
99C4×D101—82no—
100C22×C101—22no—
101C22×C101—22no—
102C22×C101—22no—
103C22×C101—41no—
104C22×C101—41no—
105C22×C101—41no—
106C22×C101—41no—
107C22×C101C2×C22⋊C411yescharacteristic
108C22×C101C4×D411yescharacteristic
109C22×C101C4×D411yescharacteristic
110C22×C101#12011yescharacteristic
111C22×C101#12111yescharacteristic
112C22×C101#12111yescharacteristic
113C22×C101#12311yescharacteristic
114C2×D101—22no—
115C2×D101—42no—
116C2×D101—42no—
117C2×D101—81no—
118C4×C101—42no—
119C4×C101—42no—
120C4×C101—81no—
121Dic101—22no—
122Dic101—42no—
123Dic101—42no—
124Dic101—81no—
125C2×C101—21no—
126C2×C101—21no—
127C2×C101—21no—
128C2×C101—21no—
129C2×C101—21no—
130C2×C101—21no—
131C2×C101#38611yescharacteristic
132C2×C101#45812yes—
133D101—42no—
134#1890—1011nomaximal, Sylow
135C101#189011yescharacteristic, complemented, Sylow
136C2×#123—1011no—
137#518—1011no—
138C2×#99—1011no—
139#381—1011no—
140#381—1011no—
141#342—1011no—
142#342D10111yescharacteristic, complemented
143C22×D4—1011no—
144C23×C4—1011no—
145#123—1014no—
146#123—2022no—
147C4×D4—2022no—
148C4×D4—2022no—
149#117—2021no—
150C2×C4⋊C4—2021no—
151C2×C22⋊C4—1011no—
152C2×C22⋊C4—1011no—
153C2×C22⋊C4—2021no—
154C2×C22⋊C4—2021no—
155C2×C22⋊C4C2×D10111yescharacteristic
156C2×C22⋊C4C2×D10111yescharacteristic
157C2×C22⋊C4C2×D10111yescharacteristic
158#99—2022no—
159#95—2022no—
160#95—2022no—
161C24—1011no—
162C24C22×D10111yescharacteristic
163C2×D4—1012no—
164C2×D4—1012no—
165C2×D4—2021no—
166C2×D4—2021no—
167C2×D4—2021no—
168C2×D4—2021no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—1011no—
172C22×C4—1012no—
173C22×C4—1012no—
174C22×C4—2022no—
175C22×C4—2021no—
176C22×C4—2021no—
177C22×C4—2021no—
178C22×C4—2021no—
179C22×C4—2021no—
180C22×C4—2021no—
181C22×C4—2021no—
182C22×C4D40412yes—
183C4⋊C4—2024no—
184C4⋊C4—2022no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—2024no—
191C22⋊C4—2022no—
192C22⋊C4—2021no—
193C22⋊C4—2021no—
194C22⋊C4—2021no—
195C22⋊C4—2021no—
196C22⋊C4C4×D10112yescomplemented
197C42—2022no—
198C23—42no—
199C23—1012no—
200C23—4042no—
201C23—2021no—
202C23—2021no—
203C23C2×C4×D10111yescharacteristic
204C23D4×D10111yescharacteristic
205C23D4×D10111yescharacteristic
206C23D4×D10111yescharacteristic
207C23C2×D40411yescharacteristic
208C23#1310807411yescharacteristic
209C23#1310807511yescharacteristic
210D4—2022no—
211D4—2022no—
212D4—2022no—
213D4—2022no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—8082no—
221C2×C4—1012no—
222C2×C4—4042no—
223C2×C4—4041no—
224C2×C4—4041no—
225C2×C4—4041no—
226C2×C4—4041no—
227C2×C4—4041no—
228C2×C4—4042no—
229C2×C4—2022no—
230C2×C4—2022no—
231C2×C4—2022no—
232C2×C4—2022no—
233C2×C4—2021no—
234C2×C4—2021no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—8081no—
243C22—4042no—
244C22—4042no—
245C22—2022no—
246C22C22⋊C4×D10111yescharacteristic
247C22#1569221711yescharacteristic
248C22C4×D40411yescharacteristic
249C22#1569222511yescharacteristic
250C22#1569222811yescharacteristic
251C22#1569223011yescharacteristic
252C22#1569224011yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—8081no—
257C4—4042no—
258C4—4042no—
259C4—2022no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—4042no—
267C2#2026505511yescharacteristic
268C2#2026536812yes—
subgroupG/H
centerC22#15692225
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101#120
prime radicalC22×C22⋊D101C22
cosocleC24×C101C23
hypercenter#342D101
hypocenterC101#1890
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #20265038C2semidirect
5. #20265038C2semidirect
6. #20264943C2semidirect
7. #342×C101C2semidirect
11. #15692239C4semidirect
40. C22×Dic101C2×C4semidirect
62. C22⋊C4×C101C2×C4semidirect
135. C101#1890semidirect
142. #342D101semidirect
196. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922392
#202656502
C2×#156921692
#202650382
#202649432
#342×C1012
#1890101