Finite Group Databasebeta

← #27140801

Subgroups — #27140801

Order 12416.

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

normal subgroups only

This group has 278 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#20261111C211yesmaximal, characteristic, complemented
2#20260935C211yesmaximal, characteristic, complemented
3#20260930C211yesmaximal, characteristic, complemented
4#20260479C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic, complemented
7#550×C97C211yesmaximal, characteristic, complemented
8C2×#13095498—21no—
9C2×#13095496—21no—
10C2×#13095496—22no—
11#15690807C2211yescharacteristic
12C4⋊#11715895C2211yescharacteristic
13C4⋊#11715895C2211yescharacteristic
14#15690730—21no—
15#15690730—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×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28#13095498—42no—
29#13095496—44no—
30#13095496—42no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C2×#11715895—21no—
34#41×C97—42no—
35#41×C97—42no—
36C2×Q8×C97D411yescharacteristic
37C2×D4×C97—22no—
38C2×D4×C97—22no—
39C2×D4×C97—21no—
40C2×D4×C97—21no—
41C2×D4×C97D411yescharacteristic
42C2×D4×C97D412yes—
43C22×C4×C97—22no—
44C22×C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—22no—
49C22⋊C4×C97—21no—
50C42×C97—21no—
51C42×C97C2311yescharacteristic
52C23×C97—21no—
53C23×C97—22no—
54C22×D97—41no—
55Q8×C97—21no—
56Q8×C97D812yescomplemented
57D4×C97—21no—
58D4×C97—21no—
59D4×C97—42no—
60D4×C97—44no—
61D4×C97—41no—
62D4×C97—41no—
63D4×C97—41no—
64D4×C97—41no—
65D4×C97—82no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—44no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75D388—42no—
76D388—41no—
77D388—41no—
78D388—41no—
79D388—41no—
80#11715895—22no—
81#11715895—41no—
82#11715895—41no—
83C22×C97—42no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97—42no—
87C22×C97—42no—
88C22×C97—41no—
89C22×C97#12011yescharacteristic
90C2×D97—81no—
91C2×D97—81no—
92C2×D97—81no—
93C4×C97—22no—
94C4×C97—21no—
95C4×C97—41no—
96C4×C97—41no—
97C4×C97C2×D811yescharacteristic
98C4×C97#13511yescharacteristic
99C4×C97AGL(1,ℤ/8)11yescharacteristic
100C4×C97AGL(1,ℤ/8)11yescharacteristic
101C2×C97—81no—
102C2×C97—82no—
103C2×C97#44711yescharacteristic
104C2×C97#44911yescharacteristic
105C2×C97#45311yescharacteristic
106D97—161no—
107#1621—971nomaximal, Sylow
108C97#162111yescharacteristic, complemented, Sylow
109#550D9711yescharacteristic, complemented
110#493—971no—
111#460—971no—
112#459—971no—
113#331—971no—
114Q8⋊C8—971no—
115D4⋊C8—971no—
116C2×#41—21no—
117C2×SD16—1941no—
118C2×D8—1941no—
119C2×D8—1942no—
120#127—21no—
121#127—971no—
122#127C2×D9711yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C2×D9711yescharacteristic
126C4×D4—21no—
127C4×D4C2×D9711yescharacteristic
128C4⋊C8—971no—
129C4⋊C8—971no—
130#102—1941no—
131#102—1941no—
132C4×C8—971no—
133#41—42no—
134#41—42no—
135C2×Q8C22⋊D9711yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—1941no—
141C2×D4—1941no—
142C2×D4—1941no—
143C2×D4C22⋊D9711yescharacteristic
144C2×D4C22⋊D9712yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—3882no—
148D8—3882no—
149D8—3884no—
150C2×C8—1941no—
151C2×C8—1941no—
152C2×C8—1941no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D9711yescharacteristic
155C4⋊C4C22⋊D9711yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C22×D9711yescharacteristic
160C23—21no—
161C23—22no—
162C23—3881no—
163Q8—21no—
164Q8#1309549612yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—82no—
174D4—3882no—
175D4—3881no—
176D4—3881no—
177D4—3881no—
178D4—3881no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C22⋊D9711yescharacteristic
186C2×C4C2×C22⋊D9711yescharacteristic
187C2×C4C2×C22⋊D9711yescharacteristic
188C8—3881no—
189C8—3881no—
190C8—1942no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—7761no—
198C22—7761no—
199C22—7761no—
200C22#1569077911yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4C2×#1309549611yescharacteristic
206C4#1569083011yescharacteristic
207C4#1569083511yescharacteristic
208C4#1569083611yescharacteristic
209C2—81no—
210C2—82no—
211C2—15521no—
212C2#2026086911yescharacteristic
213C2#2026088011yescharacteristic
214C2#2026090311yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#550×C97C2
socleC22×C97#120
hypercenter#550D97
hypocenterC97#1621
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. #20261111C2semidirect
2. #20260935C2semidirect
3. #20260930C2semidirect
4. #20260479C2semidirect
5. Q8⋊#11715895C2semidirect
6. D4⋊#11715895C2semidirect
7. #550×C97C2semidirect
56. Q8×C97D8semidirect
108. C97#1621semidirect
109. #550D97semidirect
164. Q8#13095496semidirect

Maximal subgroups

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

subgroupindex
#202611112
#202609352
#202609302
#202604792
Q8⋊#117158952
D4⋊#117158952
#550×C972
#162197