Finite Group Databasebeta

← 3−1+2×#392 #13144936

Subgroups — 3−1+2×#392 #13144936

Order 1728.

1288 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 1288 conjugacy classes of subgroups — too many to draw. Only its 301 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
1C22×D4×3−1+2C213yesmaximal, complemented
2C2×C22⋊C4×3−1+2C213yesmaximal, complemented
33−1+2×#95C211yesmaximal, characteristic, complemented
4C32×#392C311yesmaximal, characteristic
5C9×#392C313yesmaximal, complemented
6C24×3−1+2C2213yes—
7C2×D4×3−1+2—212no—
8C2×D4×3−1+2—26no—
9C22×C4×3−1+2C2211yescharacteristic
10C22×C4×3−1+2C2213yes—
11C22⋊C4×3−1+2—26no—
12C22×C32×D4C2×C313yes—
13C2×C32×C22⋊C4C2×C313yes—
14C22×D4×C9C2×C319yescomplemented
15C32×#95C2×C311yescharacteristic
16C2×C9×C22⋊C4C2×C319yescomplemented
17C9×#95C2×C313yescomplemented
18C23×3−1+2—26no—
19C23×3−1+2—26no—
20C23×3−1+2—23no—
21C23×3−1+2—23no—
22C23×3−1+2D416yescomplemented
23C23×3−1+2C2311yescharacteristic
24D4×3−1+2—224no—
25C2×C4×3−1+2—26no—
26C2×C4×3−1+2—23no—
27C2×C4×3−1+2D416yescomplemented
28C3×#392—31no—
29C3×#392C3211yescharacteristic
30C24×C32C22×C313yes—
31C2×C32×D4—212no—
32C2×C32×D4—26no—
33C22×C32×C4C22×C311yescharacteristic
34C22×C32×C4C22×C313yes—
35C24×C9C22×C319yes—
36C32×C22⋊C4—26no—
37C2×D4×C9—236no—
38C2×D4×C9—218no—
39C22×C4×C9C22×C313yes—
40C22×C4×C9C22×C319yes—
41C9×C22⋊C4—218no—
42C22×3−1+2—212no—
43C22×3−1+2—26no—
44C22×3−1+2—43no—
45C22×3−1+2—46no—
46C22×3−1+2—43no—
47C22×3−1+2C2×D413yes—
48C22×3−1+2C2×D413yes—
49C22×3−1+2#4111yescharacteristic
50C4×3−1+2—26no—
51C4×3−1+2—41no—
52C22×C3×D4—33no—
53C22×C3×D4C2×C3213yes—
54C2×C3×C22⋊C4—33no—
55C2×C3×C22⋊C4C2×C3213yes—
56C3×#95—31no—
57C3×#95C2×C3211yescharacteristic
58C23×C32—23no—
59C23×C32—26no—
60C23×C32—26no—
61C23×C32—23no—
62C23×C32C3×D416yes—
63C23×C32C23×C311yescharacteristic
64C32×D4—224no—
65C2×C32×C4—26no—
66C2×C32×C4—23no—
67C2×C32×C4C3×D416yes—
68C23×C9—29no—
69C23×C9—218no—
70C23×C9—218no—
71C23×C9—29no—
72C23×C9C3×D4118yescomplemented
73C23×C9C23×C313yes—
74D4×C9—272no—
75C2×C4×C9—218no—
76C2×C4×C9—29no—
77C2×C4×C9C3×D4118yescomplemented
78#3923−1+211yescharacteristic, direct product factor, Sylow
79C2×3−1+2—46no—
80C2×3−1+2#12013yes—
81C2×3−1+2#12113yes—
82C2×3−1+2#12711yescharacteristic
83C24×C3—33no—
84C24×C3C22×C3213yes—
85C2×C3×D4—212no—
86C2×C3×D4—26no—
87C2×C3×D4—66no—
88C2×C3×D4—612no—
89C22×C3×C4—33no—
90C22×C3×C4—31no—
91C22×C3×C4C22×C3211yescharacteristic
92C22×C3×C4C22×C3213yes—
93C3×C22⋊C4—26no—
94C3×C22⋊C4—66no—
95C22×C32—212no—
96C22×C32—26no—
97C22×C32—43no—
98C22×C32—46no—
99C22×C32—43no—
100C22×C32C2×C3×D413yes—
101C22×C32C2×C3×D413yes—
102C22×C32C3×#4111yescharacteristic
103C32×C4—26no—
104C32×C4—41no—
105C22×C9—236no—
106C22×C9—218no—
107C22×C9—49no—
108C22×C9—418no—
109C22×C9—49no—
110C22×C9C2×C3×D419yes—
111C22×C9C2×C3×D419yes—
112C22×C9C3×#4113yes—
113C4×C9—218no—
114C4×C9—43no—
115C22×D4C2×3−1+213yescomplemented
116C2×C22⋊C4C2×3−1+213yescomplemented
117#95C2×3−1+211yescharacteristic, complemented
1183−1+2#39211yescharacteristic, direct product factor, Sylow
119C23×C3—23no—
120C23×C3—26no—
121C23×C3—26no—
122C23×C3—23no—
123C23×C3—36no—
124C23×C3—31no—
125C23×C3—63no—
126C23×C3—66no—
127C23×C3—66no—
128C23×C3—63no—
129C23×C3C32×D416yes—
130C23×C3C23×C3211yescharacteristic
131C3×D4—224no—
132C3×D4—624no—
133C2×C3×C4—26no—
134C2×C3×C4—23no—
135C2×C3×C4—36no—
136C2×C3×C4—66no—
137C2×C3×C4—63no—
138C2×C3×C4C32×D416yes—
139C2×C32—46no—
140C2×C32C3×#12013yes—
141C2×C32C3×#12113yes—
142C2×C32C3×#12711yescharacteristic
143C2×C9—418no—
144C2×C9C3×#12019yes—
145C2×C9C3×#12119yes—
146C2×C9C3×#12713yes—
147C24C22×3−1+213yes—
148C2×D4—212no—
149C2×D4—26no—
150C22×C4C22×3−1+211yescharacteristic
151C22×C4C22×3−1+213yes—
152C22⋊C4—26no—
153C22×C3—212no—
154C22×C3—26no—
155C22×C3—33no—
156C22×C3—33no—
157C22×C3—31no—
158C22×C3—43no—
159C22×C3—46no—
160C22×C3—43no—
161C22×C3—612no—
162C22×C3—66no—
163C22×C3—123no—
164C22×C3—126no—
165C22×C3—123no—
166C22×C3C2×C32×D413yes—
167C22×C3C2×C32×D413yes—
168C22×C3C32×#4111yescharacteristic
169C3×C4—26no—
170C3×C4—41no—
171C3×C4—66no—
172C3×C4—121no—
173C32C3×#39211yescharacteristic
174C9C3×#39213yescomplemented
175C23—23no—
176C23—26no—
177C23—26no—
178C23—23no—
179C23D4×3−1+216yescomplemented
180C23C23×3−1+211yescharacteristic
181D4—224no—
182C2×C4—26no—
183C2×C4—23no—
184C2×C4D4×3−1+216yescomplemented
185C2×C3—33no—
186C2×C3—33no—
187C2×C3—31no—
188C2×C3—46no—
189C2×C3—126no—
190C2×C3C32×#12013yes—
191C2×C3C32×#12113yes—
192C2×C3C32×#12711yescharacteristic
193C22—212no—
194C22—26no—
195C22—43no—
196C22—46no—
197C22—43no—
198C22C2×D4×3−1+213yes—
199C22C2×D4×3−1+213yes—
200C22#41×3−1+211yescharacteristic
201C4—26no—
202C4—41no—
203C3—31no—
204C3C32×#39211yescharacteristic
205C2—46no—
206C23−1+2×#12013yes—
207C23−1+2×#12113yes—
208C23−1+2×#12711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23×C3C23×C32
prime radicalC32×#392C3
LCSG ▷ C23×C3 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D4×3−1+2C2semidirect
2. C2×C22⋊C4×3−1+2C2semidirect
3. 3−1+2×#95C2semidirect
5. C9×#392C3semidirect
14. C22×D4×C9C2×C3semidirect
16. C2×C9×C22⋊C4C2×C3semidirect
17. C9×#95C2×C3semidirect
22. C23×3−1+2D4semidirect
27. C2×C4×3−1+2D4semidirect
72. C23×C9C3×D4semidirect
77. C2×C4×C9C3×D4semidirect
78. #3923−1+2direct (N × H)
115. C22×D4C2×3−1+2semidirect
116. C2×C22⋊C4C2×3−1+2semidirect
117. #95C2×3−1+2semidirect
118. 3−1+2#392direct (N × H)
174. C9C3×#392semidirect
179. C23D4×3−1+2semidirect
184. C2×C4D4×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C22×D4×3−1+22
C2×C22⋊C4×3−1+22
3−1+2×#952
C32×#3923
C9×#3923