Finite Group Databasebeta

← #27171412

Subgroups — #27171412

Order 12928.

374 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#20265652C211yesmaximal, characteristic
2C2×#15692170C211yesmaximal, characteristic
3#20265050C212yesmaximal, complemented
4#20264943C212yesmaximal, complemented
5#386×C101C211yesmaximal, characteristic
6D4×Dic101—22no—
7D4×Dic101—22no—
8#15692214—21no—
9C2×#13108053—22no—
10C2×C22⋊Dic101—21no—
11C2×C22⋊Dic101C2212yes—
12C2×C22⋊Dic101C2212yes—
13#15692170C414yescomplemented
14#15692162—22no—
15C22×D4×C101C2211yescharacteristic
16C23×C4×C101C2211yescharacteristic
17C2×C22⋊C4×C101—21no—
18C2×C22⋊C4×C101C2211yescharacteristic
19#95×C101—21no—
20C22×Dic101—22no—
21C22×Dic101—21no—
22C22×Dic101—21no—
23C22×Dic101D412yes—
24C22×Dic101D412yes—
25C4⋊Dic101—21no—
26C4⋊Dic101—21no—
27#13108053—44no—
28C22⋊Dic101—22no—
29C22⋊Dic101—22no—
30C22⋊Dic101—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101C2×C414yescomplemented
33C4×Dic101—22no—
34C24×C101D411yescharacteristic
35C24×C101C2311yescharacteristic
36C2×D4×C101—22no—
37C2×D4×C101—22no—
38C2×D4×C101C2×C412yes—
39C2×D4×C101Q812yes—
40C22×C4×C101—21no—
41C22×C4×C101—22no—
42C22×C4×C101—21no—
43C22×C4×C101—42no—
44C22×C4×C101D411yescharacteristic
45C22⋊C4×C101—22no—
46C22⋊C4×C101—42no—
47C23×C101—21no—
48C23×C101—21no—
49C23×C101—41no—
50C23×C101—41no—
51C23×C101—41no—
52C23×C101—41no—
53C23×C101C4⋊C412yescomplemented
54C23×C101C22×C411yescharacteristic
55C23×C101C2×D412yes—
56C23×C101C2×D411yescharacteristic
57C23×C101C2×Q811yescharacteristic
58C23×C101#4112yes—
59D4×C101—24no—
60D4×C101—24no—
61C2×C4×C101—21no—
62C2×C4×C101—21no—
63C2×C4×C101—22no—
64C2×C4×C101—24no—
65C2×C4×C101—42no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101C4⋊C412yescomplemented
72C2×Dic101—24no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—42no—
76C2×Dic101—42no—
77C2×Dic101—42no—
78C2×Dic101—42no—
79C2×Dic101—81no—
80C2×Dic101—81no—
81C22×C101—22no—
82C22×C101—21no—
83C22×C101—22no—
84C22×C101—21no—
85C22×C101—22no—
86C22×C101—41no—
87C22×C101—41no—
88C22×C101—42no—
89C22×C101—42no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—81no—
93C22×C101C2×C4⋊C411yescharacteristic
94C22×C101C4×D412yes—
95C22×C101#12011yescharacteristic
96C22×C101C22⋊Q812yes—
97C22×C101#12311yescharacteristic
98C4×C101—22no—
99C4×C101—42no—
100C4×C101—81no—
101Dic101—44no—
102Dic101—42no—
103Dic101—42no—
104C2×C101—22no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101—22no—
108C2×C101—42no—
109C2×C101#38511yescharacteristic
110C2×C101B(4,𝔽2)11yescharacteristic
111C2×C101#45811yescharacteristic
112#1867—1011nomaximal, Sylow
113C101#186711yescharacteristic, complemented, Sylow
114#518—1011no—
115C2×#99—1011no—
116#386D10111yescharacteristic
117#385—1012no—
118#342—1012no—
119C22×D4C2×D10111yescharacteristic
120C23×C4C2×D10111yescharacteristic
121C4×D4—2022no—
122C4×D4—2022no—
123#117—2021no—
124C2×C4⋊C4—2022no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—1012no—
127C2×C22⋊C4—1012no—
128C2×C22⋊C4—2021no—
129C2×C22⋊C4C2×D10111yescharacteristic
130#99—1014no—
131#95—21no—
132#95—2022no—
133C24C22⋊D10111yescharacteristic
134C24C22×D10111yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4Dic20212yes—
138C2×D4C4×D10112yes—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—42no—
143C22×C4—1012no—
144C22×C4—1012no—
145C22×C4—2021no—
146C22×C4—2021no—
147C22×C4—2022no—
148C22×C4C22⋊D10111yescharacteristic
149C4⋊C4—4044no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C22⋊C4—22no—
153C22⋊C4—42no—
154C22⋊C4—1014no—
155C22⋊C4—2022no—
156C22⋊C4—2022no—
157C22⋊C4—2022no—
158C22⋊C4—2022no—
159C42—2022no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23#1310805312yescomplemented
167C23C2×C4×D10111yescharacteristic
168C23D4×D10112yes—
169C23C2×C22⋊D10111yescharacteristic
170C23C2×Dic20211yescharacteristic
171C23#1310807312yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—8081no—
185C2×C4—8081no—
186C2×C4—4042no—
187C2×C4—4042no—
188C2×C4—4042no—
189C2×C4—4042no—
190C2×C4—2024no—
191C2×C4—2022no—
192C2×C4—2022no—
193C2×C4#1310805312yescomplemented
194C22—21no—
195C22—22no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×#1310805311yescharacteristic
207C22#1569221512yes—
208C22#1569222311yescharacteristic
209C22#1569223212yes—
210C22#1569223611yescharacteristic
211C4—22no—
212C4—42no—
213C4—81no—
214C4—4044no—
215C4—4042no—
216C4—4042no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2026504911yescharacteristic
223C2#2026536311yescharacteristic
224C2#2026536611yescharacteristic
subgroupG/H
centerC22#15692223
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#386×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
cosocleC24×C101C23
hypercenter#386D101
hypocenterC101#1867
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
3. #20265050C2semidirect
4. #20264943C2semidirect
13. #15692170C4semidirect
32. C22⋊Dic101C2×C4semidirect
53. C23×C101C4⋊C4semidirect
71. C2×C4×C101C4⋊C4semidirect
113. C101#1867semidirect
166. C23#13108053semidirect
193. C2×C4#13108053semidirect

Maximal subgroups

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

subgroupindex
#202656522
C2×#156921702
#202650502
#202649432
#386×C1012
#1867101