Finite Group Databasebeta

← #27140964

Subgroups — #27140964

Order 12416.

212 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261085C211yesmaximal, characteristic, complemented
2#20261015C211yesmaximal, characteristic, complemented
3#20260940C211yesmaximal, characteristic, complemented
4#20260484C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#555×C97C211yesmaximal, characteristic
8C2×#13095499—21no—
9#15690810C2211yescharacteristic
10#15690743—21no—
11C4⋊#11715895C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13#15690734—22no—
14#15690733—21no—
15#15690731—21no—
16C4×#11715895C2211yescharacteristic
17C4⋊Q8×C97C2211yescharacteristic
18#124×C97—22no—
19#121×C97—21no—
20#121×C97—21no—
21C4×D4×C97C2211yescharacteristic
22C4×D4×C97C2211yescharacteristic
23#117×C97—21no—
24#117×C97—21no—
25C2×Dic194—21no—
26#13095499—42no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C2×Q8×C97D412yes—
33C2×D4×C97—22no—
34C2×D4×C97D411yescharacteristic
35C2×D4×C97D411yescharacteristic
36C22×C4×C97—21no—
37C22×C4×C97—21no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97—21no—
40C4⋊C4×C97D411yescharacteristic
41C4⋊C4×C97D411yescharacteristic
42C22⋊C4×C97—22no—
43C22⋊C4×C97—22no—
44C22⋊C4×C97—21no—
45C22⋊C4×C97—21no—
46C42×C97—22no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49C23×C97—21no—
50Q8×C97—42no—
51D4×C97—21no—
52D4×C97—21no—
53D4×C97—21no—
54D4×C97—21no—
55D4×C97—82no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—22no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×Dic97—41no—
67Dic194—41no—
68Dic194—41no—
69#11715895—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97#12011yescharacteristic
79C4×C97—21no—
80C4×C97—41no—
81C4×C97—41no—
82C4×C97—42no—
83C4×C97#13511yescharacteristic
84C4×C97#13511yescharacteristic
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97#13711yescharacteristic
87Dic97—81no—
88C2×C97—81no—
89C2×C97—81no—
90C2×C97#44911yescharacteristic
91C2×C97#45211yescharacteristic
92C2×C97#45511yescharacteristic
93#1646—971nomaximal, Sylow
94C97#164611yescharacteristic, complemented, Sylow
95#555D9711yescharacteristic
96#487—971no—
97#474—971no—
98#461—971no—
99#333—971no—
100D4⋊C8—971no—
101D4⋊C8—971no—
102C2×SD16—1941no—
103C4⋊Q8—971no—
104C4⋊Q8C2×D9711yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D9711yescharacteristic
109C4×D4C2×D9711yescharacteristic
110#117—21no—
111#117—21no—
112#107—1941no—
113C4⋊C8—971no—
114C4⋊C8—971no—
115#103—1942no—
116#103—1941no—
117#102—1941no—
118C4×C8—971no—
119C2×Q8—1941no—
120C2×Q8C22⋊D9712yes—
121C2×D4—22no—
122C2×D4C22⋊D9711yescharacteristic
123C2×D4C22⋊D9711yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—3882no—
127C2×C8—1941no—
128C2×C8—1941no—
129C2×C8—1941no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4C22⋊D9711yescharacteristic
135C4⋊C4C22⋊D9711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D9711yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—3881no—
146Q8—3881no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3881no—
160C2×C4C2×C22⋊D9711yescharacteristic
161C2×C4C2×C22⋊D9711yescharacteristic
162C2×C4C2×C22⋊D9711yescharacteristic
163C8—3881no—
164C8—3881no—
165C8—3881no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1569077911yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—7761no—
178C4#1569083011yescharacteristic
179C4#1569083011yescharacteristic
180C4#1569083511yescharacteristic
181C4#1569084011yescharacteristic
182C2—81no—
183C2—81no—
184C2#2026088011yescharacteristic
185C2#2026089711yescharacteristic
186C2#2026091311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#555×C97C2
socleC22×C97#120
hypercenter#555D97
hypocenterC97#1646
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20261085C2semidirect
2. #20261015C2semidirect
3. #20260940C2semidirect
4. #20260484C2semidirect
5. D4⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
94. C97#1646semidirect

Maximal subgroups

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

subgroupindex
#202610852
#202610152
#202609402
#202604842
D4⋊#117158952
#555×C972
#164697