Finite Group Databasebeta

← #44449808

Subgroups — #44449808

Order 29824.

222 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 35 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#28416344C211yesmaximal, characteristic
2#28416202C211yesmaximal, characteristic, complemented
3#323×C233C211yesmaximal, characteristic, complemented
4C22×D932C411yescharacteristic, complemented
5C22×C22⋊D233—21no—
6C2×C22⋊Dic233—21no—
7C2×#16836571—21no—
8#21418442C411yescharacteristic, complemented
9C2×C22⋊C4×C233C2211yescharacteristic
10C22⋊C8×C233—21no—
11C23×D233—21no—
12C2×D932—41no—
13C2×D932—42no—
14C2×C22⋊D233—42no—
15C2×C22⋊D233—42no—
16C2×C22⋊D233—41no—
17C2×C22⋊D233—41no—
18C22×Dic233—21no—
19C22×Dic233—41no—
20C22⋊Dic233—22no—
21C22⋊Dic233—41no—
22#16836571—42no—
23C24×C233D411yescharacteristic
24C22×C4×C233C2×C411yescharacteristic
25C22×C4×C233D411yescharacteristic
26C2×C8×C233—41no—
27C22⋊C4×C233—41no—
28C22×D233—22no—
29C22×D233—41no—
30C22×D233—41no—
31C22×D233—41no—
32C22×D233—41no—
33C22×D233—41no—
34C22×D233—41no—
35D932—42no—
36D932—42no—
37C22⋊D233—44no—
38C22⋊D233—44no—
39C2×Dic233—22no—
40C2×Dic233—41no—
41C2×Dic233—41no—
42C2×Dic233—42no—
43C2×Dic233—81no—
44C2×Dic233—81no—
45C23×C233—21no—
46C23×C233—21no—
47C23×C233—41no—
48C23×C233—41no—
49C23×C233C22⋊C411yescharacteristic
50C23×C233D811yescharacteristic
51C23×C233SD1611yescharacteristic
52C2×C4×C233—21no—
53C2×C4×C233—21no—
54C2×C4×C233—41no—
55C2×C4×C233—41no—
56C8×C233—81no—
57C22×C233—21no—
58C22×C233—21no—
59C22×C233—21no—
60C22×C233—21no—
61C22×C233—41no—
62C22×C233—41no—
63C22×C233—41no—
64C22×C233—41no—
65C22×C233—81no—
66C22×C233#9911yescharacteristic
67C22×C233#10211yescharacteristic
68C22×C233#10411yescharacteristic
69C2×D233—42no—
70C2×D233—42no—
71C2×D233—42no—
72C2×D233—81no—
73C2×D233—81no—
74C2×D233—81no—
75C2×D233—81no—
76C4×C233—41no—
77C4×C233—41no—
78Dic233—42no—
79Dic233—82no—
80C2×C233—21no—
81C2×C233—21no—
82C2×C233—41no—
83C2×C233—41no—
84C2×C233#32711yescharacteristic
85C2×C233AΣL(1,𝔽16)11yescharacteristic
86C2×C233#35311yescharacteristic
87D233—82no—
88C233#134311yescharacteristic, complemented, Sylow
89#1343—2331nomaximal, Sylow
90#392—2331no—
91#342—2331no—
92#323D23311yescharacteristic, complemented
93C22×D4—4661no—
94C22×D4—2331no—
95C2×C22⋊C4—4661no—
96C2×C22⋊C4—4661no—
97C2×C22⋊C4C2×D23311yescharacteristic
98C22⋊C8—21no—
99#95—2331no—
100C24—4661no—
101C24D93211yescharacteristic
102C2×D4—9321no—
103C2×D4—9322no—
104C2×D4—9322no—
105C2×D4—9321no—
106C2×D4—9321no—
107C2×D4—9322no—
108C22×C4—9321no—
109C22×C4—4661no—
110C22×C4C4×D23311yescharacteristic
111C22×C4C22⋊D23311yescharacteristic
112C2×C8—41no—
113C22⋊C4—41no—
114C22⋊C4—9322no—
115C22⋊C4—9321no—
116C22⋊C4—4662no—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23—9321no—
122C23—9321no—
123C23—9321no—
124C23—9321no—
125C23—9321no—
126C23—9321no—
127C23—4662no—
128C23#1683657111yescharacteristic
129C23D186411yescharacteristic
130C23#1683658311yescharacteristic
131D4—9322no—
132D4—9322no—
133D4—9324no—
134D4—9324no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—18641no—
140C2×C4—18641no—
141C2×C4—9322no—
142C2×C4—9321no—
143C2×C4—9321no—
144C2×C4—4662no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—41no—
151C22—41no—
152C22—41no—
153C22—41no—
154C22—81no—
155C22—18641no—
156C22—18641no—
157C22—18641no—
158C22—18641no—
159C22—9322no—
160C22—9322no—
161C22—9322no—
162C22#2141844911yescharacteristic
163C22#2141845711yescharacteristic
164C22#2141846111yescharacteristic
165C4—41no—
166C4—41no—
167C4—18642no—
168C4—9322no—
169C2—21no—
170C2—21no—
171C2—41no—
172C2—41no—
173C2—18642no—
174C2#2841617011yescharacteristic
175C2#2841622311yescharacteristic
176C2#2841622711yescharacteristic
subgroupG/H
centerC22#21418449
commutatorC22×C4×C233C2×C4
FrattiniC2×C22⋊C4C2×D233
Fitting#323×C233C2
socleC22×C233#99
prime radical#28416344C2
cosocleC2×C22⋊C4×C233C22
hypercenter#323D233
hypocenterC233#1343
LCSG ▷ C22×C4×C233 ▷ C23×C233 ▷ C2×C233 ▷ C233
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #28416202C2semidirect
3. #323×C233C2semidirect
4. C22×D932C4semidirect
8. #21418442C4semidirect
88. C233#1343semidirect
92. #323D233semidirect

Maximal subgroups

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

subgroupindex
#284163442
#284162022
#323×C2332
#1343233