Finite Group Databasebeta

← #13138308

Subgroups — #13138308

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#11720929C211yesmaximal, characteristic
2#11720922C211yesmaximal, characteristic
3#11720542C211yesmaximal, characteristic
4C9×#816C211yesmaximal, characteristic, complemented
5C3×#63378C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720265C211yesmaximal, characteristic, complemented
8#10590715—31nomaximal
9#10588494—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C9×C4⋊Dic3C2211yescharacteristic
15C3×C4⋊Dic9C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17#88445—21no—
18#88445—21no—
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#63871—31no—
22#63849—31no—
23#63647—31no—
24C3×#818—31no—
25C3×#816D311yescharacteristic
26#63574—31no—
27#63565—31no—
28#63508—31no—
29#63454—31no—
30C9×#103—31no—
31#63395—31no—
32(C2×D9)⋊C8—31no—
33(C2×D9)⋊C8—61no—
34#63378D311yescharacteristic, complemented
35#63367—31no—
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9D411yescharacteristic
39C4×C3⋊D9—21no—
40C4×C3⋊D9—21no—
41C2×C9×Dic3—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C2×C3×Dic9—21no—
45C9×Dic6—21no—
46C9×Dic6—21no—
47C3×C8×C9—21no—
48C3⋊(C9⋊C8)—21no—
49#4888—91no—
50#4814—91no—
51C2×C4×C3⋊D3—31no—
52C2×C3×Dic6C2×D311yescharacteristic
53C2×C32×C8C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58Dic3⋊Dic3—61no—
59#3722—61no—
60#3722—61no—
61Dic32—61no—
62C2×Q8×C9—31no—
63C2×C4×D9—31no—
64C2×C4×D9—61no—
65C2×C8×C9—21no—
66C2×C8×C9C2×D311yescharacteristic
67C9×C4⋊C4—31no—
68(C2×D9)⋊C4—61no—
69(C2×D9)⋊C4—61no—
70C4⋊Dic9C2×D311yescharacteristic
71C4⋊Dic9C2×D311yescharacteristic
72Dic9⋊C4—61no—
73C4×Dic9—61no—
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C2×C3⋊D9—21no—
78C2×C3⋊D9—21no—
79C2×C3⋊D9—21no—
80C3×C4×C9#4111yescharacteristic
81C3×C4×C9#4111yescharacteristic
82C3⋊Dic9—21no—
83C9×Dic3—41no—
84C9×Dic3—41no—
85C3×Dic9—41no—
86C3×Dic9—41no—
87#946—91no—
88#897—91no—
89#892—91no—
90#892—91no—
91C3×#107—91no—
92C3×#103—31no—
93#835—91no—
94(C2×D3)⋊C8—91no—
95(C2×D3)⋊C8—91no—
96(C2×D3)⋊C8—181no—
97#818—31no—
98#816D911yescharacteristic, complemented
99#810—91no—
100#808—91no—
101#808—91no—
102C22×C3⋊D3—31no—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—31no—
105C4×C3⋊D3—61no—
106C4×C3⋊D3—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×Dic6—21no—
112C3×Dic6—21no—
113C22×D9—31no—
114C22×D9—61no—
115C32×C8—21no—
116C3⋊(C3⋊C8)—61no—
117Q8×C9—61no—
118Q8×C9—61no—
119C2×C4×C9—21no—
120C2×C4×C9—61no—
121C2×C4×C9—61no—
122C2×C4×C9C22×D311yescharacteristic
123C2×Dic9—21no—
124C2×Dic9—21no—
125C2×Dic9—31no—
126C2×Dic9—61no—
127C4×D9—61no—
128C4×D9—61no—
129C4×D9—121no—
130C4×D9—121no—
131C8×C9—21no—
132C8×C9—41no—
133C9⋊C8—61no—
134C9⋊C8—121no—
135#485—271noSylow
136C2×C3×C9#12311yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C2×C3×C9#13711yescharacteristic
139C3⋊D9—41no—
140C2×C3×Q8—31no—
141C2×C4×D3—91no—
142C2×C4×D3—91no—
143C2×C4×D3—181no—
144C2×Dic6C2×D911yescharacteristic
145C2×C3×C8—21no—
146C2×C3×C8C2×D911yescharacteristic
147C2×C3×C8D3211yescharacteristic
148C3×C4⋊C4—31no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—91no—
151(C2×D3)⋊C4—181no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—181no—
158C4⋊Dic3C2×D911yescharacteristic
159Dic3⋊C4—181no—
160C4×Dic3—181no—
161C4×Dic3—181no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—181no—
165C22×C32D3×D411yescharacteristic
166C2×C3⋊D3—61no—
167C2×C3⋊D3—61no—
168C2×C3⋊D3—61no—
169C32×C4#23511yescharacteristic
170C32×C4#23711yescharacteristic
171C3⋊Dic3—61no—
172C3×Dic3—41no—
173C3×Dic3—41no—
174C3×Dic3—121no—
175C3×Dic3—121no—
176C22×C9—21no—
177C22×C9D3×D411yescharacteristic
178C2×D9—62no—
179C2×D9—61no—
180C2×D9—61no—
181C2×D9—61no—
182C2×D9—121no—
183C2×D9—121no—
184C4×C9—21no—
185C4×C9—21no—
186C4×C9—121no—
187C4×C9—121no—
188C4×C9#23511yescharacteristic
189C4×C9#23911yescharacteristic
190Dic9—41no—
191Dic9—41no—
192Dic9—61no—
193Dic9—121no—
194C22⋊Q8—271no—
195#117—271no—
196#107—91no—
197#106—271no—
198#103—31no—
199#103—271no—
200C22⋊C8—271no—
201C3×C9#48511yescharacteristic, complemented, Sylow
202C22×D3—91no—
203C22×D3—91no—
204C22×D3—181no—
205C3×Q8—61no—
206C3×Q8—61no—
207C2×C3×C4—21no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4C22×D911yescharacteristic
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—91no—
219C2×Dic3—91no—
220C2×Dic3—181no—
221C4×D3—181no—
222C4×D3—181no—
223C4×D3—181no—
224C4×D3—181no—
225C4×D3—361no—
226C4×D3—361no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—21no—
230C3×C8—21no—
231C3×C8—41no—
232C3⋊C8—181no—
233C3⋊C8—181no—
234C3⋊C8—361no—
235C2×C32#88311yescharacteristic
236C2×C32#91511yescharacteristic
237C2×C32#91911yescharacteristic
238C3⋊D3—121no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9#89411yescharacteristic
243C2×C9#91211yescharacteristic
244C2×C9#91811yescharacteristic
245D9—122no—
246D9—121no—
247C2×Q8—31no—
248C22×C4—271no—
249C2×C8—271no—
250C2×C8D3×D911yescharacteristic
251C4⋊C4—31no—
252C4⋊C4—91no—
253C4⋊C4—91no—
254C4⋊C4—541no—
255C22⋊C4—541no—
256C22⋊C4—541no—
257C42—541no—
258C22×C3—21no—
259C22×C3D4×D911yescharacteristic
260C22×C3#380211yescharacteristic
261C2×D3—182no—
262C2×D3—181no—
263C2×D3—181no—
264C2×D3—181no—
265C2×D3—181no—
266C2×D3—181no—
267C2×D3—181no—
268C2×D3—361no—
269C2×D3—361no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—361no—
275C3×C4—361no—
276C3×C4#369711yescharacteristic
277C3×C4#369911yescharacteristic
278C3×C4#379911yescharacteristic
279C3×C4#380411yescharacteristic
280Dic3—41no—
281Dic3—41no—
282Dic3—121no—
283Dic3—121no—
284Dic3—181no—
285Dic3—181no—
286Dic3—361no—
287C32#481411yescharacteristic
288C9—21no—
289C9#488811yescharacteristic, complemented
290C23—271no—
291Q8—61no—
292Q8—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—271no—
298C2×C4—541no—
299C2×C4—541no—
300C2×C4C2×D3×D911yescharacteristic
301C8—21no—
302C8—541no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3#6344511yescharacteristic
307C2×C3#6347711yescharacteristic
308C2×C3#6348111yescharacteristic
309C2×C3#6381011yescharacteristic
310C2×C3#6381111yescharacteristic
311C2×C3#6385411yescharacteristic
312D3—362no—
313D3—361no—
314D3—361no—
315C22—541no—
316C22—541no—
317C22—541no—
318C22#8864611yescharacteristic
319C4—121no—
320C4—121no—
321C4—361no—
322C4—361no—
323C4—541no—
324C4#8864211yescharacteristic
325C4#8866211yescharacteristic
326C3—21no—
327C3#1058849411yescharacteristic, complemented
328C3#1059071511yescharacteristic
329C2—1081no—
330C2#1172082511yescharacteristic
331C2#1172083111yescharacteristic
332C2#1172095811yescharacteristic
subgroupG/H
centerC22#88646
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×C8D3×D9
hypocenterC3×C9#485
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C9×#816C2semidirect
5. C3×#63378C2semidirect
6. #11720297C2semidirect
7. #11720265C2semidirect
34. #63378D3semidirect
98. #816D9semidirect
201. C3×C9#485semidirect
289. C9#4888semidirect
327. C3#10588494semidirect

Maximal subgroups

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

subgroupindex
#117209292
#117209222
#117205422
C9×#8162
C3×#633782
#117202972
#117202652
#105907153
#105884943