Finite Group Databasebeta

← #37573685

Subgroups — #37573685

Order 24448.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#27133289C211yesmaximal, characteristic, complemented
2#27132518C212yesmaximal
3#27132518C211yesmaximal, characteristic
4#27132429C212yesmaximal, complemented
5#342×C191C211yesmaximal, characteristic, complemented
6C24×D191C2211yescharacteristic
7#20259301—22no—
8#20259299—24no—
9C2×C22⋊Dic191C2211yescharacteristic
10C2×C22⋊Dic191C2212yes—
11C2×(C2×D191)⋊C4—24no—
12C2×(C2×D191)⋊C4—22no—
13C2×C22⋊C4×C191C2212yes—
14C2×C22⋊C4×C191C2211yescharacteristic
15C23×D191—22no—
16C23×D191—21no—
17C23×D191—21no—
18C23×D191—22no—
19C23×D191—44no—
20C23×D191Q812yescomplemented
21C2×Dic382—22no—
22C22×Dic191—21no—
23C22×Dic191D414yes—
24C4⋊Dic191—22no—
25Dic191⋊C4—24no—
26C22⋊Dic191—22no—
27C22⋊Dic191—22no—
28C22⋊Dic191—22no—
29(C2×D191)⋊C4—44no—
30(C2×D191)⋊C4—48no—
31C24×C191C2311yescharacteristic
32C22×C4×C191—22no—
33C22×C4×C191D412yes—
34C22⋊C4×C191—22no—
35C22⋊C4×C191—22no—
36C22⋊C4×C191—22no—
37C22×D191—28no—
38C22×D191—24no—
39C22×D191—24no—
40C22×D191—22no—
41C22×D191—44no—
42C22×D191—44no—
43C22×D191—48no—
44C22×D191—44no—
45C22×D191—42no—
46C22×D191—41no—
47C22×D191—42no—
48C22×D191—41no—
49Dic382—82no—
50C2×Dic191—24no—
51C2×Dic191—44no—
52C2×Dic191—42no—
53C2×Dic191—41no—
54C23×C191—42no—
55C23×C191C2×D412yes—
56C23×C191C2×D411yescharacteristic
57C23×C191C2×Q811yescharacteristic
58C23×C191#4111yescharacteristic
59C23×C191#4112yes—
60C2×C4×C191—22no—
61C2×C4×C191—42no—
62C2×C4×C191—44no—
63C2×C4×C191—42no—
64C22×C191—24no—
65C22×C191—22no—
66C22×C191—41no—
67C22×C191—41no—
68C22×C191—42no—
69C22×C191#12011yescharacteristic
70C22×C191C22⋊Q812yes—
71C22×C191C22⋊Q811yescharacteristic
72C22×C191#12412yes—
73C22×C191#12411yescharacteristic
74C2×D191—28no—
75C2×D191—24no—
76C2×D191—44no—
77C2×D191—48no—
78C2×D191—44no—
79C2×D191—42no—
80C2×D191—42no—
81C2×D191—44no—
82C4×C191—42no—
83C4×C191—82no—
84Dic191—44no—
85Dic191—81no—
86C2×C191—21no—
87C2×C191—22no—
88C2×C191—22no—
89C2×C191—21no—
90C2×C191#39311yescharacteristic
91C2×C191B(4,𝔽2)12yes—
92D191—44no—
93C191#202911yescharacteristic, complemented, Sylow
94#2029—1911nomaximal, Sylow
95#543—1911no—
96#379—1912no—
97#379—1911no—
98#342—1912no—
99#342D19111yescharacteristic, complemented
100C25—1911no—
101C22⋊Q8—3824no—
102C22⋊Q8—3822no—
103C2×C22⋊C4—3824no—
104C2×C22⋊C4—3822no—
105C2×C22⋊C4—1911no—
106C2×C22⋊C4—1912no—
107C2×C22⋊C4C2×D19112yes—
108C2×C22⋊C4C2×D19111yescharacteristic
109C24—3822no—
110C24—3821no—
111C24—3821no—
112C24—3822no—
113C24—1912no—
114C24—7644no—
115C24C22×D19111yescharacteristic
116C2×Q8—3822no—
117C22×C4—22no—
118C22×C4—3821no—
119C22×C4—1914no—
120C22×C4D76412yes—
121C4⋊C4—3824no—
122C4⋊C4—3822no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—3822no—
127C22⋊C4—3822no—
128C22⋊C4—3822no—
129C22⋊C4—7644no—
130C22⋊C4—7648no—
131C23—42no—
132C23—3828no—
133C23—3824no—
134C23—3824no—
135C23—3822no—
136C23—7644no—
137C23—7642no—
138C23—7644no—
139C23—7644no—
140C23—7648no—
141C23—7641no—
142C23—7642no—
143C23—7641no—
144C23D4×D19112yes—
145C23C2×D76411yescharacteristic
146C23Q8×D19111yescharacteristic
147C23#1569028611yescharacteristic
148C23#1569028712yes—
149Q8—15282no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—3824no—
155C2×C4—7642no—
156C2×C4—7641no—
157C2×C4—7644no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—3828no—
164C22—3824no—
165C22—7648no—
166C22—7644no—
167C22—7644no—
168C22—7642no—
169C22—7642no—
170C22—7644no—
171C22#2025929211yescharacteristic
172C22#2025930012yes—
173C22#2025930211yescharacteristic
174C22#2025930812yes—
175C22#2025931011yescharacteristic
176C4—42no—
177C4—82no—
178C4—15281no—
179C4—7644no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—7644no—
185C2#2713257611yescharacteristic
186C2#2713285112yes—
subgroupG/H
centerC22#20259292
commutator, cosocleC24×C191C23
FrattiniC24C22×D191
Fitting#342×C191C2
socleC22×C191#120
prime radicalC24×D191C22
hypercenter#342D191
hypocenterC191#2029
LCSG ▷ C24×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #27133289C2semidirect
4. #27132429C2semidirect
5. #342×C191C2semidirect
20. C23×D191Q8semidirect
93. C191#2029semidirect
99. #342D191semidirect

Maximal subgroups

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

subgroupindex
#271332892
#271325182
#271324292
#342×C1912
#2029191