Finite Group Databasebeta

← #10588472⋊C3 #13141934

Subgroups — #10588472⋊C3 #13141934

Order 1728.

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

normal subgroups only

This group has 315 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#63506⋊C3C211yesmaximal, characteristic
2#63450⋊C3C211yesmaximal, characteristic
33−1+2×#103C211yesmaximal, characteristic
4#63395⋊C3C211yesmaximal, characteristic
5#63376⋊C3C211yesmaximal, characteristic
6C8×3−1+2⋊C4C211yesmaximal, characteristic
7#63368⋊C3C211yesmaximal, characteristic
8C3×#4792—31nomaximal
9#10588472C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8—21no—
12C2×3−1+2⋊Q8C2211yescharacteristic
13C2×C8×3−1+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—
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#944—31no—
22C3×#888—31no—
23C32×#103D311yescharacteristic
24C3×#835—31no—
25C3×#816—31no—
26C3×C8×Dic3—31no—
27C3×#810—31no—
28#63506C2×C311yescharacteristic
29#63450C2×C311yescharacteristic
30C9×#103—21no—
31C9×#103C2×C311yescharacteristic
32#63395C2×C311yescharacteristic
33#63376C2×C311yescharacteristic
34C8×Dic9C2×C311yescharacteristic
35#63368C2×C311yescharacteristic
36Q8×3−1+2—21no—
37Q8×3−1+2—21no—
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4D411yescharacteristic
44C2×3−1+2⋊C4D411yescharacteristic
453−1+2⋊Q8—21no—
463−1+2⋊Q8—21no—
473−1+2⋊Q8—42no—
48C8×3−1+2—21no—
493−1+2⋊C8—21no—
50C3×#487—91no—
51#4792—31no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C3×Dic6—31no—
54C2×C3×Dic6—61no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C3×C4⋊Dic3—31no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8—31no—
63C2×Q8×C9—21no—
64C2×Q8×C9C22×C311yescharacteristic
65C2×Dic18—21no—
66C2×Dic18C22×C311yescharacteristic
67C2×C8×C9—21no—
68C2×C8×C9C22×C311yescharacteristic
69C9×C4⋊C4—21no—
70C9×C4⋊C4C22×C311yescharacteristic
71C4⋊Dic9C22×C311yescharacteristic
72Dic9⋊C4—21no—
73Dic9⋊C4—21no—
74Dic9⋊C4—21no—
75C4×Dic9C22×C311yescharacteristic
76C2×C9⋊C8C22×C311yescharacteristic
77C22×3−1+2C2×D411yescharacteristic
78C4×3−1+2—41no—
79C4×3−1+2—41no—
80C4×3−1+2#4111yescharacteristic
81C4×3−1+2#4111yescharacteristic
823−1+2⋊C4—41no—
833−1+2⋊C4—41no—
843−1+2⋊C4SD1612yes—
853−1+2⋊C4Q1612yes—
86C3×C4⋊Q8—91no—
87C3×C4⋊Q8—91no—
88#944—31no—
89#888—31no—
90C3×#103—31no—
91C3×#103—91no—
92C3×#103—91no—
93C3×#103—91no—
94C3×#103C3×D311yescharacteristic
95C3×C4×C8—91no—
96#835—31no—
97#816—31no—
98C8×Dic3—31no—
99#810—31no—
100C32×Q8—21no—
101C32×Q8—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3×Dic3—31no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C3×Dic6—61no—
110C3×Dic6—61no—
111C3×Dic6—122no—
112C32×C8—21no—
113C3×C3⋊C8—61no—
114Q8×C9—22no—
115Q8×C9—21no—
116Q8×C9—21no—
117Q8×C9—41no—
118C2×C4×C9—21no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—41no—
122C2×C4×C9—41no—
123C2×C4×C9C23×C311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—21no—
126C2×Dic9C3×D411yescharacteristic
127C2×Dic9C3×D411yescharacteristic
128Dic18—21no—
129Dic18—21no—
130Dic18—42no—
131C8×C9—21no—
132C8×C9—41no—
133C9⋊C8—21no—
134#487—91noSylow
135C2×3−1+2#12411yescharacteristic
136C2×3−1+2C2×SD1611yescharacteristic
137C2×3−1+2C2×Q1611yescharacteristic
138C2×C3×Q8—31no—
139C2×C3×Q8—91no—
140C2×C3×Q8—181no—
141C2×C3×Q8C2×C3×D311yescharacteristic
142C2×Dic6—31no—
143C2×Dic6—61no—
144C2×C3×C8—31no—
145C2×C3×C8—91no—
146C2×C3×C8C2×C3×D311yescharacteristic
147C3×C4⋊C4—31no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4—181no—
152C3×C4⋊C4C2×C3×D311yescharacteristic
153C3×C42—91no—
154C4⋊Dic3—31no—
155Dic3⋊C4—61no—
156Dic3⋊C4—61no—
157Dic3⋊C4—61no—
158C4×Dic3—31no—
159C2×C3⋊C8—31no—
160C22×C32D3×D411yescharacteristic
161C32×C4—41no—
162C32×C4—41no—
163C32×C4#23511yescharacteristic
164C32×C4#23711yescharacteristic
165C3×Dic3—32no—
166C3×Dic3—32no—
167C3×Dic3—121no—
168C3×Dic3—121no—
169C22×C9—21no—
170C22×C9C2×C3×D411yescharacteristic
171C4×C9—21no—
172C4×C9—21no—
173C4×C9—42no—
174C4×C9—41no—
175C4×C9—41no—
176C4×C9—81no—
177C4×C9C3×#4111yescharacteristic
178C4×C9C3×#4111yescharacteristic
179Dic9—41no—
180Dic9—41no—
181Dic9C3×SD1612yes—
182Dic9C3×Q1612yes—
183C4⋊Q8—91no—
184C4⋊Q8—91no—
185#103—91no—
186#103—91no—
187#103—91no—
188#103AGL(1,ℤ/9)11yescharacteristic
189C4×C8—91no—
1903−1+2#48711yescharacteristic, complemented, Sylow
191C3×Q8—21no—
192C3×Q8—21no—
193C3×Q8—61no—
194C3×Q8—61no—
195C3×Q8—181no—
196C3×Q8—181no—
197C3×Q8—362no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4C22×C3×D311yescharacteristic
208C2×Dic3—31no—
209C2×Dic3—31no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212Dic6—61no—
213Dic6—61no—
214Dic6—122no—
215C3×C8—21no—
216C3×C8—61no—
217C3×C8—181no—
218C3⋊C8—61no—
219C2×C32#88511yescharacteristic
220C2×C32D3×SD1611yescharacteristic
221C2×C32D3×Q1611yescharacteristic
222C2×C9—21no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9C3×#12411yescharacteristic
226C2×C9C2×C3×SD1611yescharacteristic
227C2×C9C2×C3×Q1611yescharacteristic
228C2×Q8—91no—
229C2×Q8—181no—
230C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
231C2×C8—91no—
232C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
233C4⋊C4—91no—
234C4⋊C4—181no—
235C4⋊C4—181no—
236C4⋊C4—181no—
237C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
238C42—91no—
239C22×C3—31no—
240C22×C3C3×D3×D411yescharacteristic
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—92no—
246C3×C4—92no—
247C3×C4—121no—
248C3×C4—121no—
249C3×C4—361no—
250C3×C4—361no—
251C3×C4C3×#23511yescharacteristic
252C3×C4C3×#23711yescharacteristic
253Dic3—32no—
254Dic3—32no—
255Dic3—121no—
256Dic3—121no—
257C32#479211yescharacteristic
258C9—21no—
259C9C3×#48711yescharacteristic, complemented
260Q8—21no—
261Q8—21no—
262Q8—181no—
263Q8—181no—
264Q8—362no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
272C8—21no—
273C8—181no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3C3×#88511yescharacteristic
278C2×C3C3×D3×SD1611yescharacteristic
279C2×C3C3×D3×Q1611yescharacteristic
280C22D4×AGL(1,ℤ/9)11yescharacteristic
281C4—41no—
282C4—41no—
283C4—92no—
284C4—92no—
285C4—361no—
286C4—361no—
287C4#3697⋊C311yescharacteristic
288C4#3699⋊C311yescharacteristic
289C3—31no—
290C3C3×#479211yescharacteristic
291C2#63447⋊C311yescharacteristic
292C2SD16×AGL(1,ℤ/9)11yescharacteristic
293C2Q16×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×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#487
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
9. #10588472C3semidirect
190. 3−1+2#487semidirect
259. C9C3×#487semidirect

Maximal subgroups

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

subgroupindex
#63506⋊C32
#63450⋊C32
3−1+2×#1032
#63395⋊C32
#63376⋊C32
C8×3−1+2⋊C42
#63368⋊C32
C3×#47923
#105884723