Finite Group Databasebeta

← #27171589

Subgroups — #27171589

Order 12928.

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

normal subgroups only

This group has 378 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
2C2×#15692211C211yesmaximal, characteristic, complemented
3C2×#15692169C211yesmaximal, characteristic
4C2×#15692169C211yesmaximal, characteristic
5#20265038C211yesmaximal, characteristic, complemented
6#20264944C211yesmaximal, characteristic, complemented
7#342×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101C2211yescharacteristic
9C23×Dic101C2211yescharacteristic
10#15692239—22no—
11#15692239—21no—
12#15692239—21no—
13#15692211—22no—
14#15692211—22no—
15C2×C4⋊Dic101C412yescomplemented
16C2×#13108053—21no—
17C2×C22⋊Dic101C2211yescharacteristic
18C2×C22⋊Dic101C2211yescharacteristic
19C2×#13108051C412yescomplemented
20C2×C4×Dic101—21no—
21#15692169—22no—
22#15692169—22no—
23#15692162—22no—
24C2×C22⋊C4×C101C2211yescharacteristic
25C2×C22⋊C4×C101C2211yescharacteristic
26C2×C22⋊C4×C101C2211yescharacteristic
27C23×D101C2×C411yescharacteristic
28C2×C22⋊D101—21no—
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C22×Dic101—21no—
35C22×Dic101—21no—
36C22×Dic101—21no—
37C22×Dic101—21no—
38C22×Dic101—21no—
39C22×Dic101—21no—
40C22×Dic101C2×C411yescharacteristic
41C22×Dic101C2×C412yes—
42C22×Dic101D412yes—
43C22×Dic101D411yescharacteristic
44C22×Dic101D411yescharacteristic
45C4⋊Dic101—22no—
46C4⋊Dic101—22no—
47#13108053—24no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101D412yes—
52#13108051—22no—
53#13108051—22no—
54C4×Dic101—24no—
55C24×C101C2311yescharacteristic
56C22×C4×C101—21no—
57C22×C4×C101—21no—
58C22×C4×C101C2×C412yes—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C22⋊C4×C101—21no—
62C22⋊C4×C101—21no—
63C22⋊C4×C101—21no—
64C22⋊C4×C101D412yes—
65C23×C101—42no—
66C23×C101C22×C411yescharacteristic
67C23×C101C2×D411yescharacteristic
68C23×C101C2×D411yescharacteristic
69C23×C101C2×D411yescharacteristic
70C23×C101C2×D411yescharacteristic
71C23×C101#4111yescharacteristic
72C23×C101#4111yescharacteristic
73C22×D101—21no—
74C22×D101—21no—
75C22×D101—21no—
76C22×D101—42no—
77C2×C4×C101—22no—
78C2×C4×C101—22no—
79C2×C4×C101—42no—
80C2×C4×C101—42no—
81C2×C4×C101—42no—
82C2×C4×C101—41no—
83C22⋊D101—41no—
84C22⋊D101—41no—
85C22⋊D101—41no—
86C22⋊D101—41no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—22no—
92C2×Dic101—22no—
93C2×Dic101—21no—
94C2×Dic101—21no—
95C2×Dic101—21no—
96C2×Dic101—21no—
97C2×Dic101—21no—
98C2×Dic101—22no—
99C2×Dic101—42no—
100C2×Dic101—42no—
101C2×Dic101—41no—
102C2×Dic101C22⋊C412yes—
103C2×Dic101C22⋊C412yes—
104C22×C101—22no—
105C22×C101—22no—
106C22×C101—22no—
107C22×C101—41no—
108C22×C101—41no—
109C22×C101—41no—
110C22×C101—41no—
111C22×C101C2×C22⋊C411yescharacteristic
112C22×C101C4×D411yescharacteristic
113C22×C101C4×D411yescharacteristic
114C22×C101#12011yescharacteristic
115C22×C101#12111yescharacteristic
116C22×C101#12111yescharacteristic
117C22×C101#12311yescharacteristic
118C2×D101—41no—
119C2×D101—42no—
120C2×D101—81no—
121C2×D101—81no—
122C4×C101—42no—
123C4×C101—42no—
124C4×C101—81no—
125Dic101—22no—
126Dic101—22no—
127Dic101—42no—
128Dic101—41no—
129Dic101—81no—
130C2×C101—21no—
131C2×C101—21no—
132C2×C101—21no—
133C2×C101—21no—
134C2×C101—21no—
135C2×C101—21no—
136C2×C101#38611yescharacteristic
137C2×C101#41012yes—
138D101—81no—
139#1899—1011nomaximal, Sylow
140C101#189911yescharacteristic, complemented, Sylow
141C2×#123—1011no—
142C2×#117—1011no—
143C2×#99—1011no—
144C2×#99—1011no—
145#381—1011no—
146#342—1011no—
147#342D10111yescharacteristic, complemented
148C22×D4—1011no—
149C23×C4—1011no—
150#123—2022no—
151#123—2021no—
152#123—2021no—
153#117—2022no—
154#117—2022no—
155C2×C4⋊C4—1012no—
156C2×C4⋊C4—2021no—
157C2×C22⋊C4—1011no—
158C2×C22⋊C4—1012no—
159C2×C22⋊C4—1011no—
160C2×C22⋊C4C2×D10111yescharacteristic
161C2×C22⋊C4C2×D10111yescharacteristic
162C2×C22⋊C4C2×D10111yescharacteristic
163C2×C42—2021no—
164#99—2022no—
165#99—2022no—
166#95—2022no—
167C24—1011no—
168C24C22×D10111yescharacteristic
169C2×D4—2021no—
170C2×D4—2021no—
171C2×D4—2021no—
172C2×D4—2021no—
173C2×D4—2021no—
174C2×D4—2021no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—1012no—
178C22×C4—1012no—
179C22×C4—1011no—
180C22×C4—1011no—
181C22×C4—1011no—
182C22×C4—2021no—
183C22×C4—2021no—
184C22×C4—2021no—
185C22×C4—2021no—
186C22×C4—2021no—
187C22×C4—2021no—
188C22×C4C4×D10112yes—
189C4⋊C4—2022no—
190C4⋊C4—2022no—
191C4⋊C4—2024no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—1012no—
198C22⋊C4—2022no—
199C22⋊C4—2022no—
200C22⋊C4—2021no—
201C22⋊C4—2021no—
202C22⋊C4—2021no—
203C22⋊C4D40412yes—
204C42—2024no—
205C23—42no—
206C23—4042no—
207C23—2021no—
208C23—2021no—
209C23—2021no—
210C23C2×C4×D10111yescharacteristic
211C23D4×D10111yescharacteristic
212C23D4×D10111yescharacteristic
213C23D4×D10111yescharacteristic
214C23C2×D40411yescharacteristic
215C23#1310807411yescharacteristic
216C23#1310807511yescharacteristic
217D4—4041no—
218D4—4041no—
219D4—4041no—
220D4—4041no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—1012no—
228C2×C4—1012no—
229C2×C4—4042no—
230C2×C4—4041no—
231C2×C4—4042no—
232C2×C4—2021no—
233C2×C4—2021no—
234C2×C4—2021no—
235C2×C4—2022no—
236C2×C4—2022no—
237C2×C4—2022no—
238C2×C4—2021no—
239C2×C4—2021no—
240C2×C4—2022no—
241C2×C4—2022no—
242C2×C4—2021no—
243C2×C4—2021no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—8081no—
252C22—8081no—
253C22—4041no—
254C22—4042no—
255C22C22⋊C4×D10111yescharacteristic
256C22#1569221711yescharacteristic
257C22C4×D40411yescharacteristic
258C22#1569222511yescharacteristic
259C22#1569222811yescharacteristic
260C22#1569223011yescharacteristic
261C22#1569224011yescharacteristic
262C4—42no—
263C4—42no—
264C4—81no—
265C4—8081no—
266C4—4042no—
267C4—4041no—
268C4—2022no—
269C4—2022no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—8081no—
277C2#2026505511yescharacteristic
278C2#2026514912yes—
subgroupG/H
centerC22C22⋊C4×D101
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×C22⋊D101C22
cosocleC24×C101C23
hypercenter#342D101
hypocenterC101#1899
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#15692211C2semidirect
5. #20265038C2semidirect
6. #20264944C2semidirect
7. #342×C101C2semidirect
15. C2×C4⋊Dic101C4semidirect
19. C2×#13108051C4semidirect
140. C101#1899semidirect
147. #342D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922392
C2×#156922112
C2×#156921692
#202650382
#202649442
#342×C1012
#1899101