Finite Group Databasebeta

← #27170159

Subgroups — #27170159

Order 12928.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265803C211yesmaximal, characteristic
2#20265513C212yesmaximal, complemented
3#20265113C211yesmaximal, characteristic
4#20264915C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic
6#15692234—21no—
7#15692234C2212yes—
8#15692232—22no—
9#15692232—22no—
10C2×C4⋊Dic101C2211yescharacteristic
11C2×#13108053—21no—
12C2×C22⋊Dic101—21no—
13C2×C22⋊Dic101—21no—
14#15692189—22no—
15#15692180—22no—
16#15692162C2211yescharacteristic
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C2×Dic202—21no—
20C2×Dic202D412yes—
21C22×Dic101—21no—
22C22×Dic101—21no—
23C22×Dic101—21no—
24C22×Dic101—21no—
25C4⋊Dic101—21no—
26C4⋊Dic101—22no—
27C4⋊Dic101D412yes—
28#13108053—22no—
29#13108053—22no—
30#13108053—42no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33C22⋊Dic101—42no—
34C24×C101D411yescharacteristic
35C22×C4×C101D411yescharacteristic
36C22×C4×C101C2311yescharacteristic
37C2×C8×C101—22no—
38C22⋊C4×C101—22no—
39C23×C101—21no—
40C23×C101—21no—
41C23×C101—41no—
42C23×C101—41no—
43C23×C101Q1612yes—
44C23×C101C2×D411yescharacteristic
45C2×C4×C101—21no—
46C2×C4×C101—41no—
47C2×C4×C101—41no—
48C2×C4×C101C2×D412yes—
49C2×Dic101—22no—
50C2×Dic101—22no—
51C2×Dic101—22no—
52C2×Dic101—41no—
53C2×Dic101—41no—
54C2×Dic101—41no—
55C2×Dic101—41no—
56C2×Dic101—41no—
57C2×Dic101—81no—
58C2×Dic101—81no—
59Dic202—42no—
60Dic202—81no—
61C8×C101—42no—
62C22×C101—21no—
63C22×C101—21no—
64C22×C101—22no—
65C22×C101—42no—
66C22×C101—41no—
67C22×C101—41no—
68C22×C101—81no—
69C22×C101#12011yescharacteristic
70C22×C101C2×Q1611yescharacteristic
71C22×C101#13711yescharacteristic
72C4×C101—22no—
73C4×C101—41no—
74Dic101—42no—
75Dic101—42no—
76Dic101—42no—
77Dic101—81no—
78C2×C101—21no—
79C2×C101—21no—
80C2×C101—42no—
81C2×C101#45111yescharacteristic
82C2×C101#45311yescharacteristic
83C2×C101#45811yescharacteristic
84#1614—1011nomaximal, Sylow
85C101#161411yescharacteristic, complemented, Sylow
86#543—1011no—
87#484—1012no—
88#399—1011no—
89#328—1012no—
90#323D10111yescharacteristic
91C22⋊Q8—1012no—
92C22⋊Q8—2022no—
93C22⋊Q8—2022no—
94C22⋊Q8—2021no—
95C2×C4⋊C4—1011no—
96C2×C4⋊C4—2021no—
97C2×C22⋊C4—2021no—
98C2×C22⋊C4—2021no—
99C2×C22⋊C4C2×D10111yescharacteristic
100#107—2022no—
101#103—2022no—
102C22⋊C8C2×D10112yes—
103#95—1011no—
104C24D40411yescharacteristic
105C2×Q8—1012no—
106C2×Q8—2021no—
107C22×C4—2021no—
108C22×C4—2021no—
109C22×C4—2021no—
110C22×C4—2021no—
111C22×C4D40411yescharacteristic
112C22×C4C22×D10111yescharacteristic
113C2×C8—22no—
114C4⋊C4—1012no—
115C4⋊C4—4042no—
116C4⋊C4—2022no—
117C4⋊C4—2021no—
118C4⋊C4—2022no—
119C4⋊C4—2022no—
120C22⋊C4—22no—
121C22⋊C4—4042no—
122C22⋊C4—2022no—
123C22⋊C4—2022no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic40412yes—
129C23C2×D40411yescharacteristic
130Q8—8081no—
131Q8—4042no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—8081no—
136C2×C4—8081no—
137C2×C4—4041no—
138C2×C4—4041no—
139C2×C4—4041no—
140C2×C4—4041no—
141C2×C4—4041no—
142C2×C4—2022no—
143C2×C4—2022no—
144C2×C4—2022no—
145C2×C4D4×D10112yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1569222511yescharacteristic
155C22C2×Dic40411yescharacteristic
156C22#1569228811yescharacteristic
157C4—22no—
158C4—41no—
159C4—8081no—
160C4—4042no—
161C4—4042no—
162C4—4042no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2026533511yescharacteristic
167C2#2026534711yescharacteristic
168C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radicalC24×C101D4
hypercenter#323D101
hypocenterC101#1614
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265513C2semidirect
4. #20264915C2semidirect
85. C101#1614semidirect

Maximal subgroups

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

subgroupindex
#202658032
#202655132
#202651132
#202649152
#323×C1012
#1614101