Finite Group Databasebeta

← #27142288

Subgroups — #27142288

Order 12416.

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

normal subgroups only

This group has 308 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#20261306C211yesmaximal, characteristic
2#20260719C211yesmaximal, characteristic
3C2×#15690728C212yesmaximal, complemented
4#20260507C212yesmaximal, complemented
5#568×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×#13095495C2212yes—
8#15690808—21no—
9#15690797—21no—
10#15690791C414yes—
11#15690789—22no—
12#15690769C2211yescharacteristic
13#15690729—21no—
14#15690728—24no—
15#15690727—21no—
16C2×#41×C97C2211yescharacteristic
17#131×C97—24no—
18C2×M16×C97—22no—
19C2×M16×C97C2212yes—
20C2×Dic194—21no—
21C2×Dic194—21no—
22C2×Dic194—22no—
23C2×Dic194—22no—
24C22×Dic97C2×C412yes—
25#13095495—22no—
26#13095495—22no—
27C2×#11715895—22no—
28C4⋊Dic97—21no—
29#13095490—24no—
30C22⋊Dic97—24no—
31C22⋊Dic97C2×C412yes—
32C4×Dic97—21no—
33#41×C97—22no—
34#41×C97—22no—
35C2×Q8×C97D411yescharacteristic
36C2×D4×C97C2×C412yes—
37C2×D4×C97D411yescharacteristic
38C22×C4×C97D412yes—
39C22×C4×C97C2311yescharacteristic
40M16×C97—24no—
41M16×C97—22no—
42M16×C97D414yes—
43C2×C8×C97—24no—
44C2×C8×C97—22no—
45C23×C97C22⋊C412yes—
46C23×C97C22×C411yescharacteristic
47Q8×C97—22no—
48D4×C97—24no—
49D4×C97—22no—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97C22⋊C412yes—
53C2×C4×C97C2×D412yes—
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97#4112yes—
57C2×Dic97—22no—
58C2×Dic97—22no—
59C2×Dic97—24no—
60Dic194—41no—
61Dic194—42no—
62Dic194—41no—
63C8×C97—24no—
64#11715895—42no—
65C22×C97—22no—
66C22×C97—41no—
67C22×C97—42no—
68C22×C97C2×C22⋊C411yescharacteristic
69C22×C97C4×D412yes—
70C4×C97—22no—
71C4×C97#12011yescharacteristic
72C4×C97#12112yes—
73C4×C97#12311yescharacteristic
74Dic97—42no—
75Dic97—42no—
76C2×C97—21no—
77C2×C97—22no—
78C2×C97—42no—
79C2×C97#38611yescharacteristic
80#1915—971nomaximal, Sylow
81C97#191511yescharacteristic, complemented, Sylow
82#568D9711yescharacteristic
83#536—971no—
84#413—971no—
85C2×#101—972no—
86#344—972no—
87C2×#41C2×D9711yescharacteristic
88C22×Q8—971no—
89#131—24no—
90C2×M16—22no—
91C2×M16—972no—
92C2×M16C2×D9712yes—
93C4⋊Q8—1941no—
94#124—1941no—
95#123—974no—
96C22⋊Q8—1942no—
97#117—971no—
98#101—1941no—
99#101—1944no—
100#100—1941no—
101#41—22no—
102#41—22no—
103C2×Q8—1942no—
104C2×Q8—1941no—
105C2×Q8—1941no—
106C2×Q8—1942no—
107C2×Q8D38811yescharacteristic
108C2×D4C4×D9712yes—
109C2×D4D38811yescharacteristic
110C22×C4—972no—
111C22×C4C22⋊D9712yes—
112C22×C4C22×D9711yescharacteristic
113M16—24no—
114M16—22no—
115M16—1942no—
116M16—1942no—
117M16C22⋊D9714yes—
118C2×C8—24no—
119C2×C8—22no—
120C2×C8—1942no—
121C4⋊C4—1944no—
122C4⋊C4—1941no—
123C22⋊C4—972no—
124C22⋊C4—1944no—
125C42—1941no—
126C23#1309548812yes—
127C23C2×C4×D9711yescharacteristic
128Q8—22no—
129Q8—3881no—
130Q8—3882no—
131Q8—3881no—
132D4—24no—
133D4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—1942no—
137C2×C4—1942no—
138C2×C4—1944no—
139C2×C4#1309548812yes—
140C2×C4C2×C22⋊D9712yes—
141C2×C4C2×C22⋊D9711yescharacteristic
142C2×C4C2×D38811yescharacteristic
143C2×C4#1309551112yes—
144C8—24no—
145C8—3882no—
146C22—22no—
147C22—41no—
148C22—42no—
149C22C2×#1309548811yescharacteristic
150C22C4×C22⋊D9712yes—
151C4—22no—
152C4—3882no—
153C4—3882no—
154C4#1569077911yescharacteristic
155C4#1569078412yes—
156C4#1569079311yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2026061411yescharacteristic
subgroupG/H
centerC2#20260614
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#568×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#568D97
hypocenterC97#1915
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
3. C2×#15690728C2semidirect
4. #20260507C2semidirect
81. C97#1915semidirect

Maximal subgroups

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

subgroupindex
#202613062
#202607192
C2×#156907282
#202605072
#568×C972
#191597