Finite Group Databasebeta

← #27182593

Subgroups — #27182593

Order 12928.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265798C212yesmaximal, complemented
2#20265473C212yesmaximal, complemented
3#20265458C212yesmaximal, complemented
4#20265423C212yesmaximal, complemented
5#20265418C211yesmaximal, characteristic, complemented
6#20265413C211yesmaximal, characteristic
7#20265400C212yesmaximal, complemented
8#20265393C212yesmaximal, complemented
9#424×C101C211yesmaximal, characteristic, complemented
10C2×Dic404C2211yescharacteristic
11C2×#13108063C2212yes—
12C2×D808C2211yescharacteristic
13#15692255C2212yes—
14#15692250—22no—
15#15692243C2212yes—
16#15692241—22no—
17#15692237—22no—
18#15692234C2212yes—
19#15692233—22no—
20#15692229C2212yes—
21C4×Dic202C2212yes—
22C4×D404C2212yes—
23#15692189C2211yescharacteristic
24#15692189C2212yes—
25#15692189C2211yescharacteristic
26#15692180C2212yes—
27#15692180C2212yes—
28#15692177C2212yes—
29#15692177C2212yes—
30C2×M16×C101C2211yescharacteristic
31C22×C8×C101C2211yescharacteristic
32#117×C101C2211yescharacteristic
33C4⋊C8×C101C2212yes—
34C4⋊C8×C101C2212yes—
35C2×Dic202—22no—
36C2×Dic202C2312yes—
37C2×D404C2312yes—
38C2×C22⋊D101—22no—
39C2×C4×D101—22no—
40Dic404—41no—
41#13108063—42no—
42D808—41no—
43C4⋊Dic101—22no—
44C4⋊Dic101C2312yes—
45C4⋊Dic101C2312yes—
46#13108053—22no—
47#13108053—22no—
48C22⋊Dic101—22no—
49#13108051—22no—
50#13108051—22no—
51#13108051—22no—
52#13108051—22no—
53C4×Dic101—22no—
54C22×C4×C101C2311yescharacteristic
55M16×C101—41no—
56C2×C8×C101—41no—
57C2×C8×C101C2311yescharacteristic
58C2×C8×C101C2311yescharacteristic
59C2×C8×C101C2312yes—
60C4⋊C4×C101D411yescharacteristic
61C4⋊C4×C101D411yescharacteristic
62C22⋊C4×C101D412yes—
63C42×C101C2312yes—
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101C2×D412yes—
69C2×C4×C101C2×D412yes—
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2411yescharacteristic
72C22⋊D101—82no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—22no—
76D404—22no—
77D404—22no—
78C4×D101—82no—
79Dic202—22no—
80Dic202—22no—
81Dic202—82no—
82C8×C101—21no—
83C8×C101—21no—
84C8×C101—22no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101C22×D411yescharacteristic
89C2×D101—42no—
90C2×D101—42no—
91C2×D101—42no—
92C4×C101—21no—
93C4×C101—22no—
94C4×C101—22no—
95C4×C101H(5,𝔽2)11yescharacteristic
96C4×C1012−1+411yescharacteristic
97Dic101—42no—
98Dic101—42no—
99Dic101—42no—
100C2×C101—41no—
101C2×C101#53711yescharacteristic
102C2×C101#57611yescharacteristic
103C2×C101#57811yescharacteristic
104D101—82no—
105#3262—1011nomaximal, Sylow
106C101#326211yescharacteristic, complemented, Sylow
107#542—1012no—
108#477—1012no—
109#474—1012no—
110#468—1012no—
111#467—1011no—
112#466—1011no—
113#464—1012no—
114#463—1012no—
115#424D10111yescharacteristic, complemented
116C2×Q16—1011no—
117C2×SD16—1012no—
118C2×D8—1011no—
119C2×M16C2×D10111yescharacteristic
120C22×C8C2×D10111yescharacteristic
121C4⋊Q8—1012no—
122#126—2022no—
123#124—1012no—
124#124—2022no—
125#123—2022no—
126C22⋊Q8—1012no—
127C22⋊Q8—2022no—
128#121—1012no—
129C4×Q8—1012no—
130C4×D4—1012no—
131#117C2×D10111yescharacteristic
132#107—1011no—
133#107—1011no—
134#107—1012no—
135C4⋊C8C2×D10112yes—
136C4⋊C8C2×D10112yes—
137#103—1012no—
138#103—1012no—
139#102—1012no—
140#102—1012no—
141C2×Q8—1012no—
142C2×Q8—2022no—
143C2×D4—1012no—
144C2×D4—2022no—
145C22×C4—2022no—
146C22×C4C22×D10111yescharacteristic
147Q16—4041no—
148SD16—4042no—
149D8—4041no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D10111yescharacteristic
153C2×C8C22×D10112yes—
154C2×C8C22×D10111yescharacteristic
155C4⋊C4—1012no—
156C4⋊C4—1012no—
157C4⋊C4—2022no—
158C4⋊C4—2022no—
159C4⋊C4—2022no—
160C4⋊C4D40411yescharacteristic
161C4⋊C4D40411yescharacteristic
162C22⋊C4—2022no—
163C22⋊C4—2022no—
164C22⋊C4—2022no—
165C22⋊C4—2022no—
166C22⋊C4—2022no—
167C22⋊C4D40412yes—
168C42—2022no—
169C42C22×D10112yes—
170C23—2022no—
171C23C2×D40411yescharacteristic
172Q8—8082no—
173Q8—2022no—
174Q8—2022no—
175D4—8082no—
176D4—2022no—
177D4—2022no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—8082no—
181C2×C4—2022no—
182C2×C4—2022no—
183C2×C4—2022no—
184C2×C4C2×D40412yes—
185C2×C4C2×D40412yes—
186C2×C4C2×D40411yescharacteristic
187C2×C4C23×D10111yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—4042no—
195C22—4042no—
196C22—4042no—
197C22C22×D40411yescharacteristic
198C4—22no—
199C4—21no—
200C4—22no—
201C4—4042no—
202C4—4042no—
203C4—4042no—
204C4#1569230011yescharacteristic
205C4#1569230311yescharacteristic
206C2—41no—
207C2—8082no—
208C2#2026575811yescharacteristic
209C2#2026598211yescharacteristic
210C2#2026599411yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#424×C101C2
socleC22×C101C22×D4
prime radical#20265413C2
hypercenter#424D101
hypocenterC101#3262
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #20265798C2semidirect
2. #20265473C2semidirect
3. #20265458C2semidirect
4. #20265423C2semidirect
5. #20265418C2semidirect
7. #20265400C2semidirect
8. #20265393C2semidirect
9. #424×C101C2semidirect
106. C101#3262semidirect
115. #424D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202654732
#202654582
#202654232
#202654182
#202654132
#202654002
#202653932
#424×C1012
#3262101