Finite Group Databasebeta

← #11895142

Subgroups — #11895142

Order 1152.

1110 conjugacy classes of subgroups; only the 311 normal subgroups are shown.

normal subgroups only

This group has 1110 conjugacy classes of subgroups — too many to draw. Only its 311 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
1C32×#557C211yesmaximal, characteristic, complemented
2Q8×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3#10592782C213yesmaximal
4#10592778C213yesmaximal, complemented
5C4×#3755C213yesmaximal
6Q8×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#10592719C213yesmaximal, complemented
8#83688—34nomaximal
9C2×Q8×C3⋊D3—21no—
10C32×C4⋊Q8C2213yes—
11C32×#125—23no—
12C32×C4×Q8—21no—
13C32×C4×Q8C2213yes—
14C32×C4×Q8C2211yescharacteristic
15C2×#3755C2213yes—
16#64106—23no—
17C4×C4⋊(C3⋊D3)C2213yes—
18#64078C2213yes—
19#63662C2213yes—
20#63662C2216yes—
21#63636C2211yescharacteristic
22#63635C2213yes—
23C4⋊(C3⋊(C3⋊C8))C2213yes—
24C4×C3⋊(C3⋊C8)C2213yes—
25C3×#557D314yescomplemented
26Q8×D12—34no—
27#5083—312no—
28#5079—312no—
29C4×#215—312no—
30Q8×C3⋊C8—34no—
31#5020—312no—
32C2×C32×Q8D411yescharacteristic
33C2×C32×Q8C2313yes—
34Q8×C3⋊D3—44no—
35C2×C4⋊(C3⋊D3)C2311yescharacteristic
36C2×C4×C3⋊D3—23no—
37C2×#634—23no—
38C32×C4⋊C4—23no—
39C32×C4⋊C4—23no—
40C32×C4⋊C4—23no—
41C32×C4⋊C4D413yes—
42C32×C4⋊C4C2311yescharacteristic
43C32×C42—23no—
44C32×C42C2313yes—
45#3755—26no—
46#3752—23no—
47C4⋊(C3⋊Dic3)—23no—
48C4⋊(C3⋊Dic3)C2313yes—
49C2×C3⋊(C3⋊C8)C2311yescharacteristic
50C2×C3⋊(C3⋊C8)C2313yes—
51#3359—91noSylow
52C2×D3×Q8—64no—
53C3×C4⋊Q8C2×D3112yes—
54C3×#125—212no—
55C3×C4×Q8—24no—
56C3×C4×Q8C2×D3112yes—
57C3×C4×Q8C2×D314yes—
58C2×#215—312no—
59#897—612no—
60C4×D12—312no—
61#869—312no—
62#835—324no—
63#835—312no—
64#809—34no—
65#808—312no—
66C4⋊(C3⋊C8)—312no—
67C4×C3⋊C8—312no—
68C22×C3⋊D3—21no—
69C32×Q8—23no—
70C32×Q8—23no—
71C32×Q8SD1614yescomplemented
72C2×C32×C4—21no—
73C2×C32×C4—23no—
74C2×C32×C4C2×D413yes—
75C2×C32×C4C2×D413yes—
76C2×C32×C4C2411yescharacteristic
77C2×C3⋊Dic3—23no—
78C4⋊(C3⋊D3)—21no—
79C4⋊(C3⋊D3)#4112yes—
80C4×C3⋊D3—46no—
81#634—46no—
82C3⋊(C3⋊C8)—23no—
83C3⋊(C3⋊C8)—21no—
84#557C3⋊D311yescharacteristic, complemented
85D4×Q8—91no—
86#487—93no—
87#475—93no—
88#461—93no—
89C8×Q8—91no—
90C4×SD16—93no—
91C2×C3×Q8#6714yes—
92C2×C3×Q8C22×D3112yes—
93D3×Q8—1216no—
94C2×D12—34no—
95C2×C4×D3—612no—
96C2×Dic6—612no—
97C3×C4⋊C4—212no—
98C3×C4⋊C4—212no—
99C3×C4⋊C4—212no—
100C3×C4⋊C4#67112yes—
101C3×C4⋊C4C22×D314yes—
102C3×C42—212no—
103C3×C42C22×D3112yes—
104#215—624no—
105(C2×D3)⋊C4—612no—
106C4⋊Dic3—312no—
107C4⋊Dic3—612no—
108C2×C3⋊C8—34no—
109C2×C3⋊C8—312no—
110C22×C32C22×D411yescharacteristic
111C2×C3⋊D3—22no—
112C2×C3⋊D3—41no—
113C2×C3⋊D3—41no—
114C32×C4—23no—
115C32×C4—43no—
116C32×C4—41no—
117C32×C4C2×SD1616yes—
118C32×C4C2×#4111yescharacteristic
119C32×C42−1+411yescharacteristic
120C3⋊Dic3—43no—
121C22×Q8—181no—
122C2×SD16—93no—
123C4⋊Q8—93no—
124C4⋊Q8C2×C3⋊D313yes—
125#125—23no—
126C22⋊Q8—183no—
127C4×Q8—21no—
128C4×Q8C2×C3⋊D313yes—
129C4×Q8C2×C3⋊D311yescharacteristic
130C4×D4—93no—
131#106—93no—
132C4⋊C8—93no—
133#103—93no—
134#103—96no—
135#102—91no—
136C4×C8—93no—
137C22×D3—64no—
138C3×Q8—212no—
139C3×Q8—212no—
140C3×Q8#215116yescomplemented
141C2×C3×C4—24no—
142C2×C3×C4—212no—
143C2×C3×C4C2×#67112yes—
144C2×C3×C4C2×#67112yes—
145C2×C3×C4C23×D314yes—
146C2×Dic3—612no—
147D12—38no—
148D12—64no—
149C4×D3—1224no—
150Dic6—1224no—
151C3⋊C8—612no—
152C3⋊C8—64no—
153C2×C32#54811yescharacteristic
154C2×C32C22×SD1611yescharacteristic
155C2×C32#57811yescharacteristic
156C3⋊D3—42no—
157C2×Q8—183no—
158C2×Q8—364no—
159C2×Q8#63811yescharacteristic
160C2×Q8C22×C3⋊D313yes—
161C2×D4—91no—
162C22×C4—183no—
163SD16—186no—
164C2×C8—91no—
165C2×C8—93no—
166C4⋊C4—23no—
167C4⋊C4—23no—
168C4⋊C4—23no—
169C4⋊C4—93no—
170C4⋊C4—183no—
171C4⋊C4#63813yes—
172C4⋊C4C22×C3⋊D311yescharacteristic
173C22⋊C4—183no—
174C42—23no—
175C42C22×C3⋊D313yes—
176C22×C3C22×#6714yes—
177C2×D3—68no—
178C2×D3—124no—
179C2×D3—124no—
180C3×C4—212no—
181C3×C4—412no—
182C3×C4—44no—
183C3×C4C2×#215124yes—
184C3×C4D3×#4114yes—
185C3×C4#101014yes—
186Dic3—1212no—
187C32#335911yescharacteristic, complemented, Sylow
188C23—181no—
189Q8—23no—
190Q8—23no—
191Q8—366no—
192Q8#375514yescomplemented
193D4—92no—
194D4—181no—
195C2×C4—21no—
196C2×C4—23no—
197C2×C4—183no—
198C2×C4—366no—
199C2×C4C2×#63813yes—
200C2×C4C2×#63813yes—
201C2×C4C23×C3⋊D311yescharacteristic
202C8—183no—
203C8—181no—
204C2×C3C22×#21514yes—
205C2×C3#583114yes—
206C2×C3#583814yes—
207D3—128no—
208C22—182no—
209C22—361no—
210C22—361no—
211C22C22×#63811yescharacteristic
212C4—23no—
213C4—43no—
214C4—41no—
215C4—363no—
216C4C2×#375516yes—
217C4#41×C3⋊D311yescharacteristic
218C4#6436711yescharacteristic
219C3#8368814yescomplemented
220C2—362no—
221C2C22×#375511yescharacteristic
222C2#1059630211yescharacteristic
223C2#1059630911yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#557C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenter#557C3⋊D3
hypocenterC32#3359
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
4. #10592778C2semidirect
6. Q8×C3⋊(C3⋊C8)C2semidirect
7. #10592719C2semidirect
25. C3×#557D3semidirect
71. C32×Q8SD16semidirect
84. #557C3⋊D3semidirect
140. C3×Q8#215semidirect
187. C32#3359semidirect
192. Q8#3755semidirect
219. C3#83688semidirect

Maximal subgroups

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

subgroupindex
C32×#5572
Q8×C4⋊(C3⋊D3)2
#105927822
#105927782
C4×#37552
Q8×C3⋊(C3⋊C8)2
#105927192
#836883