Finite Group Databasebeta

← #10589518⋊C3 #13146891

Subgroups — #10589518⋊C3 #13146891

Order 1728.

278 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 38 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
1Q8×#1094C211yesmaximal, characteristic
2#1094⋊Q8C211yesmaximal, characteristic
3#1094⋊C8C211yesmaximal, characteristic
4C3×#5228—31nomaximal
5#10589518C311yesmaximal, characteristic, complemented
6C4×#1094—21no—
7C4×#1094C2211yescharacteristic
8#1094⋊C4—21no—
9#3674⋊C3—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×C3×#216—31no—
12C2×C3×#216—31no—
13C3×#924—31no—
14Q8×C22⋊C9—21no—
15Q8×C22⋊C9C2×C311yescharacteristic
16(C22⋊C9)⋊Q8C2×C311yescharacteristic
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20(C22⋊C9)⋊C8C2×C311yescharacteristic
21C2×#1094D411yescharacteristic
22Q8×3−1+2—41no—
233−1+2⋊Q8—41no—
243−1+2⋊C8—41no—
25C3×#451—91no—
26#5228—31no—
27C2×C32×Q8—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C22×C32×C4—21no—
31C22×C32×C4C2×D311yescharacteristic
32C2×C3×Dic6—31no—
33C3×#217—61no—
34C3×#216—62no—
35C3×#216—61no—
36C3×#216—61no—
37C3×C4⋊Dic3—31no—
38C3×Dic3⋊C4—61no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41C4×C22⋊C9—22no—
42C4×C22⋊C9—21no—
43C4×C22⋊C9—21no—
44C4×C22⋊C9C22×C311yescharacteristic
45(C22⋊C9)⋊C4—21no—
46#3674—41no—
47#1094Q1611yescharacteristic, complemented
48C4×3−1+2—41no—
49C4×3−1+2—81no—
503−1+2⋊C4—81no—
51C22×C3×Q8—31no—
52C22×C3×Q8C3×D311yescharacteristic
53C2×C3×Q16—91no—
54C2×C3×Q16—91no—
55C3×C22⋊Q8—91no—
56C2×#216—31no—
57C2×#216—31no—
58#924—31no—
59C3×#103—91no—
60C3×#103—91no—
61C3×C22⋊C8—91no—
62#835—31no—
63#835—31no—
64#830—31no—
65C23×C32#6711yescharacteristic
66C32×Q8—32no—
67C32×Q8—31no—
68C32×Q8—61no—
69C32×Q8S411yescharacteristic
70C2×C32×C4—31no—
71C2×C32×C4—31no—
72C2×C32×C4—61no—
73C2×C32×C4—61no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C3×Dic6—61no—
77C3×Dic6—61no—
78C2×C22⋊C9—21no—
79C2×C22⋊C9C3×D411yescharacteristic
80C3×C3⋊C8—61no—
81C3×C3⋊C8—61no—
82Q8×C9—41no—
83Q8×C9—81no—
84Dic18—41no—
85C9⋊C8—41no—
86#451—91noSylow
87C2×3−1+2—41no—
88C2×C3×Q8—31no—
89C2×C3×Q8—31no—
90C2×C3×Q8—31no—
91C2×C3×Q8—31no—
92C2×C3×Q8—61no—
93C2×C3×Q8—61no—
94C2×C3×Q8—61no—
95C2×C3×Q8—61no—
96C2×C3×Q8—62no—
97C2×C3×Q8—91no—
98C22×C3×C4—21no—
99C22×C3×C4—31no—
100C22×C3×C4—61no—
101C22×C3×C4C2×C3×D311yescharacteristic
102C2×Dic6—31no—
103C3×Q16—182no—
104C3×Q16—181no—
105C3×Q16—181no—
106C2×C3×C8—91no—
107C2×C3×C8—91no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—181no—
110C3×C22⋊C4—181no—
111#217—61no—
112#216—62no—
113#216—61no—
114#216—61no—
115C4⋊Dic3—31no—
116Dic3⋊C4—61no—
117C2×C3⋊C8—31no—
118C2×C3⋊C8—31no—
119C22×C32—31no—
120C22×C32—31no—
121C22×C32#21611yescharacteristic
122C32×C4—21no—
123C32×C4—31no—
124C32×C4—61no—
125C32×C4C2×S411yescharacteristic
126C3×Dic3—121no—
127C3×Dic3—121no—
128C22⋊C9—21no—
129C22⋊C9C3×Q1611yescharacteristic, complemented
130C4×C9—41no—
131C4×C9—81no—
132C4×C9—82no—
133C4×C9—81no—
134Dic9—81no—
135C22×Q8AGL(1,ℤ/9)11yescharacteristic
136C2×Q16—91no—
137C2×Q16—91no—
138C22⋊Q8—91no—
139#103—91no—
140#103—91no—
141C22⋊C8—91no—
1423−1+2—41noSylow
143C23×C3—31no—
144C23×C3C3×#6711yescharacteristic
145C3×Q8—32no—
146C3×Q8—32no—
147C3×Q8—31no—
148C3×Q8—31no—
149C3×Q8—31no—
150C3×Q8—61no—
151C3×Q8—62no—
152C3×Q8—61no—
153C3×Q8—61no—
154C3×Q8—62no—
155C3×Q8—181no—
156C3×Q8—181no—
157C3×Q8C3×S411yescharacteristic
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—61no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174Dic6—61no—
175Dic6—61no—
176C3×C8—181no—
177C3×C8—181no—
178C3⋊C8—61no—
179C3⋊C8—61no—
180C2×C32—31no—
181C2×C32—31no—
182C2×C32GL(2,ℤ/4)11yescharacteristic
183C2×C9—41no—
184C2×C9—81no—
185C2×Q8—31no—
186C2×Q8—31no—
187C2×Q8—61no—
188C2×Q8—91no—
189C22×C4—21no—
190C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
191Q16—182no—
192Q16—181no—
193Q16—181no—
194C2×C8—91no—
195C2×C8—91no—
196C4⋊C4—91no—
197C4⋊C4—181no—
198C22⋊C4—181no—
199C22×C3—31no—
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—61no—
205C22×C3—61no—
206C22×C3C3×#21611yescharacteristic
207C3×C4—21no—
208C3×C4—31no—
209C3×C4—31no—
210C3×C4—31no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—62no—
214C3×C4—61no—
215C3×C4—61no—
216C3×C4—361no—
217C3×C4—361no—
218C3×C4C2×C3×S411yescharacteristic
219Dic3—121no—
220Dic3—121no—
221C32#541611yescharacteristic
222C9—41no—
223C9—81no—
224C23#611⋊C311yescharacteristic
225Q8—32no—
226Q8—31no—
227Q8—61no—
228Q8—181no—
229Q8—181no—
230Q8((C22⋊C9)⋊C2)⋊C311yescharacteristic
231C2×C4—31no—
232C2×C4—31no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—181no—
236C2×C4—181no—
237C8—181no—
238C8—181no—
239C2×C3—31no—
240C2×C3—31no—
241C2×C3—31no—
242C2×C3—31no—
243C2×C3—31no—
244C2×C3—61no—
245C2×C3—61no—
246C2×C3C3×GL(2,ℤ/4)11yescharacteristic
247C22—31no—
248C22—31no—
249C22#3674⋊C311yescharacteristic, complemented
250C4—21no—
251C4—31no—
252C4—61no—
253C4—361no—
254C4—361no—
255C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
256C3—31no—
257C3C3×#541611yescharacteristic
258C2—31no—
259C2—31no—
260C2#63694⋊C311yescharacteristic
subgroupG/H
centerC2#63694⋊C3
commutator, cosocleC4×C22⋊C9C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×#67
prime radicalC23×C32#67
hypercenterQ8((C22⋊C9)⋊C2)⋊C3
hypocenterC22⋊C9C3×Q16
LCSG ▷ C4×C22⋊C9 ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
5. #10589518C3semidirect
47. #1094Q16semidirect
129. C22⋊C9C3×Q16semidirect
249. C22#3674⋊C3semidirect

Maximal subgroups

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

subgroupindex
Q8×#10942
#1094⋊Q82
#1094⋊C82
C3×#52283
#105895183
#3674⋊C34