Finite Group Databasebeta

← 3−1+2×#328 #13135987

Subgroups — 3−1+2×#328 #13135987

Order 1728.

320 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 119 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×C22⋊Q8C211yesmaximal, characteristic
23−1+2×C22⋊C8C211yesmaximal, characteristic
33−1+2×#95C211yesmaximal, characteristic
4C32×#328C311yesmaximal, characteristic
5C9×#328C313yesmaximal, complemented
6C2×Q8×3−1+2C411yescharacteristic, complemented
7C22×C4×3−1+2—21no—
8C22×C4×3−1+2C2211yescharacteristic
9C2×C8×3−1+2—21no—
10C4⋊C4×3−1+2—21no—
11C4⋊C4×3−1+2C411yescharacteristic, complemented
12C22⋊C4×3−1+2—21no—
13C32×C22⋊Q8C2×C311yescharacteristic
14C32×C22⋊C8C2×C311yescharacteristic
15C32×#95C2×C311yescharacteristic
16C9×C22⋊Q8C2×C313yes—
17C9×C22⋊C8C2×C313yes—
18C9×#95C2×C313yes—
19C23×3−1+2D411yescharacteristic
20Q8×3−1+2—41no—
21C2×C4×3−1+2—22no—
22C2×C4×3−1+2—21no—
23C2×C4×3−1+2—21no—
24C2×C4×3−1+2—41no—
25C2×C4×3−1+2—41no—
26C2×C4×3−1+2—41no—
27C2×C4×3−1+2C2×C411yescharacteristic
28C2×C4×3−1+2D411yescharacteristic
29C8×3−1+2—41no—
30C3×#328—31no—
31C3×#328C3211yescharacteristic
32C2×C32×Q8C3×C411yescharacteristic
33C22×C32×C4—21no—
34C22×C32×C4C22×C311yescharacteristic
35C2×C32×C8—21no—
36C32×C4⋊C4—21no—
37C32×C4⋊C4C3×C411yescharacteristic
38C32×C22⋊C4—21no—
39C2×Q8×C9C3×C413yescomplemented
40C22×C4×C9—23no—
41C22×C4×C9C22×C313yes—
42C2×C8×C9—23no—
43C9×C4⋊C4—23no—
44C9×C4⋊C4C3×C413yescomplemented
45C9×C22⋊C4—23no—
46C22×3−1+2—21no—
47C22×3−1+2—21no—
48C22×3−1+2C22⋊C411yescharacteristic
49C22×3−1+2SD1611yescharacteristic
50C22×3−1+2Q1611yescharacteristic
51C4×3−1+2—21no—
52C4×3−1+2—21no—
53C4×3−1+2—42no—
54C4×3−1+2—41no—
55C4×3−1+2—41no—
56C3×C22⋊Q8—31no—
57C3×C22⋊Q8C2×C3211yescharacteristic
58C3×C22⋊C8—31no—
59C3×C22⋊C8C2×C3211yescharacteristic
60C3×#95—31no—
61C3×#95C2×C3211yescharacteristic
62C23×C32C3×D411yescharacteristic
63C32×Q8—41no—
64C2×C32×C4—22no—
65C2×C32×C4—21no—
66C2×C32×C4—21no—
67C2×C32×C4—41no—
68C2×C32×C4—41no—
69C2×C32×C4—41no—
70C2×C32×C4C2×C3×C411yescharacteristic
71C2×C32×C4C3×D411yescharacteristic
72C23×C9C3×D413yes—
73C32×C8—41no—
74Q8×C9—43no—
75C2×C4×C9—26no—
76C2×C4×C9—23no—
77C2×C4×C9—23no—
78C2×C4×C9—43no—
79C2×C4×C9—43no—
80C2×C4×C9—43no—
81C2×C4×C9C2×C3×C413yes—
82C2×C4×C9C3×D413yes—
83C8×C9—43no—
84#3283−1+211yescharacteristic, direct product factor, Sylow
85C2×3−1+2—21no—
86C2×3−1+2—21no—
87C2×3−1+2#9911yescharacteristic
88C2×3−1+2#10311yescharacteristic
89C2×3−1+2#10411yescharacteristic
90C2×C3×Q8—31no—
91C2×C3×Q8C32×C411yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—31no—
94C22×C3×C4—61no—
95C22×C3×C4C22×C3211yescharacteristic
96C2×C3×C8—21no—
97C2×C3×C8—61no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—61no—
101C3×C4⋊C4C32×C411yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—61no—
104C22×C32—21no—
105C22×C32—21no—
106C22×C32C3×C22⋊C411yescharacteristic
107C22×C32C3×SD1611yescharacteristic
108C22×C32C3×Q1611yescharacteristic
109C32×C4—21no—
110C32×C4—21no—
111C32×C4—42no—
112C32×C4—41no—
113C32×C4—41no—
114C22×C9—23no—
115C22×C9—23no—
116C22×C9C3×C22⋊C413yes—
117C22×C9C3×SD1613yes—
118C22×C9C3×Q1613yes—
119C4×C9—23no—
120C4×C9—23no—
121C4×C9—46no—
122C4×C9—43no—
123C4×C9—43no—
124C22⋊Q8C2×3−1+211yescharacteristic
125C22⋊C8C2×3−1+211yescharacteristic
126#95C2×3−1+211yescharacteristic
1273−1+2#32811yescharacteristic, direct product factor, Sylow
128C23×C3—31no—
129C23×C3C32×D411yescharacteristic
130C3×Q8—41no—
131C3×Q8—121no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—62no—
143C2×C3×C4—121no—
144C2×C3×C4—121no—
145C2×C3×C4—121no—
146C2×C3×C4C2×C32×C411yescharacteristic
147C2×C3×C4C32×D411yescharacteristic
148C3×C8—41no—
149C3×C8—121no—
150C2×C32—21no—
151C2×C32—21no—
152C2×C32C3×#9911yescharacteristic
153C2×C32C3×#10311yescharacteristic
154C2×C32C3×#10411yescharacteristic
155C2×C9—23no—
156C2×C9—23no—
157C2×C9C3×#9913yes—
158C2×C9C3×#10313yes—
159C2×C9C3×#10413yes—
160C2×Q8C4×3−1+211yescharacteristic, complemented
161C22×C4—21no—
162C22×C4C22×3−1+211yescharacteristic
163C2×C8—21no—
164C4⋊C4—21no—
165C4⋊C4C4×3−1+211yescharacteristic, complemented
166C22⋊C4—21no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—31no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—61no—
173C22×C3—61no—
174C22×C3C32×C22⋊C411yescharacteristic
175C22×C3C32×SD1611yescharacteristic
176C22×C3C32×Q1611yescharacteristic
177C3×C4—21no—
178C3×C4—21no—
179C3×C4—42no—
180C3×C4—41no—
181C3×C4—41no—
182C3×C4—61no—
183C3×C4—61no—
184C3×C4—121no—
185C3×C4—121no—
186C3×C4—122no—
187C32C3×#32811yescharacteristic
188C9C3×#32813yescomplemented
189C23D4×3−1+211yescharacteristic
190Q8—41no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4C2×C4×3−1+211yescharacteristic
198C2×C4D4×3−1+211yescharacteristic
199C8—41no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—61no—
206C2×C3—61no—
207C2×C3C32×#9911yescharacteristic
208C2×C3C32×#10311yescharacteristic
209C2×C3C32×#10411yescharacteristic
210C22—21no—
211C22—21no—
212C22C22⋊C4×3−1+211yescharacteristic
213C22SD16×3−1+211yescharacteristic
214C22Q16×3−1+211yescharacteristic
215C4—21no—
216C4—21no—
217C4—42no—
218C4—41no—
219C4—41no—
220C3—31no—
221C3C32×#32811yescharacteristic
222C2—21no—
223C2—21no—
224C23−1+2×#9911yescharacteristic
225C23−1+2×#10311yescharacteristic
226C23−1+2×#10411yescharacteristic
subgroupG/H
center, socleC22×C3C32×C22⋊C4
commutatorC2×C3×C4C2×C32×C4
Frattini, cosocleC22×C3×C4C22×C32
prime radicalC23×C32C3×D4
LCSG ▷ C2×C3×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C4×3−1+2 ◁ G

Complements

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

NH ≅ G/Ntype
5. C9×#328C3semidirect
6. C2×Q8×3−1+2C4semidirect
11. C4⋊C4×3−1+2C4semidirect
39. C2×Q8×C9C3×C4semidirect
44. C9×C4⋊C4C3×C4semidirect
84. #3283−1+2direct (N × H)
127. 3−1+2#328direct (N × H)
160. C2×Q8C4×3−1+2semidirect
165. C4⋊C4C4×3−1+2semidirect
188. C9C3×#328semidirect

Maximal subgroups

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

subgroupindex
3−1+2×C22⋊Q82
3−1+2×C22⋊C82
3−1+2×#952
C32×#3283
C9×#3283