Finite Group Databasebeta

← #10589521⋊C3 #13146897

Subgroups — #10589521⋊C3 #13146897

Order 1728.

310 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 45 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×#6204C211yesmaximal, characteristic
2C2×#3688⋊C3C211yesmaximal, characteristic, complemented
3#63421⋊C3C211yesmaximal, characteristic, complemented
4C3×#5228—31nomaximal
5#10589521C311yesmaximal, characteristic, complemented
6C2×#6204—21no—
7C2×#6204C2211yescharacteristic
8#3688⋊C3—21no—
9#3676⋊C3—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×C3×#216—31no—
12C2×C3×#216—31no—
13C3×#924—31no—
14C22×Q8⋊C9—21no—
15C22×Q8⋊C9C2×C311yescharacteristic
16C2×#3688C2×C311yescharacteristic, complemented
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20#63421C2×C311yescharacteristic, complemented
21C23×3−1+2—41no—
22C2×3−1+2⋊C4—41no—
23C2×3−1+2⋊C4—41no—
24#6204D411yescharacteristic
25C3×#451—91no—
26#5228—31no—
27C2×C32×Q8—21no—
28C2×C32×Q8—31no—
29C2×C32×Q8—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C22×C32×C4—31no—
32C2×C3×Dic6—31no—
33C3×#217—61no—
34C3×#216—61no—
35C3×#216—62no—
36C3×#216—61no—
37C3×C4⋊Dic3—31no—
38C3×Dic3⋊C4—61no—
39C2×C3×C3⋊C8—31no—
40C2×C3×C3⋊C8—31no—
41C2×Q8⋊C9—22no—
42C2×Q8⋊C9—21no—
43C2×Q8⋊C9—21no—
44C2×Q8⋊C9C22×C311yescharacteristic
45#3688—21no—
46#3676—41no—
47C22×3−1+2—41no—
48C22×3−1+2—42no—
49C22×3−1+2—81no—
50C22×3−1+2—81no—
513−1+2⋊C4—81no—
523−1+2⋊C4—81no—
53C22×C3×Q8—31no—
54C22×C3×Q8C3×D311yescharacteristic
55C2×C3×Q16—91no—
56C2×C3×Q16—91no—
57C3×C22⋊Q8—91no—
58C2×#216—31no—
59C2×#216—31no—
60#924—31no—
61C3×#103—91no—
62C3×#103—91no—
63C3×C22⋊C8—91no—
64#835—31no—
65#835—31no—
66#830—31no—
67C23×C32S411yescharacteristic
68C32×Q8—31no—
69C32×Q8—32no—
70C32×Q8—61no—
71C32×Q8#6711yescharacteristic
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C2×C32×C4—61no—
75C2×C32×C4—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C3×Dic6—61no—
79C3×Dic6—61no—
80C23×C9—41no—
81C23×C9—81no—
82C3×C3⋊C8—61no—
83C3×C3⋊C8—61no—
84C2×Dic9—41no—
85C2×Dic9—41no—
86Q8⋊C9—21no—
87Q8⋊C9C3×D411yescharacteristic
88#451—91noSylow
89C2×3−1+2—41no—
90C2×3−1+2—41no—
91C2×3−1+2—41no—
92C2×3−1+2—82no—
93C2×C3×Q8—21no—
94C2×C3×Q8—31no—
95C2×C3×Q8—31no—
96C2×C3×Q8—31no—
97C2×C3×Q8—61no—
98C2×C3×Q8—61no—
99C2×C3×Q8—62no—
100C2×C3×Q8—61no—
101C2×C3×Q8—61no—
102C2×C3×Q8—91no—
103C2×C3×Q8C2×C3×D311yescharacteristic
104C22×C3×C4—31no—
105C22×C3×C4—31no—
106C22×C3×C4—61no—
107C2×Dic6—31no—
108C3×Q16—181no—
109C3×Q16—182no—
110C3×Q16—181no—
111C2×C3×C8—91no—
112C2×C3×C8—91no—
113C3×C4⋊C4—91no—
114C3×C4⋊C4—181no—
115C3×C22⋊C4—181no—
116#217—61no—
117#216—61no—
118#216—62no—
119#216—61no—
120C4⋊Dic3—31no—
121Dic3⋊C4—61no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32#22612yes—
127C22×C32C2×S411yescharacteristic
128C32×C4—31no—
129C32×C4—31no—
130C32×C4—61no—
131C3×Dic3—121no—
132C3×Dic3—121no—
133C22×C9—41no—
134C22×C9—42no—
135C22×C9—81no—
136C22×C9—81no—
137C22×C9—82no—
138C22×C9—82no—
139C22×C9—81no—
140C22×C9—82no—
141Dic9—81no—
142Dic9—81no—
143C22×Q8AGL(1,ℤ/9)11yescharacteristic
144C2×Q16—91no—
145C2×Q16—91no—
146C22⋊Q8—91no—
147#103—91no—
148#103—91no—
149C22⋊C8—91no—
1503−1+2—41noSylow
151C23×C3—31no—
152C23×C3C3×S411yescharacteristic
153C3×Q8—31no—
154C3×Q8—32no—
155C3×Q8—31no—
156C3×Q8—32no—
157C3×Q8—31no—
158C3×Q8—61no—
159C3×Q8—62no—
160C3×Q8—61no—
161C3×Q8—61no—
162C3×Q8—62no—
163C3×Q8—181no—
164C3×Q8—181no—
165C3×Q8C3×#6711yescharacteristic
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182Dic6—61no—
183Dic6—61no—
184C3×C8—181no—
185C3×C8—181no—
186C3⋊C8—61no—
187C3⋊C8—61no—
188C2×C32—22no—
189C2×C32C2×#22611yescharacteristic
190C2×C32#98311yescharacteristic
191C2×C32GL(2,ℤ/4)11yescharacteristic
192C2×C9—41no—
193C2×C9—41no—
194C2×C9—41no—
195C2×C9—82no—
196C2×C9—84no—
197C2×C9—81no—
198C2×C9—81no—
199C2×C9—81no—
200C2×Q8—21no—
201C2×Q8—31no—
202C2×Q8—61no—
203C2×Q8—91no—
204C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
205C22×C4—31no—
206Q16—181no—
207Q16—182no—
208Q16—181no—
209C2×C8—91no—
210C2×C8—91no—
211C4⋊C4—91no—
212C4⋊C4—181no—
213C22⋊C4—181no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—32no—
217C22×C3—31no—
218C22×C3—61no—
219C22×C3—61no—
220C22×C3C3×#22612yes—
221C22×C3C2×C3×S411yescharacteristic
222C3×C4—31no—
223C3×C4—31no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—61no—
227C3×C4—61no—
228C3×C4—61no—
229C3×C4—62no—
230C3×C4—61no—
231C3×C4—361no—
232C3×C4—361no—
233Dic3—121no—
234Dic3—121no—
235C32#541911yescharacteristic
236C9—41no—
237C9—81no—
238C23((C22⋊C9)⋊C2)⋊C311yescharacteristic
239Q8—31no—
240Q8—32no—
241Q8—61no—
242Q8—181no—
243Q8—181no—
244Q8#611⋊C311yescharacteristic
245C2×C4—31no—
246C2×C4—31no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—181no—
250C2×C4—181no—
251C8—181no—
252C8—181no—
253C2×C3—22no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—31no—
257C2×C3—62no—
258C2×C3C2×C3×#22611yescharacteristic
259C2×C3C3×#98311yescharacteristic
260C2×C3C3×GL(2,ℤ/4)11yescharacteristic
261C22—21no—
262C22—21no—
263C22#3688⋊C312yescomplemented
264C22C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
265C4—31no—
266C4—31no—
267C4—61no—
268C4—361no—
269C4—361no—
270C3—31no—
271C3C3×#541911yescharacteristic
272C2—22no—
273C2C2×#3688⋊C311yescharacteristic
274C2#63689⋊C311yescharacteristic
275C2#63694⋊C311yescharacteristic
subgroupG/H
centerC22C2×((C22⋊C9)⋊C2)⋊C3
commutator, cosocleC2×Q8⋊C9C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC22×C32×Q8D3
prime radicalC23×C32S4
hypercenterC23((C22⋊C9)⋊C2)⋊C3
hypocenterQ8⋊C9C3×D4
LCSG ▷ C2×Q8⋊C9 ▷ Q8⋊C9
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#3688⋊C3C2semidirect
3. #63421⋊C3C2semidirect
5. #10589521C3semidirect
16. C2×#3688C2×C3semidirect
20. #63421C2×C3semidirect
263. C22#3688⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#62042
C2×#3688⋊C32
#63421⋊C32
C3×#52283
#105895213
#3676⋊C34