Finite Group Databasebeta

← #15672522

Subgroups — #15672522

Order 2944.

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

normal subgroups only

This group has 256 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#13093089C211yesmaximal, characteristic
2#13092791C211yesmaximal, characteristic, complemented
3#13092704C211yesmaximal, characteristic, complemented
4#13092160C211yesmaximal, characteristic, complemented
5Q8⋊(C23⋊C8)C211yesmaximal, characteristic, complemented
6#13092144C211yesmaximal, characteristic, complemented
7C23×#477C211yesmaximal, characteristic, complemented
8C2×#67292—21no—
9C23×C4⋊Q8C2211yescharacteristic
10C4×Q8×C23C2211yescharacteristic
11C2×#67274—22no—
12#10625066—22no—
13#10625066—21no—
14#10625045—21no—
15#10625045C2211yescharacteristic
16C4×D92—21no—
17C4×D92C2211yescharacteristic
18C23×#107—21no—
19C23×C4⋊C8C2211yescharacteristic
20C23×#103—21no—
21#10624991—21no—
22#10624990—21no—
23C4⋊(C23⋊C8)C2211yescharacteristic
24C4×C23⋊C8C2211yescharacteristic
25C2×Q8×C23—21no—
26C2×Q8×C23D411yescharacteristic
27#67292—42no—
28#67292—42no—
29C2×D92—22no—
30C2×D92—21no—
31C2×D92—21no—
32C2×D92—22no—
33C2×D92D411yescharacteristic
34C2×D92D412yes—
35C2×C4×D23—22no—
36C2×C4×D23—21no—
37C2×C8×C23—21no—
38C4⋊C4×C23—21no—
39C4⋊C4×C23—21no—
40C4⋊C4×C23—21no—
41C4⋊C4×C23D411yescharacteristic
42C42×C23—21no—
43C42×C23C2311yescharacteristic
44#67274—44no—
45(C2×D23)⋊C4—22no—
46(C2×D23)⋊C4—21no—
47C4⋊Dic23D411yescharacteristic
48C2×C23⋊C8—21no—
49C2×C23⋊C8—21no—
50C22×D23—21no—
51C22×D23—22no—
52Q8×C23—21no—
53Q8×C23—42no—
54Q8×C23SD1612yescomplemented
55C2×C4×C23—21no—
56C2×C4×C23—21no—
57C2×C4×C23—41no—
58C2×C4×C23C2×D411yescharacteristic
59C2×C4×C23C2×D411yescharacteristic
60C2×C4×C23C2×D411yescharacteristic
61C2×Dic23—21no—
62D92—21no—
63D92—21no—
64D92—42no—
65D92—44no—
66D92—41no—
67D92—41no—
68D92—41no—
69D92—41no—
70D92—82no—
71C4×D23—44no—
72C4×D23—41no—
73C4×D23—41no—
74C8×C23—41no—
75C23⋊C8—22no—
76C23⋊C8—41no—
77#1631—231nomaximal, Sylow
78C22×C23#12011yescharacteristic
79C2×D23—42no—
80C2×D23—41no—
81C2×D23—41no—
82C2×D23—42no—
83C2×D23—42no—
84C2×D23—41no—
85C4×C23—22no—
86C4×C23—21no—
87C4×C23—41no—
88C4×C23—81no—
89C4×C23C2×SD1611yescharacteristic
90C4×C23#13511yescharacteristic
91C4×C23AGL(1,ℤ/8)11yescharacteristic
92C4×C23#13711yescharacteristic
93Dic23—41no—
94#550—231no—
95#492—231no—
96#477D2311yescharacteristic, complemented
97#476—231no—
98#331—231no—
99Q8⋊C8—231no—
100D4⋊C8—231no—
101C2×C23#45011yescharacteristic
102C2×C23#45211yescharacteristic
103C2×C23#45311yescharacteristic
104D23—81no—
105D23—82no—
106C2×#41—461no—
107C2×SD16—462no—
108C4⋊Q8C2×D2311yescharacteristic
109#127—231no—
110#127—461no—
111#121—462no—
112#121—461no—
113C4×Q8C2×D2311yescharacteristic
114C4×D4—231no—
115C4×D4—461no—
116#107—21no—
117#106—461no—
118C4⋊C8—231no—
119C4⋊C8C2×D2311yescharacteristic
120#103—21no—
121#102—461no—
122C4×C8—231no—
123C23#163111yescharacteristic, complemented, Sylow
124#41—922no—
125#41—922no—
126C2×Q8—21no—
127C2×Q8C22⋊D2311yescharacteristic
128C2×D4—231no—
129C2×D4—232no—
130C2×D4—462no—
131C2×D4—462no—
132C2×D4—461no—
133C2×D4—461no—
134C22×C4—462no—
135C22×C4—461no—
136SD16—924no—
137C2×C8—21no—
138C2×C8—461no—
139C2×C8—461no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—231no—
144C4⋊C4C22⋊D2311yescharacteristic
145C22⋊C4—462no—
146C22⋊C4—461no—
147C42—21no—
148C42C22×D2311yescharacteristic
149C23—461no—
150C23—462no—
151Q8—21no—
152Q8—42no—
153Q8#6727412yescomplemented
154D4—461no—
155D4—461no—
156D4—922no—
157D4—924no—
158D4—921no—
159D4—921no—
160D4—921no—
161D4—921no—
162D4—1842no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—461no—
167C2×C4—924no—
168C2×C4—921no—
169C2×C4—921no—
170C2×C4D4×D2311yescharacteristic
171C2×C4D4×D2311yescharacteristic
172C2×C4C2×C22⋊D2311yescharacteristic
173C8—41no—
174C8—462no—
175C8—921no—
176C22—922no—
177C22—921no—
178C22—921no—
179C22—922no—
180C22—922no—
181C22—921no—
182C22#1062510811yescharacteristic
183C4—22no—
184C4—21no—
185C4—41no—
186C4—81no—
187C4—921no—
188C4#1062508911yescharacteristic
189C4#1062509011yescharacteristic
190C4C2×#6727411yescharacteristic
191C4#1062512011yescharacteristic
192C2—1841no—
193C2—1842no—
194C2#1309256711yescharacteristic
195C2#1309257811yescharacteristic
196C2#1309258311yescharacteristic
subgroupG/H
centerC22#10625108
commutator, cosocleC42×C23C23
FrattiniC42C22×D23
FittingC23×#477C2
socleC22×C23#120
prime radical#13093089C2
hypercenter#477D23
hypocenterC23#1631
LCSG ▷ C42×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #13092791C2semidirect
3. #13092704C2semidirect
4. #13092160C2semidirect
5. Q8⋊(C23⋊C8)C2semidirect
6. #13092144C2semidirect
7. C23×#477C2semidirect
54. Q8×C23SD16semidirect
96. #477D23semidirect
123. C23#1631semidirect
153. Q8#67274semidirect

Maximal subgroups

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

subgroupindex
#130930892
#130927912
#130927042
#130921602
Q8⋊(C23⋊C8)2
#130921442
C23×#4772
#163123