Finite Group Databasebeta

← #13140708

Subgroups — #13140708

Order 1728.

338 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721854C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic
3C9×#835C211yesmaximal, characteristic
4C3×#63395C211yesmaximal, characteristic
5#11720277C211yesmaximal, characteristic
6#11720275C211yesmaximal, characteristic
7#11720269C211yesmaximal, characteristic
8#10591616—31nomaximal
9#10588477—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12#88534—21no—
13#88534—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64153—31no—
22#63872—31no—
23C3×#835—31no—
24C3×#835D311yescharacteristic
25#63567—31no—
26#63567—31no—
27#63555—31no—
28#63506—31no—
29#63506—61no—
30#63452—31no—
31C9×#103—31no—
32#63395D311yescharacteristic
33#63376—31no—
34#63375—31no—
35#63368—31no—
36C3×Q8×C9—21no—
37C3×Q8×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43#6226—21no—
44#6226—21no—
45C2×C9×Dic3—21no—
46C2×C3×Dic9—21no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4797—91no—
50#4797—91no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×#634—31no—
53#3750—61no—
54#3750—61no—
55C4×C3⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62C2×Q8×C9—21no—
63C2×Q8×C9C2×D311yescharacteristic
64C2×Dic18—31no—
65C2×Dic18—61no—
66C2×C8×C9—31no—
67C9×C4⋊C4—31no—
68C4⋊Dic9—61no—
69C4⋊Dic9C2×D311yescharacteristic
70Dic9⋊C4—61no—
71Dic9⋊C4—61no—
72Dic9⋊C4—61no—
73Dic9⋊C4—62no—
74Dic9⋊C4—62no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C3×C4×C9—41no—
80C3×C4×C9#4111yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—21no—
83C3⋊Dic9—21no—
84C3⋊Dic9—41no—
85C9×Dic3—41no—
86C3×Dic9—41no—
87#944—91no—
88#944—91no—
89#944—181no—
90#890—91no—
91#890—91no—
92C3×#103—31no—
93C3×#103—91no—
94#835—31no—
95#835D911yescharacteristic
96#816—91no—
97#816—91no—
98#815—91no—
99#815—91no—
100#810—91no—
101#810—91no—
102C32×Q8—21no—
103C32×Q8—21no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—31no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—61no—
109#634—61no—
110#634—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—61no—
113C3×C3⋊C8—21no—
114C3×C3⋊C8—61no—
115Q8×C9—21no—
116Q8×C9—21no—
117Q8×C9—41no—
118Q8×C9—41no—
119C2×C4×C9—22no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—61no—
123C2×C4×C9C22×D311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9—31no—
126C2×Dic9—31no—
127C2×Dic9—61no—
128C2×Dic9—61no—
129C2×Dic9—62no—
130C2×Dic9—61no—
131Dic18—61no—
132Dic18—61no—
133Dic18—121no—
134Dic18—121no—
135C8×C9—61no—
136C9⋊C8—21no—
137#491—271noSylow
138C2×C3×C9#12411yescharacteristic
139C2×C3×C9#13711yescharacteristic
140C2×C3×C9#13711yescharacteristic
141C2×C3×Q8—21no—
142C2×C3×Q8C2×D911yescharacteristic
143C2×C3×Q8D3211yescharacteristic
144C2×Dic6—91no—
145C2×Dic6—91no—
146C2×Dic6—181no—
147C2×C3×C8—31no—
148C2×C3×C8—91no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—91no—
151C4⋊Dic3—31no—
152C4⋊Dic3—181no—
153C4⋊Dic3—181no—
154C4⋊Dic3C2×D911yescharacteristic
155Dic3⋊C4—181no—
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—182no—
162Dic3⋊C4—182no—
163C4×Dic3—91no—
164C4×Dic3—91no—
165C4×Dic3—181no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8C2×D911yescharacteristic
168C22×C32D3×D411yescharacteristic
169C32×C4—41no—
170C32×C4#23511yescharacteristic
171C32×C4#23711yescharacteristic
172C3⋊Dic3—61no—
173C3⋊Dic3—61no—
174C3⋊Dic3—121no—
175C3×Dic3—41no—
176C3×Dic3—121no—
177C22×C9—21no—
178C22×C9D3×D411yescharacteristic
179C4×C9—21no—
180C4×C9—21no—
181C4×C9—42no—
182C4×C9—41no—
183C4×C9—121no—
184C4×C9#23511yescharacteristic
185C4×C9#23711yescharacteristic
186Dic9—41no—
187Dic9—62no—
188Dic9—62no—
189Dic9—61no—
190Dic9—61no—
191Dic9—122no—
192Dic9—121no—
193C4⋊Q8—271no—
194#125—271no—
195#103—31no—
196#103—91no—
197#103—271no—
198#103—271no—
199#97—271no—
200C3×C9#49111yescharacteristic, complemented, Sylow
201C3×Q8—21no—
202C3×Q8—21no—
203C3×Q8—21no—
204C3×Q8—21no—
205C3×Q8—41no—
206C3×Q8—41no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—61no—
212C2×C3×C4—181no—
213C2×C3×C4C22×D911yescharacteristic
214C2×C3×C4C2×D3211yescharacteristic
215C2×Dic3—21no—
216C2×Dic3—61no—
217C2×Dic3—91no—
218C2×Dic3—91no—
219C2×Dic3—91no—
220C2×Dic3—91no—
221C2×Dic3—181no—
222C2×Dic3—181no—
223C2×Dic3—182no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226Dic6—181no—
227Dic6—181no—
228Dic6—181no—
229Dic6—181no—
230Dic6—361no—
231Dic6—361no—
232C3×C8—61no—
233C3×C8—181no—
234C3⋊C8—21no—
235C3⋊C8—61no—
236C2×C32#88511yescharacteristic
237C2×C32#91511yescharacteristic
238C2×C32#91811yescharacteristic
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9#88511yescharacteristic
243C2×C9#91511yescharacteristic
244C2×C9#91811yescharacteristic
245C2×Q8—271no—
246C2×Q8D3×D911yescharacteristic
247C2×C8—31no—
248C2×C8—91no—
249C4⋊C4—31no—
250C4⋊C4—91no—
251C4⋊C4—541no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C4⋊C4—541no—
255C42—271no—
256C22×C3—21no—
257C22×C3D4×D911yescharacteristic
258C22×C3#381111yescharacteristic
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—42no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—121no—
265C3×C4—361no—
266C3×C4#369711yescharacteristic
267C3×C4#369911yescharacteristic
268C3×C4#380411yescharacteristic
269C3×C4#380411yescharacteristic
270Dic3—41no—
271Dic3—121no—
272Dic3—182no—
273Dic3—182no—
274Dic3—181no—
275Dic3—181no—
276Dic3—181no—
277Dic3—181no—
278Dic3—362no—
279Dic3—361no—
280Dic3—361no—
281C32#479711yescharacteristic
282C9—21no—
283C9#479711yescharacteristic, complemented
284Q8—21no—
285Q8—21no—
286Q8—541no—
287Q8—541no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—181no—
291C2×C4—271no—
292C2×C4—271no—
293C2×C4—541no—
294C2×C4C2×D3×D911yescharacteristic
295C8—61no—
296C8—181no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3#6344711yescharacteristic
301C2×C3#6347711yescharacteristic
302C2×C3#6348011yescharacteristic
303C2×C3#6395111yescharacteristic
304C2×C3#6395111yescharacteristic
305C2×C3#6395311yescharacteristic
306C22#8866811yescharacteristic
307C4—41no—
308C4—121no—
309C4—361no—
310C4—541no—
311C4—541no—
312C4—1081no—
313C4#8865811yescharacteristic
314C4#8866211yescharacteristic
315C3—21no—
316C3#1058847711yescharacteristic, complemented
317C3#1059161611yescharacteristic
318C2#1172121711yescharacteristic
319C2#1172122511yescharacteristic
320C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
hypercenterC2×Q8D3×D9
hypocenterC3×C9#491
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
200. C3×C9#491semidirect
283. C9#4797semidirect
316. C3#10588477semidirect

Maximal subgroups

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

subgroupindex
#117218542
#117209372
C9×#8352
C3×#633952
#117202772
#117202752
#117202692
#105916163
#105884773