Finite Group Databasebeta

← #10588555⋊C3 #13142235

Subgroups — #10588555⋊C3 #13142235

Order 1728.

306 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 94 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
1#63450⋊C3C211yesmaximal, characteristic
2#63449⋊C3C211yesmaximal, characteristic
33−1+2×#107C211yesmaximal, characteristic
4#63376⋊C3C211yesmaximal, characteristic
5(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic
6#63368⋊C3C211yesmaximal, characteristic
7#63366⋊C3C211yesmaximal, characteristic
8C3×#4875—31nomaximal
9#10588555C311yesmaximal, characteristic, complemented
10C2×3−1+2⋊Q8—21no—
11C2×3−1+2⋊Q8C2211yescharacteristic
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16(3−1+2⋊C4)⋊C4—21no—
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#888—31no—
22C3×#887—31no—
23C32×#107D311yescharacteristic
24C3×#816—31no—
25C3×Dic3⋊C8—31no—
26C3×#810—31no—
27C3×#807—31no—
28#63450C2×C311yescharacteristic
29#63449C2×C311yescharacteristic
30C9×#107—21no—
31C9×#107C2×C311yescharacteristic
32#63376C2×C311yescharacteristic
33Dic9⋊C8C2×C311yescharacteristic
34#63368C2×C311yescharacteristic
35#63366C2×C311yescharacteristic
36C2×C4×3−1+2—21no—
37C2×C4×3−1+2—21no—
38C2×C4×3−1+2C2311yescharacteristic
39C2×3−1+2⋊C4—21no—
40C2×3−1+2⋊C4—21no—
41C2×3−1+2⋊C4D411yescharacteristic
42C2×3−1+2⋊C4D411yescharacteristic
433−1+2⋊Q8—21no—
443−1+2⋊Q8—42no—
453−1+2⋊Q8Q812yescomplemented
46C8×3−1+2—21no—
473−1+2⋊C8—21no—
48C3×#477—91no—
49#4875—31no—
50C2×C3×Dic6—31no—
51C2×C3×Dic6—61no—
52C2×C32×C8C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C3×C4⋊Dic3—31no—
56C3×Dic3⋊C4—61no—
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—31no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—31no—
61C2×Dic18—21no—
62C2×Dic18C22×C311yescharacteristic
63C2×C8×C9—21no—
64C2×C8×C9C22×C311yescharacteristic
65C9×C4⋊C4—21no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C9×C4⋊C4C22×C311yescharacteristic
69C4⋊Dic9C22×C311yescharacteristic
70Dic9⋊C4—21no—
71Dic9⋊C4—21no—
72C4×Dic9—21no—
73C4×Dic9C22×C311yescharacteristic
74C2×C9⋊C8C22×C311yescharacteristic
75C22×3−1+2C2×D411yescharacteristic
76C4×3−1+2—41no—
77C4×3−1+2—41no—
78C4×3−1+2C2×Q811yescharacteristic
79C4×3−1+2#4111yescharacteristic
803−1+2⋊C4—22no—
813−1+2⋊C4—21no—
823−1+2⋊C4—41no—
833−1+2⋊C4Q1612yes—
84C3×C4⋊Q8—91no—
85C3×C4×Q8—91no—
86#888—31no—
87#887—31no—
88C3×#107—31no—
89C3×#107—91no—
90C3×#107C3×D311yescharacteristic
91C3×C4⋊C8—91no—
92C3×#103—91no—
93C3×#103—91no—
94#816—31no—
95Dic3⋊C8—31no—
96#810—31no—
97#807—31no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3×Dic3—31no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C3×Dic6—32no—
106C3×Dic6—61no—
107C3×Dic6—122no—
108C32×C8—21no—
109C3×C3⋊C8—61no—
110C2×C4×C9—21no—
111C2×C4×C9—21no—
112C2×C4×C9—21no—
113C2×C4×C9—41no—
114C2×C4×C9—41no—
115C2×C4×C9C23×C311yescharacteristic
116C2×Dic9—21no—
117C2×Dic9—21no—
118C2×Dic9C3×D411yescharacteristic
119C2×Dic9C3×D411yescharacteristic
120Dic18—21no—
121Dic18—42no—
122Dic18C3×Q812yescomplemented
123C8×C9—22no—
124C8×C9—21no—
125C9⋊C8—21no—
126#477—91noSylow
127C2×3−1+2C22⋊Q811yescharacteristic
128C2×3−1+2C2×Q1611yescharacteristic
129C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
130C2×C3×Q8—91no—
131C2×C3×Q8—181no—
132C2×Dic6—31no—
133C2×Dic6—61no—
134C2×C3×C8—31no—
135C2×C3×C8—91no—
136C2×C3×C8C2×C3×D311yescharacteristic
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—181no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4C2×C3×D311yescharacteristic
143C3×C4⋊C4C2×C3×D311yescharacteristic
144C3×C42—91no—
145C3×C42—181no—
146C4⋊Dic3—31no—
147Dic3⋊C4—61no—
148Dic3⋊C4—61no—
149C4×Dic3—31no—
150C4×Dic3—61no—
151C2×C3⋊C8—31no—
152C22×C32D3×D411yescharacteristic
153C32×C4—41no—
154C32×C4—41no—
155C32×C4#23511yescharacteristic
156C32×C4D3×Q811yescharacteristic
157C3×Dic3—32no—
158C3×Dic3—62no—
159C3×Dic3—61no—
160C3×Dic3—121no—
161C22×C9—21no—
162C22×C9C2×C3×D411yescharacteristic
163C4×C9—21no—
164C4×C9—21no—
165C4×C9—41no—
166C4×C9—41no—
167C4×C9—81no—
168C4×C9—81no—
169C4×C9C2×C3×Q811yescharacteristic
170C4×C9C3×#4111yescharacteristic
171Dic9—22no—
172Dic9—21no—
173Dic9—41no—
174Dic9C3×Q1612yes—
175C4⋊Q8—91no—
176C4×Q8—91no—
177#107—91no—
178#107AGL(1,ℤ/9)11yescharacteristic
179C4⋊C8—91no—
180#103—91no—
181#103—91no—
1823−1+2#47711yescharacteristic, complemented, Sylow
183C3×Q8—92no—
184C3×Q8—181no—
185C3×Q8—362no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—91no—
192C2×C3×C4—91no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C22×C3×D311yescharacteristic
196C2×Dic3—31no—
197C2×Dic3—31no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200Dic6—32no—
201Dic6—61no—
202Dic6—122no—
203C3×C8—21no—
204C3×C8—61no—
205C3×C8—181no—
206C3⋊C8—61no—
207C2×C32#89611yescharacteristic
208C2×C32#91111yescharacteristic
209C2×C32D3×Q1611yescharacteristic
210C2×C9—21no—
211C2×C9—21no—
212C2×C9—21no—
213C2×C9C3×C22⋊Q811yescharacteristic
214C2×C9C2×C3×Q1611yescharacteristic
215C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
216C2×Q8—91no—
217C2×Q8—181no—
218C2×C8—91no—
219C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
220C4⋊C4—91no—
221C4⋊C4—181no—
222C4⋊C4—181no—
223C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
224C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
225C42—91no—
226C42—181no—
227C22×C3—31no—
228C22×C3C3×D3×D411yescharacteristic
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4—92no—
234C3×C4—121no—
235C3×C4—121no—
236C3×C4—181no—
237C3×C4—182no—
238C3×C4—361no—
239C3×C4C3×#23511yescharacteristic
240C3×C4C3×D3×Q811yescharacteristic
241Dic3—32no—
242Dic3—62no—
243Dic3—61no—
244Dic3—121no—
245C32#487511yescharacteristic
246C9—21no—
247C9C3×#47711yescharacteristic, complemented
248Q8—92no—
249Q8—181no—
250Q8—362no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—91no—
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
258C8—21no—
259C8—181no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3C3×#89611yescharacteristic
264C2×C3C3×#91111yescharacteristic
265C2×C3C3×D3×Q1611yescharacteristic
266C22D4×AGL(1,ℤ/9)11yescharacteristic
267C4—41no—
268C4—41no—
269C4—92no—
270C4—182no—
271C4—181no—
272C4—361no—
273C4#3697⋊C311yescharacteristic
274C4Q8×AGL(1,ℤ/9)11yescharacteristic
275C3—31no—
276C3C3×#487511yescharacteristic
277C2#63458⋊C311yescharacteristic
278C2#63473⋊C311yescharacteristic
279C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#107AGL(1,ℤ/9)
hypocenterC9C3×#477
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
9. #10588555C3semidirect
45. 3−1+2⋊Q8Q8semidirect
122. Dic18C3×Q8semidirect
182. 3−1+2#477semidirect
247. C9C3×#477semidirect

Maximal subgroups

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

subgroupindex
#63450⋊C32
#63449⋊C32
3−1+2×#1072
#63376⋊C32
(3−1+2⋊C4)⋊C82
#63368⋊C32
#63366⋊C32
C3×#48753
#105885553