Finite Group Databasebeta

← #27140863

Subgroups — #27140863

Order 12416.

256 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 61 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#20261107C211yesmaximal, characteristic, complemented
2#20261030C211yesmaximal, characteristic, complemented
3#20261025C211yesmaximal, characteristic, complemented
4#20260479C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#550×C97C211yesmaximal, characteristic
8C2×#13095499—22no—
9#15690810C2211yescharacteristic
10#15690743—21no—
11#15690741—21no—
12C4⋊#11715895C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14#15690734—21no—
15#15690731—21no—
16C4×#11715895C2211yescharacteristic
17C2×#41×C97—21no—
18#127×C97—21no—
19#127×C97C2211yescharacteristic
20#121×C97—22no—
21#121×C97—21no—
22C4×Q8×C97C2211yescharacteristic
23C4×D4×C97—21no—
24C4×D4×C97C2211yescharacteristic
25C2×Dic194—21no—
26#13095499—44no—
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32#41×C97—42no—
33#41×C97—42no—
34C2×Q8×C97D411yescharacteristic
35C2×D4×C97—22no—
36C2×D4×C97—22no—
37C2×D4×C97—21no—
38C2×D4×C97—21no—
39C2×D4×C97D411yescharacteristic
40C2×D4×C97D412yes—
41C22×C4×C97—22no—
42C22×C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97D411yescharacteristic
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—22no—
47C22⋊C4×C97—21no—
48C42×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—21no—
51C23×C97—22no—
52Q8×C97—21no—
53Q8×C97SD1612yescomplemented
54D4×C97—21no—
55D4×C97—21no—
56D4×C97—42no—
57D4×C97—44no—
58D4×C97—41no—
59D4×C97—41no—
60D4×C97—41no—
61D4×C97—41no—
62D4×C97—82no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—44no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×Dic97—41no—
73Dic194—42no—
74#11715895—22no—
75#11715895—41no—
76#11715895—41no—
77C22×C97—42no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—42no—
81C22×C97—42no—
82C22×C97—41no—
83C22×C97#12011yescharacteristic
84C4×C97—22no—
85C4×C97—21no—
86C4×C97—41no—
87C4×C97—41no—
88C4×C97C2×SD1611yescharacteristic
89C4×C97#13511yescharacteristic
90C4×C97AGL(1,ℤ/8)11yescharacteristic
91C4×C97#13711yescharacteristic
92Dic97—81no—
93C2×C97—81no—
94C2×C97—82no—
95C2×C97#45011yescharacteristic
96C2×C97#45211yescharacteristic
97C2×C97#45311yescharacteristic
98#1631—971nomaximal, Sylow
99C97#163111yescharacteristic, complemented, Sylow
100#550D9711yescharacteristic
101#492—971no—
102#477—971no—
103#476—971no—
104#331—971no—
105Q8⋊C8—971no—
106D4⋊C8—971no—
107C2×#41—21no—
108C2×SD16—1942no—
109C4⋊Q8—971no—
110#127—21no—
111#127C2×D9711yescharacteristic
112#121—22no—
113#121—21no—
114C4×Q8C2×D9711yescharacteristic
115C4×D4—21no—
116C4×D4C2×D9711yescharacteristic
117#107—1941no—
118#106—1941no—
119C4⋊C8—971no—
120C4⋊C8—971no—
121#103—1941no—
122#102—1941no—
123C4×C8—971no—
124#41—42no—
125#41—42no—
126C2×Q8—1941no—
127C2×Q8C22⋊D9711yescharacteristic
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊D9711yescharacteristic
133C2×D4C22⋊D9712yes—
134C22×C4—22no—
135C22×C4—21no—
136SD16—3884no—
137C2×C8—1941no—
138C2×C8—1941no—
139C2×C8—1941no—
140C4⋊C4—21no—
141C4⋊C4—1941no—
142C4⋊C4—1941no—
143C4⋊C4C22⋊D9711yescharacteristic
144C4⋊C4C22⋊D9711yescharacteristic
145C22⋊C4—22no—
146C22⋊C4—21no—
147C42—21no—
148C42C22×D9711yescharacteristic
149C23—21no—
150C23—22no—
151Q8—21no—
152Q8—3882no—
153Q8#1309549912yescomplemented
154D4—21no—
155D4—21no—
156D4—42no—
157D4—44no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—82no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3881no—
170C2×C4C2×C22⋊D9711yescharacteristic
171C2×C4C2×C22⋊D9711yescharacteristic
172C2×C4C2×C22⋊D9711yescharacteristic
173C8—3881no—
174C8—3881no—
175C8—1942no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22#1569077911yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—7761no—
188C4C2×#1309549911yescharacteristic
189C4#1569083011yescharacteristic
190C4#1569083611yescharacteristic
191C4#1569084011yescharacteristic
192C2—81no—
193C2—82no—
194C2#2026088611yescharacteristic
195C2#2026089711yescharacteristic
196C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#550×C97C2
socleC22×C97#120
hypercenter#550D97
hypocenterC97#1631
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #20261107C2semidirect
2. #20261030C2semidirect
3. #20261025C2semidirect
4. #20260479C2semidirect
5. Q8⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
53. Q8×C97SD16semidirect
99. C97#1631semidirect
153. Q8#13095499semidirect

Maximal subgroups

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

subgroupindex
#202611072
#202610302
#202610252
#202604792
Q8⋊#117158952
D4⋊#117158952
#550×C972
#163197