Finite Group Databasebeta

← 3−1+2×#488 #13145224

Subgroups — 3−1+2×#488 #13145224

Order 1728.

384 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 175 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
13−1+2×#125C211yesmaximal, characteristic, complemented
23−1+2×#124C211yesmaximal, characteristic
33−1+2×#103C212yesmaximal, complemented
43−1+2×#102C212yesmaximal
5C4×C8×3−1+2C211yesmaximal, characteristic, complemented
6C32×#488C311yesmaximal, characteristic
7C9×#488C313yesmaximal, complemented
8C2×Q8×3−1+2C2211yescharacteristic
9C2×D4×3−1+2C2211yescharacteristic
10C2×C8×3−1+2C2212yes—
11C4⋊C4×3−1+2—22no—
12C4⋊C4×3−1+2C2212yes—
13C22⋊C4×3−1+2—22no—
14C42×3−1+2C2211yescharacteristic
15C32×#125C2×C311yescharacteristic
16C32×#124C2×C311yescharacteristic
17C32×#103C2×C312yes—
18C32×#102C2×C312yes—
19C32×C4×C8C2×C311yescharacteristic
20C9×#125C2×C313yescomplemented
21C9×#124C2×C313yes—
22C9×#103C2×C316yescomplemented
23C9×#102C2×C316yes—
24C4×C8×C9C2×C313yescomplemented
25C23×3−1+2—21no—
26Q8×3−1+2—22no—
27D4×3−1+2—22no—
28C2×C4×3−1+2—21no—
29C2×C4×3−1+2—22no—
30C2×C4×3−1+2D412yes—
31C2×C4×3−1+2C2311yescharacteristic
32C8×3−1+2—22no—
33C3×#488—31no—
34C3×#488C3211yescharacteristic
35C2×C32×Q8C22×C311yescharacteristic
36C2×C32×D4C22×C311yescharacteristic
37C2×C32×C8C22×C312yes—
38C32×C4⋊C4—22no—
39C32×C4⋊C4C22×C312yes—
40C32×C22⋊C4—22no—
41C32×C42C22×C311yescharacteristic
42C2×Q8×C9C22×C313yes—
43C2×D4×C9C22×C313yes—
44C2×C8×C9C22×C316yes—
45C9×C4⋊C4—26no—
46C9×C4⋊C4C22×C316yes—
47C9×C22⋊C4—26no—
48C42×C9C22×C313yes—
49C22×3−1+2—41no—
50C22×3−1+2—42no—
51C22×3−1+2C2×D411yescharacteristic
52C4×3−1+2—22no—
53C4×3−1+2—41no—
54C4×3−1+2—42no—
55C4×3−1+2#4112yes—
56C3×#125—31no—
57C3×#125C2×C3211yescharacteristic
58C3×#124—31no—
59C3×#124C2×C3211yescharacteristic
60C3×#103—32no—
61C3×#103C2×C3212yes—
62C3×#102—32no—
63C3×#102C2×C3212yes—
64C3×C4×C8—31no—
65C3×C4×C8C2×C3211yescharacteristic
66C23×C32—21no—
67C32×Q8—22no—
68C32×D4—22no—
69C2×C32×C4—21no—
70C2×C32×C4—22no—
71C2×C32×C4C3×D412yes—
72C2×C32×C4C23×C311yescharacteristic
73C23×C9—23no—
74C32×C8—22no—
75Q8×C9—26no—
76D4×C9—26no—
77C2×C4×C9—23no—
78C2×C4×C9—26no—
79C2×C4×C9C3×D416yes—
80C2×C4×C9C23×C313yes—
81C8×C9—26no—
82#4883−1+211yescharacteristic, direct product factor, Sylow
83C2×3−1+2—81no—
84C2×3−1+2#12411yescharacteristic
85C2×3−1+2#13512yes—
86C2×C3×Q8—31no—
87C2×C3×Q8C22×C3211yescharacteristic
88C2×C3×D4—31no—
89C2×C3×D4C22×C3211yescharacteristic
90C2×C3×C8—32no—
91C2×C3×C8C22×C3212yes—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—32no—
94C3×C4⋊C4—62no—
95C3×C4⋊C4C22×C3212yes—
96C3×C22⋊C4—22no—
97C3×C22⋊C4—62no—
98C3×C42—31no—
99C3×C42C22×C3211yescharacteristic
100C22×C32—41no—
101C22×C32—42no—
102C22×C32C2×C3×D411yescharacteristic
103C32×C4—22no—
104C32×C4—41no—
105C32×C4—42no—
106C32×C4C3×#4112yes—
107C22×C9—43no—
108C22×C9—46no—
109C22×C9C2×C3×D413yes—
110C4×C9—26no—
111C4×C9—43no—
112C4×C9—46no—
113C4×C9C3×#4116yes—
114#125C2×3−1+211yescharacteristic, complemented
115#124C2×3−1+211yescharacteristic
116#103C2×3−1+212yescomplemented
117#102C2×3−1+212yes—
118C4×C8C2×3−1+211yescharacteristic, complemented
1193−1+2#48811yescharacteristic, direct product factor, Sylow
120C23×C3—21no—
121C23×C3—61no—
122C3×Q8—22no—
123C3×Q8—62no—
124C3×D4—22no—
125C3×D4—62no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—32no—
129C2×C3×C4—31no—
130C2×C3×C4—61no—
131C2×C3×C4—62no—
132C2×C3×C4C32×D412yes—
133C2×C3×C4C23×C3211yescharacteristic
134C3×C8—22no—
135C3×C8—62no—
136C2×C32—81no—
137C2×C32C3×#12411yescharacteristic
138C2×C32C3×#13512yes—
139C2×C9—83no—
140C2×C9C3×#12413yes—
141C2×C9C3×#13516yes—
142C2×Q8C22×3−1+211yescharacteristic
143C2×D4C22×3−1+211yescharacteristic
144C2×C8C22×3−1+212yes—
145C4⋊C4—22no—
146C4⋊C4C22×3−1+212yes—
147C22⋊C4—22no—
148C42C22×3−1+211yescharacteristic
149C22×C3—31no—
150C22×C3—41no—
151C22×C3—42no—
152C22×C3—121no—
153C22×C3—122no—
154C22×C3C2×C32×D411yescharacteristic
155C3×C4—22no—
156C3×C4—32no—
157C3×C4—41no—
158C3×C4—42no—
159C3×C4—62no—
160C3×C4—121no—
161C3×C4—122no—
162C3×C4C32×#4112yes—
163C32C3×#48811yescharacteristic
164C9C3×#48813yescomplemented
165C23—21no—
166Q8—22no—
167D4—22no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4D4×3−1+212yes—
171C2×C4C23×3−1+211yescharacteristic
172C8—22no—
173C2×C3—31no—
174C2×C3—32no—
175C2×C3—81no—
176C2×C3—241no—
177C2×C3C32×#12411yescharacteristic
178C2×C3C32×#13512yes—
179C22—41no—
180C22—42no—
181C22C2×D4×3−1+211yescharacteristic
182C4—22no—
183C4—41no—
184C4—42no—
185C4#41×3−1+212yes—
186C3—31no—
187C3C32×#48811yescharacteristic
188C2—81no—
189C23−1+2×#12411yescharacteristic
190C23−1+2×#13512yes—
subgroupG/H
center, socleC22×C3C2×C32×D4
commutator, Frattini, cosocleC2×C3×C4C23×C32
prime radicalC2×C32×D4C22×C3
LCSG ▷ C2×C3×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C42×3−1+2 ◁ G

Complements

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

NH ≅ G/Ntype
1. 3−1+2×#125C2semidirect
3. 3−1+2×#103C2semidirect
5. C4×C8×3−1+2C2semidirect
7. C9×#488C3semidirect
20. C9×#125C2×C3semidirect
22. C9×#103C2×C3semidirect
24. C4×C8×C9C2×C3semidirect
82. #4883−1+2direct (N × H)
114. #125C2×3−1+2semidirect
116. #103C2×3−1+2semidirect
118. C4×C8C2×3−1+2semidirect
119. 3−1+2#488direct (N × H)
164. C9C3×#488semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1252
3−1+2×#1242
3−1+2×#1032
3−1+2×#1022
C4×C8×3−1+22
C32×#4883
C9×#4883