Finite Group Databasebeta

← #44507863

Subgroups — #44507863

Order 30848.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#29508224C211yesmaximal, characteristic
2#29507701C211yesmaximal, characteristic, complemented
3#29507680C212yesmaximal, complemented
4#29507488C212yesmaximal, complemented
5#568×C241C211yesmaximal, characteristic, complemented
6C22×C4×D241C2211yescharacteristic
7C2×#16838754C2212yes—
8#21440623C412yes—
9#21440618—21no—
10#21440613—21no—
11#21440610C412yes—
12C4×C22⋊D241—22no—
13#21440601C2211yescharacteristic
14#21440561—21no—
15#21440559—21no—
16#21440554—24no—
17C2×#41×C241C2211yescharacteristic
18#131×C241—24no—
19C2×M16×C241—22no—
20C2×M16×C241C2212yes—
21C23×D241C2×C411yescharacteristic
22C2×C22⋊D241—22no—
23C22×Dic241C2×C411yescharacteristic
24C2×C4×D241—22no—
25C2×C4×D241—21no—
26C2×C4×D241—21no—
27C2×C4×D241—22no—
28#16838754—22no—
29#16838754—22no—
30C2×#14515349—22no—
31C4⋊Dic241—21no—
32#16838749—22no—
33C22⋊Dic241—22no—
34C22⋊Dic241C2×C412yes—
35#16838747—22no—
36C4×Dic241—21no—
37#41×C241—22no—
38#41×C241—22no—
39C2×Q8×C241D411yescharacteristic
40C2×D4×C241C2×C412yes—
41C2×D4×C241D411yescharacteristic
42C22×C4×C241D412yes—
43C22×C4×C241C2311yescharacteristic
44M16×C241—24no—
45M16×C241—22no—
46M16×C241—22no—
47C2×C8×C241—24no—
48C2×C8×C241D414yescomplemented
49C22×D241—21no—
50C22×D241—22no—
51C22×D241—42no—
52C22⋊D241—44no—
53C2×Dic241—22no—
54C2×Dic241—21no—
55C2×Dic241—22no—
56C4×D241—41no—
57C4×D241—42no—
58C4×D241—41no—
59#14515349—42no—
60C23×C241C22⋊C412yes—
61C23×C241C22×C411yescharacteristic
62Q8×C241—22no—
63D4×C241—24no—
64D4×C241—22no—
65C2×C4×C241—22no—
66C2×C4×C241—22no—
67C2×C4×C241C22⋊C412yes—
68C2×C4×C241C2×D412yes—
69C2×C4×C241C2×D411yescharacteristic
70C2×C4×C241C2×D411yescharacteristic
71C2×C4×C241#4112yes—
72C8×C241—24no—
73C22×C241—22no—
74C22×C241—41no—
75C22×C241—42no—
76C22×C241C2×C22⋊C411yescharacteristic
77C22×C241C4×D412yes—
78C2×D241—22no—
79C2×D241—24no—
80C2×D241—41no—
81C2×D241—42no—
82C2×D241—41no—
83C4×C241—22no—
84C4×C241#12011yescharacteristic
85C4×C241#12112yes—
86C4×C241#12311yescharacteristic
87Dic241—42no—
88Dic241—41no—
89C2×C241—21no—
90C2×C241—22no—
91C2×C241—42no—
92C2×C241#38611yescharacteristic
93D241—42no—
94C241#189111yescharacteristic, complemented, Sylow
95#1891—2411nomaximal, Sylow
96#568D24111yescharacteristic, complemented
97#525—2411no—
98#413—2411no—
99#407—2412no—
100#344—2412no—
101C2×#41C2×D24111yescharacteristic
102C23×C4—2411no—
103#131—24no—
104C2×M16—22no—
105C2×M16—2412no—
106C2×M16C2×D24112yes—
107#123—2412no—
108C22⋊Q8—4821no—
109#121—4821no—
110#120—2412no—
111C4×D4—4822no—
112#117—2411no—
113#101—4821no—
114#100—4821no—
115C22⋊C8—4824no—
116C24—2411no—
117#41—22no—
118#41—22no—
119C2×Q8D96411yescharacteristic
120C2×D4—4822no—
121C2×D4C4×D24112yes—
122C2×D4D96411yescharacteristic
123C22×C4—4822no—
124C22×C4—4821no—
125C22×C4—4821no—
126C22×C4—4822no—
127C22×C4—2411no—
128C22×C4C22⋊D24112yes—
129C22×C4C22×D24111yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—4822no—
134M16—4822no—
135C2×C8—24no—
136C2×C8—4822no—
137C2×C8C22⋊D24114yescomplemented
138C4⋊C4—4822no—
139C4⋊C4—4821no—
140C22⋊C4—4822no—
141C22⋊C4—4822no—
142C22⋊C4—2412no—
143C42—4821no—
144C23—9642no—
145C23—4821no—
146C23—4822no—
147C23#1683874712yes—
148C23C2×C4×D24111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—9644no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—9641no—
156C2×C4—9642no—
157C2×C4—9641no—
158C2×C4—4822no—
159C2×C4—4821no—
160C2×C4—4822no—
161C2×C4#1683874712yes—
162C2×C4C2×C22⋊D24112yes—
163C2×C4C2×C22⋊D24111yescharacteristic
164C2×C4C2×D96411yescharacteristic
165C2×C4#1683877012yes—
166C8—24no—
167C8—9642no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—9641no—
172C22—9642no—
173C22—9641no—
174C22—4822no—
175C22—4824no—
176C22C2×#1683874711yescharacteristic
177C22C4×C22⋊D24112yes—
178C4—22no—
179C4—9642no—
180C4—9641no—
181C4#2144061111yescharacteristic
182C4#2144061612yes—
183C4#2144062511yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—9642no—
188C2#2950759611yescharacteristic
subgroupG/H
centerC4#21440611
commutatorC23×C241C22×C4
FrattiniC22×C4C22×D241
Fitting#568×C241C2
socleC2×C241#386
prime radical#29508224C2
cosocleC22×C4×C241C23
hypercenter#568D241
hypocenterC241#1891
LCSG ▷ C23×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #29507701C2semidirect
3. #29507680C2semidirect
4. #29507488C2semidirect
5. #568×C241C2semidirect
48. C2×C8×C241D4semidirect
94. C241#1891semidirect
96. #568D241semidirect
137. C2×C8C22⋊D241semidirect

Maximal subgroups

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

subgroupindex
#295082242
#295077012
#295076802
#295074882
#568×C2412
#1891241