Finite Group Databasebeta

← #13137810

Subgroups — #13137810

Order 1728.

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

normal subgroups only

This group has 344 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#11721746C211yesmaximal, characteristic, complemented
2#11720951C211yesmaximal, characteristic
3C9×#810C211yesmaximal, characteristic, complemented
4C3×#63369C211yesmaximal, characteristic
5#11720298C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic
7#11720271C211yesmaximal, characteristic, complemented
8#10590539—31nomaximal
9#10588480—31nomaximal
10C2×C3×D36C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic9)—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20(C2×D9)⋊Dic3—21no—
21#64099—31no—
22#63863—31no—
23C3×#810D311yescharacteristic
24C3×#809—31no—
25#63566—31no—
26#63563—31no—
27#63553—31no—
28#63509—31no—
29#63452—31no—
30#63452—61no—
31C9×#103—31no—
32#63395—31no—
33#63380—31no—
34C8×Dic9—31no—
35#63369D311yescharacteristic
36C22×C3×D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42C2×C9×Dic3—21no—
43C3×D36—21no—
44C3×D36—21no—
45C9×Dic6—21no—
46C9×Dic6—21no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4800—91no—
50#4768—91no—
51C2×C3×D12—31no—
52C2×C3×Dic6C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—61no—
56#3750—61no—
57C4×C3⋊Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60(C2×D3)⋊Dic3—61no—
61(C2×D3)⋊Dic3—61no—
62C2×Q8×C9—31no—
63C2×D36C2×D311yescharacteristic
64C2×C8×C9—31no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4C2×D311yescharacteristic
67(C2×D9)⋊C4—61no—
68(C2×D9)⋊C4—61no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9—61no—
72C4⋊Dic9—62no—
73Dic9⋊C4—61no—
74Dic9⋊C4—62no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3×D9—41no—
80C2×C3×D9—41no—
81C2×C3×D9—41no—
82C3×C4×C9—41no—
83C3×C4×C9#4111yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—21no—
87C3⋊Dic9—41no—
88C9×Dic3—41no—
89#947—91no—
90#936—91no—
91#890—91no—
92#890—91no—
93#890—181no—
94C3×#103—31no—
95C3×#102—91no—
96#835—91no—
97#832—91no—
98#821—91no—
99#816—91no—
100C8×Dic3—91no—
101C8×Dic3—91no—
102#810D911yescharacteristic, complemented
103#809—31no—
104C22×C3×D3—61no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110C2×C3×Dic3—21no—
111C3×D12—61no—
112C3×D12—61no—
113C3×Dic6—21no—
114C3×Dic6—21no—
115C22×D9—21no—
116C3×C3⋊C8—21no—
117C3×C3⋊C8—61no—
118Q8×C9—61no—
119Q8×C9—61no—
120C2×C4×C9—22no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—61no—
124C2×C4×C9C22×D311yescharacteristic
125C2×Dic9—31no—
126C2×Dic9—31no—
127C2×Dic9—61no—
128C2×Dic9—61no—
129C2×Dic9—62no—
130C2×Dic9—61no—
131D36—21no—
132D36—21no—
133C8×C9—61no—
134C9⋊C8—21no—
135#488—271noSylow
136C2×C3×C9#12411yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C2×C3×C9#13511yescharacteristic
139C3×D9—81no—
140C2×C3×Q8—31no—
141C2×C3×D4—91no—
142C2×D12—31no—
143C2×Dic6C2×D911yescharacteristic
144C2×C3×C8—31no—
145C2×C3×C8—91no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4C2×D911yescharacteristic
148C3×C4⋊C4D3211yescharacteristic
149#217—181no—
150#217—181no—
151(C2×D3)⋊C4—181no—
152(C2×D3)⋊C4—181no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158C4⋊Dic3—182no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161Dic3⋊C4—182no—
162C4×Dic3—91no—
163C4×Dic3—91no—
164C4×Dic3—181no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8C2×D911yescharacteristic
167C22×C32D3×D411yescharacteristic
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C32×C4—41no—
172C32×C4#23511yescharacteristic
173C32×C4#23711yescharacteristic
174C3⋊Dic3—61no—
175C3⋊Dic3—61no—
176C3⋊Dic3—121no—
177C3×Dic3—41no—
178C22×C9—21no—
179C22×C9D3×D411yescharacteristic
180C2×D9—41no—
181C2×D9—41no—
182C2×D9—41no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—42no—
186C4×C9—41no—
187C4×C9—121no—
188C4×C9#23511yescharacteristic
189C4×C9#23711yescharacteristic
190Dic9—61no—
191Dic9—61no—
192Dic9—121no—
193Dic9—121no—
194Dic9—122no—
195Dic9—121no—
196#125—271no—
197#124—271no—
198#103—31no—
199#103—271no—
200#102—91no—
201#102—271no—
202C4×C8—271no—
203C3×C9#48811yescharacteristic, complemented, Sylow
204C23×C3—181no—
205C22×D3—61no—
206C3×Q8—61no—
207C3×Q8—61no—
208C3×D4—181no—
209C3×D4—181no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—61no—
215C2×C3×C4C22×D911yescharacteristic
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—91no—
219C2×Dic3—91no—
220C2×Dic3—91no—
221C2×Dic3—91no—
222C2×Dic3—181no—
223C2×Dic3—181no—
224C2×Dic3—182no—
225C2×Dic3—181no—
226C2×Dic3—181no—
227D12—61no—
228D12—61no—
229Dic6—21no—
230Dic6—21no—
231C3×C8—61no—
232C3×C8—181no—
233C3⋊C8—21no—
234C3⋊C8—61no—
235C2×C32#88511yescharacteristic
236C2×C32#91611yescharacteristic
237C2×C32#91911yescharacteristic
238C3×D3—241no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9#88511yescharacteristic
243C2×C9#91211yescharacteristic
244C2×C9#91611yescharacteristic
245D9—81no—
246C2×Q8—31no—
247C2×D4—91no—
248C2×C8—31no—
249C2×C8—91no—
250C4⋊C4—271no—
251C4⋊C4—541no—
252C4⋊C4—541no—
253C4⋊C4D3×D911yescharacteristic
254C22⋊C4—541no—
255C22⋊C4—541no—
256C42—271no—
257C22×C3—21no—
258C22×C3—361no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3D4×D911yescharacteristic
262C22×C3#381111yescharacteristic
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—121no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—42no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—121no—
272C3×C4#369711yescharacteristic
273C3×C4#369911yescharacteristic
274C3×C4#379511yescharacteristic
275C3×C4#379511yescharacteristic
276Dic3—41no—
277Dic3—181no—
278Dic3—181no—
279Dic3—181no—
280Dic3—181no—
281Dic3—361no—
282Dic3—361no—
283Dic3—362no—
284Dic3—361no—
285Dic3—361no—
286C32#480011yescharacteristic
287C9—21no—
288C9#476811yescharacteristic, complemented
289C23—181no—
290Q8—61no—
291Q8—61no—
292D4—181no—
293D4—181no—
294C2×C4—21no—
295C2×C4—61no—
296C2×C4—271no—
297C2×C4—271no—
298C2×C4—541no—
299C2×C4C2×D3×D911yescharacteristic
300C8—61no—
301C8—181no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—721no—
306C2×C3#6344711yescharacteristic
307C2×C3#6347811yescharacteristic
308C2×C3#6348111yescharacteristic
309C2×C3#6391311yescharacteristic
310C2×C3#6393611yescharacteristic
311C2×C3#6395011yescharacteristic
312D3—241no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22#8866811yescharacteristic
317C4—41no—
318C4—121no—
319C4—541no—
320C4—541no—
321C4—1081no—
322C4#8863811yescharacteristic
323C4#8864111yescharacteristic
324C3—21no—
325C3#1058848011yescharacteristic, complemented
326C3#1059053911yescharacteristic
327C2—721no—
328C2#1172104211yescharacteristic
329C2#1172118611yescharacteristic
330C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#488
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721746C2semidirect
3. C9×#810C2semidirect
5. #11720298C2semidirect
7. #11720271C2semidirect
102. #810D9semidirect
203. C3×C9#488semidirect
288. C9#4768semidirect
325. C3#10588480semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117209512
C9×#8102
C3×#633692
#117202982
#117202832
#117202712
#105905393
#105884803