Finite Group Databasebeta

← #1996×C233 #44448028

Subgroups — #1996×C233 #44448028

Order 29824.

324 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 116 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×C22⋊Q8×C233C211yesmaximal, characteristic
2#380×C233C211yesmaximal, characteristic
3C4×C22⋊C4×C233C211yesmaximal, characteristic
4#342×C233C214yesmaximal
5C22×Q8×C233C411yescharacteristic, complemented
6C23×C4×C233C2211yescharacteristic
7C22⋊Q8×C233—21no—
8C22⋊Q8×C233—21no—
9C22⋊Q8×C233—21no—
10C22⋊Q8×C233—21no—
11C2×C4⋊C4×C233—21no—
12C2×C4⋊C4×C233C411yescharacteristic, complemented
13C2×C4⋊C4×C233C412yescomplemented
14C2×C22⋊C4×C233C2212yes—
15C2×C22⋊C4×C233C2212yes—
16C2×C22⋊C4×C233C2212yes—
17C2×C42×C233—21no—
18#95×C233—21no—
19#95×C233—21no—
20C24×C233C2311yescharacteristic
21C2×Q8×C233—21no—
22C2×Q8×C233—21no—
23C2×Q8×C233—42no—
24C22×C4×C233—22no—
25C22×C4×C233—22no—
26C22×C4×C233—42no—
27C22×C4×C233C2×C411yescharacteristic
28C22×C4×C233C2×C412yes—
29C22×C4×C233C2×C412yes—
30C22×C4×C233C2×C411yescharacteristic
31C22×C4×C233D411yescharacteristic
32C22×C4×C233D411yescharacteristic
33C22×C4×C233D411yescharacteristic
34C22×C4×C233D411yescharacteristic
35C4⋊C4×C233—21no—
36C4⋊C4×C233—21no—
37C4⋊C4×C233—22no—
38C4⋊C4×C233—22no—
39C4⋊C4×C233—42no—
40C22⋊C4×C233—22no—
41C22⋊C4×C233—22no—
42C22⋊C4×C233—22no—
43C22⋊C4×C233—24no—
44C22⋊C4×C233Q814yescomplemented
45C42×C233—42no—
46C23×C233—41no—
47C23×C233—41no—
48C23×C233C22×C411yescharacteristic
49C23×C233C2×D411yescharacteristic
50C23×C233C2×D411yescharacteristic
51C23×C233C2×Q812yes—
52C23×C233#4112yes—
53Q8×C233—42no—
54Q8×C233—42no—
55C2×C4×C233—22no—
56C2×C4×C233—22no—
57C2×C4×C233—22no—
58C2×C4×C233—22no—
59C2×C4×C233—22no—
60C2×C4×C233—24no—
61C2×C4×C233—41no—
62C2×C4×C233—41no—
63C2×C4×C233—41no—
64C2×C4×C233—42no—
65C2×C4×C233—42no—
66C2×C4×C233—42no—
67C2×C4×C233—42no—
68C2×C4×C233—42no—
69C2×C4×C233—42no—
70C2×C4×C233—42no—
71C2×C4×C233C22⋊C411yescharacteristic
72C2×C4×C233C22⋊C411yescharacteristic
73C2×C4×C233C22⋊C411yescharacteristic
74C2×C4×C233C22⋊C411yescharacteristic
75C22×C233—21no—
76C22×C233—21no—
77C22×C233—21no—
78C22×C233—21no—
79C22×C233—21no—
80C22×C233—21no—
81C22×C233—42no—
82C22×C233—42no—
83C22×C233C2×C22⋊C411yescharacteristic
84C22×C233C4×Q812yes—
85C22×C233C22⋊Q812yes—
86C22×C233#12411yescharacteristic
87C22×C233C4⋊Q811yescharacteristic
88C4×C233—21no—
89C4×C233—21no—
90C4×C233—21no—
91C4×C233—44no—
92C4×C233—42no—
93C4×C233—42no—
94C4×C233—82no—
95C4×C233#9912yes—
96C2×C233—22no—
97C2×C233—21no—
98C2×C233—22no—
99C2×C233—21no—
100C2×C233#39111yescharacteristic
101C2×C233C2×#9911yescharacteristic
102C2×C233#41011yescharacteristic
103C233#199611yescharacteristic, direct product factor, Sylow
104#1996C23311yesmaximal, characteristic, direct product factor, Sylow
105C2×C22⋊Q8C2×C23311yescharacteristic
106#380C2×C23311yescharacteristic
107C4×C22⋊C4C2×C23311yescharacteristic
108#342C2×C23314yes—
109C22×Q8C4×C23311yescharacteristic, complemented
110C23×C4C22×C23311yescharacteristic
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C4×C23311yescharacteristic, complemented
117C2×C4⋊C4C4×C23312yescomplemented
118C2×C22⋊C4C22×C23312yes—
119C2×C22⋊C4C22×C23312yes—
120C2×C22⋊C4C22×C23312yes—
121C2×C42—21no—
122#95—21no—
123#95—21no—
124C24C23×C23311yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—42no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—42no—
131C22×C4C2×C4×C23311yescharacteristic
132C22×C4C2×C4×C23312yes—
133C22×C4C2×C4×C23312yes—
134C22×C4C2×C4×C23311yescharacteristic
135C22×C4D4×C23311yescharacteristic
136C22×C4D4×C23311yescharacteristic
137C22×C4D4×C23311yescharacteristic
138C22×C4D4×C23311yescharacteristic
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—24no—
148C22⋊C4Q8×C23314yescomplemented
149C42—42no—
150C23—41no—
151C23—41no—
152C23C22×C4×C23311yescharacteristic
153C23C2×D4×C23311yescharacteristic
154C23C2×D4×C23311yescharacteristic
155C23C2×Q8×C23312yes—
156C23#41×C23312yes—
157Q8—42no—
158Q8—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4C22⋊C4×C23311yescharacteristic
176C2×C4C22⋊C4×C23311yescharacteristic
177C2×C4C22⋊C4×C23311yescharacteristic
178C2×C4C22⋊C4×C23311yescharacteristic
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22C2×C22⋊C4×C23311yescharacteristic
188C22C4×Q8×C23312yes—
189C22C22⋊Q8×C23312yes—
190C22#124×C23311yescharacteristic
191C22C4⋊Q8×C23311yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—44no—
196C4—42no—
197C4—42no—
198C4—82no—
199C4#99×C23312yes—
200C2—22no—
201C2—21no—
202C2—22no—
203C2—21no—
204C2#391×C23311yescharacteristic
205C2C2×#99×C23311yescharacteristic
206C2#410×C23311yescharacteristic
subgroupG/H
center, socleC22×C233C2×C22⋊C4
commutatorC23C22×C4×C233
Frattini, cosocleC24C23×C233
prime radicalC24×C233C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C233 ◁ C23×C4×C233 ◁ G

Complements

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

NH ≅ G/Ntype
5. C22×Q8×C233C4semidirect
12. C2×C4⋊C4×C233C4semidirect
13. C2×C4⋊C4×C233C4semidirect
44. C22⋊C4×C233Q8semidirect
103. C233#1996direct (N × H)
104. #1996C233direct (N × H)
109. C22×Q8C4×C233semidirect
116. C2×C4⋊C4C4×C233semidirect
117. C2×C4⋊C4C4×C233semidirect
148. C22⋊C4Q8×C233semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×C2332
#380×C2332
C4×C22⋊C4×C2332
#342×C2332
#1996233