Finite Group Databasebeta

← 3−1+2×#123 #11721906

Subgroups — 3−1+2×#123 #11721906

Order 864.

312 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×D4×3−1+2C211yesmaximal, characteristic
2C22×C4×3−1+2C211yesmaximal, characteristic, complemented
3C4⋊C4×3−1+2C212yesmaximal, complemented
4C22⋊C4×3−1+2C212yesmaximal, complemented
5C22⋊C4×3−1+2C211yesmaximal, characteristic, complemented
6C32×#123C311yesmaximal, characteristic
7C9×#123C313yesmaximal, complemented
8C23×3−1+2C2211yescharacteristic
9C23×3−1+2C2211yescharacteristic
10D4×3−1+2—22no—
11C2×C4×3−1+2—22no—
12C2×C4×3−1+2C2212yes—
13C2×C4×3−1+2C2211yescharacteristic
14C2×C4×3−1+2C2212yes—
15C2×C32×D4C2×C311yescharacteristic
16C22×C32×C4C2×C311yescharacteristic
17C32×C4⋊C4C2×C312yes—
18C32×C22⋊C4C2×C311yescharacteristic
19C32×C22⋊C4C2×C312yes—
20C2×D4×C9C2×C313yes—
21C22×C4×C9C2×C313yescomplemented
22C9×C4⋊C4C2×C316yescomplemented
23C9×C22⋊C4C2×C313yescomplemented
24C9×C22⋊C4C2×C316yescomplemented
25C22×3−1+2—22no—
26C22×3−1+2—21no—
27C22×3−1+2—22no—
28C22×3−1+2D412yes—
29C22×3−1+2C2311yescharacteristic
30C4×3−1+2—22no—
31C4×3−1+2—21no—
32C4×3−1+2—22no—
33C3×#123—31no—
34C3×#123C3211yescharacteristic
35C23×C32C22×C311yescharacteristic
36C23×C32C22×C311yescharacteristic
37C32×D4—22no—
38C2×C32×C4—22no—
39C2×C32×C4C22×C312yes—
40C2×C32×C4C22×C312yes—
41C2×C32×C4C22×C311yescharacteristic
42C23×C9C22×C313yes—
43C23×C9C22×C313yes—
44D4×C9—26no—
45C2×C4×C9—26no—
46C2×C4×C9C22×C316yes—
47C2×C4×C9C22×C316yes—
48C2×C4×C9C22×C313yes—
49C2×3−1+2—22no—
50C2×3−1+2—41no—
51C2×3−1+2C2×D411yescharacteristic
52C2×3−1+2#4112yes—
53C2×C3×D4—31no—
54C2×C3×D4C2×C3211yescharacteristic
55C22×C3×C4—31no—
56C22×C3×C4C2×C3211yescharacteristic
57C3×C4⋊C4—32no—
58C3×C4⋊C4C2×C3212yes—
59C3×C22⋊C4—31no—
60C3×C22⋊C4—32no—
61C3×C22⋊C4C2×C3211yescharacteristic
62C3×C22⋊C4C2×C3212yes—
63C22×C32—22no—
64C22×C32—21no—
65C22×C32—22no—
66C22×C32C3×D412yes—
67C22×C32C23×C311yescharacteristic
68C32×C4—22no—
69C32×C4—21no—
70C32×C4—22no—
71C22×C9—26no—
72C22×C9—23no—
73C22×C9—26no—
74C22×C9C3×D416yes—
75C22×C9C23×C313yes—
76C4×C9—26no—
77C4×C9—23no—
78C4×C9—26no—
79#1233−1+211yescharacteristic, direct product factor, Sylow
803−1+2#12311yescharacteristic, direct product factor, Sylow
81C23×C3—31no—
82C23×C3—31no—
83C23×C3C22×C3211yescharacteristic
84C23×C3C22×C3211yescharacteristic
85C3×D4—22no—
86C3×D4—62no—
87C2×C3×C4—22no—
88C2×C3×C4—32no—
89C2×C3×C4—32no—
90C2×C3×C4—31no—
91C2×C3×C4—62no—
92C2×C3×C4C22×C3212yes—
93C2×C3×C4C22×C3212yes—
94C2×C3×C4C22×C3211yescharacteristic
95C2×C32—22no—
96C2×C32—41no—
97C2×C32C2×C3×D411yescharacteristic
98C2×C32C3×#4112yes—
99C2×C9—26no—
100C2×C9—43no—
101C2×C9C2×C3×D413yes—
102C2×C9C3×#4116yes—
103C2×D4C2×3−1+211yescharacteristic
104C22×C4C2×3−1+211yescharacteristic, complemented
105C4⋊C4C2×3−1+212yescomplemented
106C22⋊C4C2×3−1+211yescharacteristic, complemented
107C22⋊C4C2×3−1+212yescomplemented
108C22×C3—22no—
109C22×C3—21no—
110C22×C3—22no—
111C22×C3—32no—
112C22×C3—31no—
113C22×C3—62no—
114C22×C3—61no—
115C22×C3—62no—
116C22×C3C32×D412yes—
117C22×C3C23×C3211yescharacteristic
118C3×C4—22no—
119C3×C4—21no—
120C3×C4—22no—
121C3×C4—62no—
122C3×C4—61no—
123C3×C4—62no—
124C32C3×#12311yescharacteristic
125C9C3×#12313yescomplemented
126C23C22×3−1+211yescharacteristic
127C23C22×3−1+211yescharacteristic
128D4—22no—
129C2×C4—22no—
130C2×C4C22×3−1+212yes—
131C2×C4C22×3−1+212yes—
132C2×C4C22×3−1+211yescharacteristic
133C2×C3—22no—
134C2×C3—31no—
135C2×C3—32no—
136C2×C3—41no—
137C2×C3—62no—
138C2×C3—121no—
139C2×C3C2×C32×D411yescharacteristic
140C2×C3C32×#4112yes—
141C22—22no—
142C22—21no—
143C22—22no—
144C22D4×3−1+212yes—
145C22C23×3−1+211yescharacteristic
146C4—22no—
147C4—21no—
148C4—22no—
149C3—31no—
150C3C32×#12311yescharacteristic
151C2—22no—
152C2—41no—
153C2C2×D4×3−1+211yescharacteristic
154C2#41×3−1+212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22×C3C23×C32
prime radicalC2×C32×D4C2×C3
LCSG ▷ C22×C3 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C4×3−1+2C2semidirect
3. C4⋊C4×3−1+2C2semidirect
4. C22⋊C4×3−1+2C2semidirect
5. C22⋊C4×3−1+2C2semidirect
7. C9×#123C3semidirect
21. C22×C4×C9C2×C3semidirect
22. C9×C4⋊C4C2×C3semidirect
23. C9×C22⋊C4C2×C3semidirect
24. C9×C22⋊C4C2×C3semidirect
79. #1233−1+2direct (N × H)
80. 3−1+2#123direct (N × H)
104. C22×C4C2×3−1+2semidirect
105. C4⋊C4C2×3−1+2semidirect
106. C22⋊C4C2×3−1+2semidirect
107. C22⋊C4C2×3−1+2semidirect
125. C9C3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×3−1+22
C22×C4×3−1+22
C4⋊C4×3−1+22
C22⋊C4×3−1+22
C32×#1233
C9×#1233