Finite Group Databasebeta

← #10589519⋊C3 #13146893

Subgroups — #10589519⋊C3 #13146893

Order 1728.

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

normal subgroups only

This group has 302 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, complemented
2C4⋊(((C22⋊C9)⋊C2)⋊C3)C211yesmaximal, characteristic
3#1094⋊C8C211yesmaximal, characteristic, complemented
4C3×#5227—31nomaximal
5#10589519C311yesmaximal, characteristic, complemented
6C2×((C22⋊C9)⋊C2)⋊C3—21no—
7C4×#1094—21no—
8C4×#1094C2211yescharacteristic
9#3675⋊C3—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C2×C3×#215—31no—
12C2×C3×#215—31no—
13C3×#930—31no—
14Q8×C22⋊C9—21no—
15Q8×C22⋊C9C2×C311yescharacteristic, complemented
16C4⋊((C22⋊C9)⋊C2)C2×C311yescharacteristic
17C3×#835—31no—
18C3×#835—31no—
19C3×#830—31no—
20(C22⋊C9)⋊C8C2×C311yescharacteristic, complemented
21C2×#1094D411yescharacteristic
22((C22⋊C9)⋊C2)⋊C3—41no—
23Q8×3−1+2—41no—
24C4⋊AGL(1,ℤ/9)—41no—
253−1+2⋊C8—41no—
26C3×#448—91no—
27#5227—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C22×C32×C4—21no—
32C22×C32×C4C2×D311yescharacteristic
33C2×C3×#67—61no—
34C2×C3×D12—31no—
35C2×(C22⋊C9)⋊C2—21no—
36C3×#215—62no—
37C3×#215—61no—
38C3×#215—61no—
39C3×(C2×D3)⋊C4—61no—
40C3×C4⋊Dic3—31no—
41C2×C3×C3⋊C8—31no—
42C2×C3×C3⋊C8—31no—
43C4×C22⋊C9—22no—
44C4×C22⋊C9—21no—
45C4×C22⋊C9—21no—
46C4×C22⋊C9C22×C311yescharacteristic
47#3675—41no—
48C2×AGL(1,ℤ/9)—81no—
49#1094SD1611yescharacteristic, complemented
50C4×3−1+2—41no—
51C4×3−1+2—81no—
52C22×C3×Q8—31no—
53C22×C3×Q8C3×D311yescharacteristic
54C2×C3×SD16—91no—
55C2×C3×SD16—91no—
56C3×#121—91no—
57C2×#215—31no—
58C2×#215—31no—
59#930—31no—
60C3×#103—91no—
61C3×#103—91no—
62C3×C22⋊C8—91no—
63#835—31no—
64#835—31no—
65#830—31no—
66C23×C32#6711yescharacteristic
67C22×C3×D3—61no—
68C32×Q8—32no—
69C32×Q8—31no—
70C32×Q8—61no—
71C32×Q8S411yescharacteristic
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C2×C32×C4—61no—
75C2×C32×C4—61no—
76C3×#67—121no—
77C3×#67—121no—
78C2×C3×Dic3—61no—
79C3×D12—61no—
80C3×D12—61no—
81C2×C22⋊C9—21no—
82C2×C22⋊C9C3×D411yescharacteristic
83(C22⋊C9)⋊C2—41no—
84C3×C3⋊C8—61no—
85C3×C3⋊C8—61no—
86Q8×C9—41no—
87Q8×C9—81no—
88D36—41no—
89C9⋊C8—41no—
90#448—91noSylow
91C2×3−1+2—41no—
92AGL(1,ℤ/9)—161no—
93C2×C3×Q8—31no—
94C2×C3×Q8—31no—
95C2×C3×Q8—31no—
96C2×C3×Q8—31no—
97C2×C3×Q8—61no—
98C2×C3×Q8—61no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—62no—
102C2×C3×D4—91no—
103C2×C3×D4—181no—
104C22×C3×C4—21no—
105C22×C3×C4—31no—
106C22×C3×C4—61no—
107C22×C3×C4C2×C3×D311yescharacteristic
108C2×#67—61no—
109C2×D12—31no—
110C3×SD16—182no—
111C3×SD16—181no—
112C3×SD16—181no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C3×C4⋊C4—91no—
116C3×C22⋊C4—181no—
117#215—62no—
118#215—61no—
119#215—61no—
120(C2×D3)⋊C4—61no—
121C4⋊Dic3—31no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—31no—
124C22×C32—31no—
125C22×C32—31no—
126C22×C32#21511yescharacteristic
127C2×C3×D3—121no—
128C2×C3×D3—121no—
129C2×C3×D3—121no—
130C32×C4—21no—
131C32×C4—31no—
132C32×C4—61no—
133C32×C4C2×S411yescharacteristic
134C3×Dic3—121no—
135C2×D9—81no—
136C22⋊C9—21no—
137C22⋊C9C3×SD1611yescharacteristic, complemented
138C4×C9—41no—
139C4×C9—81no—
140C4×C9—82no—
141C4×C9—81no—
142C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
143C2×SD16—91no—
144C2×SD16—91no—
145#121—91no—
146#103—91no—
147#103—91no—
148C22⋊C8—91no—
1493−1+2—41noSylow
150C23×C3—31no—
151C23×C3—181no—
152C23×C3C3×#6711yescharacteristic
153C22×D3—61no—
154C3×Q8—32no—
155C3×Q8—32no—
156C3×Q8—31no—
157C3×Q8—31no—
158C3×Q8—31no—
159C3×Q8—61no—
160C3×Q8—62no—
161C3×Q8—61no—
162C3×Q8—61no—
163C3×Q8—62no—
164C3×Q8C3×S411yescharacteristic
165C3×D4—181no—
166C3×D4—181no—
167C3×D4—361no—
168C3×D4—361no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—181no—
182#67—121no—
183#67—121no—
184C2×Dic3—61no—
185D12—61no—
186D12—61no—
187C3×C8—181no—
188C3×C8—181no—
189C3⋊C8—61no—
190C3⋊C8—61no—
191C2×C32—31no—
192C2×C32—31no—
193C2×C32GL(2,ℤ/4)11yescharacteristic
194C3×D3—241no—
195C2×C9—41no—
196C2×C9—81no—
197D9—161no—
198C2×Q8—31no—
199C2×Q8—31no—
200C2×Q8—61no—
201C2×D4—91no—
202C2×D4—181no—
203C22×C4—21no—
204C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
205SD16—182no—
206SD16—181no—
207SD16—181no—
208C2×C8—91no—
209C2×C8—91no—
210C4⋊C4—91no—
211C22⋊C4—181no—
212C22×C3—31no—
213C22×C3—31no—
214C22×C3—31no—
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—361no—
220C22×C3—361no—
221C22×C3—361no—
222C22×C3C3×#21511yescharacteristic
223C2×D3—121no—
224C2×D3—121no—
225C2×D3—121no—
226C3×C4—21no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—62no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—361no—
236C3×C4C2×C3×S411yescharacteristic
237Dic3—121no—
238C32#541711yescharacteristic
239C9—41no—
240C9—81no—
241C23—181no—
242C23#611⋊C311yescharacteristic
243Q8—32no—
244Q8—31no—
245Q8—61no—
246Q8((C22⋊C9)⋊C2)⋊C311yescharacteristic, complemented
247D4—181no—
248D4—181no—
249D4—361no—
250D4—361no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—181no—
256C8—181no—
257C8—181no—
258C2×C3—31no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—721no—
266C2×C3C3×GL(2,ℤ/4)11yescharacteristic
267D3—241no—
268C22—31no—
269C22—31no—
270C22—361no—
271C22—361no—
272C22—361no—
273C22#3675⋊C311yescharacteristic, complemented
274C4—21no—
275C4—31no—
276C4—61no—
277C4—361no—
278C4C2×((C22⋊C9)⋊C2)⋊C311yescharacteristic
279C3—31no—
280C3C3×#541711yescharacteristic
281C2—31no—
282C2—31no—
283C2—721no—
284C2#63694⋊C311yescharacteristic
subgroupG/H
centerC2#63694⋊C3
commutator, cosocleC4×C22⋊C9C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×#67
prime radicalC4⋊(((C22⋊C9)⋊C2)⋊C3)C2
hypercenterQ8((C22⋊C9)⋊C2)⋊C3
hypocenterC22⋊C9C3×SD16
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
1. Q8×#1094C2semidirect
3. #1094⋊C8C2semidirect
5. #10589519C3semidirect
15. Q8×C22⋊C9C2×C3semidirect
20. (C22⋊C9)⋊C8C2×C3semidirect
49. #1094SD16semidirect
137. C22⋊C9C3×SD16semidirect
142. C22×Q8AGL(1,ℤ/9)semidirect
246. Q8((C22⋊C9)⋊C2)⋊C3semidirect
273. C22#3675⋊C3semidirect

Maximal subgroups

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

subgroupindex
Q8×#10942
C4⋊(((C22⋊C9)⋊C2)⋊C3)2
#1094⋊C82
C3×#52273
#105895193
#3675⋊C34