Finite Group Databasebeta

← #10588477⋊C3 #13141940

Subgroups — #10588477⋊C3 #13141940

Order 1728.

291 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 86 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#63452⋊C3C211yesmaximal, characteristic
33−1+2×#103C211yesmaximal, characteristic
4#63395⋊C3C211yesmaximal, characteristic
5#63376⋊C3C211yesmaximal, characteristic
6#63375⋊C3C211yesmaximal, characteristic
7#63368⋊C3C211yesmaximal, characteristic
8C3×#4797—31nomaximal
9#10588477C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8C2211yescharacteristic
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊(3−1+2⋊C4)—21no—
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×#890—31no—
23C32×#103D311yescharacteristic
24C3×#835—31no—
25C3×#816—31no—
26C3×#815—31no—
27C3×#810—31no—
28#63506C2×C311yescharacteristic
29#63452C2×C311yescharacteristic
30C9×#103—21no—
31C9×#103C2×C311yescharacteristic
32#63395C2×C311yescharacteristic
33#63376C2×C311yescharacteristic
34#63375C2×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—
47C8×3−1+2—21no—
483−1+2⋊C8—21no—
49C3×#491—91no—
50#4797—31no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C3×Dic6—31no—
53C2×C32×C8C2×D311yescharacteristic
54C32×C4⋊C4C2×D311yescharacteristic
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—61no—
57C3×Dic3⋊C4—61no—
58C3×Dic3⋊C4—61no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×Q8×C9—21no—
63C2×Q8×C9C22×C311yescharacteristic
64C2×Dic18C22×C311yescharacteristic
65C2×C8×C9—21no—
66C2×C8×C9C22×C311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4C22×C311yescharacteristic
69C4⋊Dic9—21no—
70C4⋊Dic9C22×C311yescharacteristic
71Dic9⋊C4—21no—
72Dic9⋊C4—21no—
73Dic9⋊C4—21no—
74C4×Dic9C22×C311yescharacteristic
75C2×C9⋊C8C22×C311yescharacteristic
76C22×3−1+2C2×D411yescharacteristic
77C4×3−1+2—41no—
78C4×3−1+2—41no—
79C4×3−1+2#4111yescharacteristic
80C4×3−1+2#4111yescharacteristic
813−1+2⋊C4—21no—
823−1+2⋊C4—21no—
833−1+2⋊C4—41no—
843−1+2⋊C4—41no—
85C3×C4⋊Q8—91no—
86C3×#125—91no—
87#944—31no—
88#890—31no—
89C3×#103—31no—
90C3×#103—91no—
91C3×#103—91no—
92C3×#103—91no—
93C3×#103C3×D311yescharacteristic
94C3×#97—91no—
95#835—31no—
96#816—31no—
97#815—31no—
98#810—31no—
99C32×Q8—21no—
100C32×Q8—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3×Dic3—31no—
105C2×C3×Dic3—31no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C3×Dic6—61no—
109C3×Dic6—61no—
110C32×C8—21no—
111C3×C3⋊C8—61no—
112Q8×C9—22no—
113Q8×C9—21no—
114Q8×C9—21no—
115Q8×C9—41no—
116C2×C4×C9—21no—
117C2×C4×C9—21no—
118C2×C4×C9—21no—
119C2×C4×C9—41no—
120C2×C4×C9—41no—
121C2×C4×C9C23×C311yescharacteristic
122C2×Dic9—21no—
123C2×Dic9—21no—
124C2×Dic9C3×D411yescharacteristic
125C2×Dic9C3×D411yescharacteristic
126Dic18—21no—
127Dic18—21no—
128C8×C9—21no—
129C8×C9—41no—
130C9⋊C8—21no—
131#491—91noSylow
132C2×3−1+2#12411yescharacteristic
133C2×3−1+2#13711yescharacteristic
134C2×3−1+2#13711yescharacteristic
135C2×C3×Q8—31no—
136C2×C3×Q8—91no—
137C2×C3×Q8C2×C3×D311yescharacteristic
138C2×Dic6—31no—
139C2×C3×C8—31no—
140C2×C3×C8—91no—
141C2×C3×C8C2×C3×D311yescharacteristic
142C3×C4⋊C4—31no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—181no—
145C3×C4⋊C4—181no—
146C3×C4⋊C4—181no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4C2×C3×D311yescharacteristic
149C3×C42—91no—
150C4⋊Dic3—31no—
151C4⋊Dic3—61no—
152Dic3⋊C4—61no—
153Dic3⋊C4—61no—
154Dic3⋊C4—61no—
155C4×Dic3—31no—
156C2×C3⋊C8—31no—
157C22×C32D3×D411yescharacteristic
158C32×C4—41no—
159C32×C4—41no—
160C32×C4#23511yescharacteristic
161C32×C4#23711yescharacteristic
162C3×Dic3—61no—
163C3×Dic3—61no—
164C3×Dic3—121no—
165C3×Dic3—121no—
166C22×C9—21no—
167C22×C9C2×C3×D411yescharacteristic
168C4×C9—21no—
169C4×C9—21no—
170C4×C9—42no—
171C4×C9—41no—
172C4×C9—41no—
173C4×C9—81no—
174C4×C9C3×#4111yescharacteristic
175C4×C9C3×#4111yescharacteristic
176Dic9—21no—
177Dic9—21no—
178Dic9—41no—
179Dic9—41no—
180C4⋊Q8—91no—
181#125—91no—
182#103—91no—
183#103—91no—
184#103—91no—
185#103AGL(1,ℤ/9)11yescharacteristic
186#97—91no—
1873−1+2#49111yescharacteristic, complemented, Sylow
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—61no—
191C3×Q8—61no—
192C3×Q8—181no—
193C3×Q8—181no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4C22×C3×D311yescharacteristic
204C2×Dic3—31no—
205C2×Dic3—31no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208Dic6—61no—
209Dic6—61no—
210C3×C8—21no—
211C3×C8—61no—
212C3×C8—181no—
213C3⋊C8—61no—
214C2×C32#88511yescharacteristic
215C2×C32#91511yescharacteristic
216C2×C32#91811yescharacteristic
217C2×C9—21no—
218C2×C9—21no—
219C2×C9—21no—
220C2×C9C3×#12411yescharacteristic
221C2×C9C3×#13711yescharacteristic
222C2×C9C3×#13711yescharacteristic
223C2×Q8—91no—
224C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
225C2×C8—91no—
226C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
227C4⋊C4—91no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C4⋊C4—181no—
231C4⋊C4—181no—
232C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
233C42—91no—
234C22×C3—31no—
235C22×C3C3×D3×D411yescharacteristic
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—121no—
241C3×C4—121no—
242C3×C4—181no—
243C3×C4—181no—
244C3×C4—361no—
245C3×C4—361no—
246C3×C4C3×#23511yescharacteristic
247C3×C4C3×#23711yescharacteristic
248Dic3—61no—
249Dic3—61no—
250Dic3—121no—
251Dic3—121no—
252C32#479711yescharacteristic
253C9—21no—
254C9C3×#49111yescharacteristic, complemented
255Q8—21no—
256Q8—21no—
257Q8—181no—
258Q8—181no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—181no—
264C2×C4—181no—
265C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
266C8—21no—
267C8—181no—
268C2×C3—31no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3C3×#88511yescharacteristic
272C2×C3C3×#91511yescharacteristic
273C2×C3C3×#91811yescharacteristic
274C22D4×AGL(1,ℤ/9)11yescharacteristic
275C4—41no—
276C4—41no—
277C4—181no—
278C4—181no—
279C4—361no—
280C4—361no—
281C4#3697⋊C311yescharacteristic
282C4#3699⋊C311yescharacteristic
283C3—31no—
284C3C3×#479711yescharacteristic
285C2#63447⋊C311yescharacteristic
286C2#63477⋊C311yescharacteristic
287C2#63480⋊C311yescharacteristic
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×#491
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. #10588477C3semidirect
187. 3−1+2#491semidirect
254. C9C3×#491semidirect

Maximal subgroups

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

subgroupindex
#63506⋊C32
#63452⋊C32
3−1+2×#1032
#63395⋊C32
#63376⋊C32
#63375⋊C32
#63368⋊C32
C3×#47973
#105884773