Finite Group Databasebeta

← #27171413

Subgroups — #27171413

Order 12928.

374 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15692170C211yesmaximal, characteristic
2#20265057C211yesmaximal, characteristic
3#20265049C212yesmaximal, complemented
4#20264944C212yesmaximal, complemented
5#518×C101C211yesmaximal, characteristic
6C23×Dic101C2211yescharacteristic
7C2×#13108053—22no—
8C2×C22⋊Dic101—21no—
9C2×C22⋊Dic101C2211yescharacteristic
10C2×C22⋊Dic101C2212yes—
11#15692170C414yescomplemented
12#15692162—22no—
13#15692162—21no—
14C22×D4×C101C2211yescharacteristic
15C4×D4×C101—22no—
16C4×D4×C101—22no—
17#117×C101—21no—
18C2×C22⋊C4×C101—21no—
19C2×C22⋊C4×C101C2212yes—
20C22×Dic101—21no—
21C22×Dic101—22no—
22C22×Dic101—22no—
23C22×Dic101—21no—
24C22×Dic101—42no—
25#13108053—44no—
26C22⋊Dic101—22no—
27C22⋊Dic101—22no—
28C22⋊Dic101—42no—
29C22⋊Dic101C2×C414yescomplemented
30C24×C101D411yescharacteristic
31C24×C101C2311yescharacteristic
32C2×D4×C101—22no—
33C2×D4×C101—22no—
34C2×D4×C101C2×C412yes—
35C2×D4×C101Q812yes—
36C22×C4×C101—21no—
37C22×C4×C101—21no—
38C22×C4×C101D411yescharacteristic
39C22×C4×C101D412yes—
40C22×C4×C101D412yes—
41C4⋊C4×C101—21no—
42C4⋊C4×C101—21no—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—22no—
45C22⋊C4×C101—22no—
46C42×C101—22no—
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×D411yescharacteristic
56C23×C101C2×D412yes—
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—22no—
65C2×C4×C101—42no—
66C2×C4×C101—42no—
67C2×C4×C101—81no—
68C2×C4×C101—81no—
69C2×C4×C101C4⋊C412yescomplemented
70C2×Dic101—22no—
71C2×Dic101—24no—
72C2×Dic101—24no—
73C2×Dic101—42no—
74C2×Dic101—42no—
75C2×Dic101—42no—
76C2×Dic101—41no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—41no—
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—42no—
101Dic101—44no—
102Dic101—42no—
103Dic101—81no—
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#518D10111yescharacteristic
115C2×#99—1011no—
116#386—1011no—
117#385—1012no—
118#342—1012no—
119C22×D4C2×D10111yescharacteristic
120C23×C4—1011no—
121C4×D4—22no—
122C4×D4—22no—
123#117—21no—
124C2×C4⋊C4—2022no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—1012no—
127C2×C22⋊C4—1011no—
128C2×C22⋊C4—2021no—
129C2×C22⋊C4C2×D10112yes—
130#99—1014no—
131#95—2022no—
132#95—2021no—
133C24C22⋊D10111yescharacteristic
134C24C22×D10111yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4Dic20212yes—
138C2×D4C4×D10112yes—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—4042no—
142C22×C4—2021no—
143C22×C4—2022no—
144C22×C4—2021no—
145C22×C4—2022no—
146C22×C4C22⋊D10111yescharacteristic
147C22×C4C22⋊D10112yes—
148C22×C4C22⋊D10112yes—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—4044no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—1014no—
156C22⋊C4—4042no—
157C22⋊C4—2022no—
158C22⋊C4—2022no—
159C42—22no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23#1310805312yescomplemented
167C23C2×C4×D10111yescharacteristic
168C23C2×C22⋊D10111yescharacteristic
169C23C2×C22⋊D10112yes—
170C23C2×Dic20211yescharacteristic
171C23#1310807412yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—4041no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—4042no—
186C2×C4—4042no—
187C2×C4—4041no—
188C2×C4—4042no—
189C2×C4—4041no—
190C2×C4—2022no—
191C2×C4—2024no—
192C2×C4—2024no—
193C2×C4#1310805312yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×#1310805311yescharacteristic
207C22C4×C22⋊D10112yes—
208C22#1569222411yescharacteristic
209C22#1569223412yes—
210C22#1569223811yescharacteristic
211C4—22no—
212C4—42no—
213C4—42no—
214C4—8081no—
215C4—4044no—
216C4—4042no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#2026505011yescharacteristic
223C2#2026536411yescharacteristic
224C2#2026536711yescharacteristic
subgroupG/H
centerC22#15692224
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#518×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
cosocleC24×C101C23
hypercenter#518D101
hypocenterC101#1867
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #20265049C2semidirect
4. #20264944C2semidirect
11. #15692170C4semidirect
29. C22⋊Dic101C2×C4semidirect
53. C23×C101C4⋊C4semidirect
69. 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
C2×#156921702
#202650572
#202650492
#202649442
#518×C1012
#1867101