Finite Group Databasebeta

← #13137586

Subgroups — #13137586

Order 1728.

392 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721744C211yesmaximal, characteristic, complemented
2#11720998C211yesmaximal, characteristic
3C3×#63369C211yesmaximal, characteristic
4C9×#808C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10590472—31nomaximal
9#10588550—31nomaximal
10C2×#6226—21no—
11C2×C3×D36C2211yescharacteristic
12C2×Dic3×D9—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21#64097—31no—
22#63898—31no—
23C3×#809—31no—
24C3×#808D311yescharacteristic
25#63573—31no—
26#63572—31no—
27#63563—31no—
28#63455—31no—
29#63450—31no—
30#63450—61no—
31C9×#106—31no—
32#63380—31no—
33Dic9⋊C8—31no—
34#63369D311yescharacteristic
35#63367—31no—
36C22×C3×D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42#6226—42no—
43C2×C9×Dic3—21no—
44C3×D36—21no—
45C3×D36Q812yescomplemented
46Dic3×D9—42no—
47C9×C3⋊C8—21no—
48C3×C9⋊C8—21no—
49#4870—91no—
50#4761—91no—
51C2×#634—61no—
52C2×C3×D12—31no—
53C2×D3×Dic3—61no—
54C32×C4⋊C4C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)—31no—
56#3750—61no—
57C4×C3⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61(C2×D3)⋊Dic3—61no—
62C2×D36C2×D311yescharacteristic
63C2×C4×D9—61no—
64C2×Dic18—61no—
65C2×Dic18—62no—
66C2×C8×C9—31no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4—31no—
69C9×C4⋊C4C2×D311yescharacteristic
70(C2×D9)⋊C4—61no—
71C4⋊Dic9—31no—
72C4⋊Dic9—61no—
73Dic9⋊C4—61no—
74Dic9⋊C4—62no—
75C4×Dic9—31no—
76C4×Dic9—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×C3×D9—22no—
80C2×C3×D9—41no—
81C2×C3×D9—41no—
82C3×C4×C9—41no—
83C3×C4×C9C2×Q811yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87C3⋊Dic9SD1612yes—
88C9×Dic3—41no—
89D4×Dic3—91no—
90#893—91no—
91#888—91no—
92#888—91no—
93#888—181no—
94C3×#106—31no—
95C3×#102—91no—
96#832—91no—
97#821—91no—
98#817—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#809—31no—
102#808—91no—
103#808D911yescharacteristic, complemented
104C22×C3×D3—61no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110#634—122no—
111C2×C3×Dic3—21no—
112C3×D12—32no—
113C3×D12—61no—
114D3×Dic3—122no—
115C22×D9—21no—
116C3×C3⋊C8—21no—
117C3×C3⋊C8—61no—
118C2×C4×C9—22no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—61no—
122C2×C4×C9C22×D311yescharacteristic
123C2×Dic9—31no—
124C2×Dic9—31no—
125C2×Dic9—61no—
126C2×Dic9—61no—
127C2×Dic9—62no—
128C2×Dic9—61no—
129D36—21no—
130D36Dic612yescomplemented
131C4×D9—122no—
132Dic18—124no—
133Dic18—122no—
134C8×C9—61no—
135C9⋊C8—21no—
136#476—271noSylow
137C2×C3×C9C22⋊Q811yescharacteristic
138C2×C3×C9C2×SD1611yescharacteristic
139C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
140C3×D9—42no—
141C2×C3×D4—91no—
142C22×Dic3—181no—
143C2×D12—31no—
144C2×C4×D3—181no—
145C2×Dic6—181no—
146C2×Dic6—181no—
147C2×Dic6—182no—
148C2×C3×C8—31no—
149C2×C3×C8—91no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4C2×D911yescharacteristic
153C3×C4⋊C4D3211yescharacteristic
154#217—181no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—91no—
157C4⋊Dic3—91no—
158C4⋊Dic3—181no—
159C4⋊Dic3C2×D911yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—182no—
163C4×Dic3—91no—
164C4×Dic3—91no—
165C4×Dic3—181no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8C2×D911yescharacteristic
168C22×C32D3×D411yescharacteristic
169C2×C3×D3—62no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C32×C4—41no—
173C32×C4#23511yescharacteristic
174C32×C4D3×Q811yescharacteristic
175C3⋊Dic3—32no—
176C3⋊Dic3—61no—
177C3⋊Dic3—121no—
178C3×Dic3—41no—
179C22×C9—21no—
180C22×C9D3×D411yescharacteristic
181C2×D9—22no—
182C2×D9—41no—
183C2×D9—41no—
184C4×C9—21no—
185C4×C9—21no—
186C4×C9—42no—
187C4×C9—41no—
188C4×C9—121no—
189C4×C9C2×Dic611yescharacteristic
190C4×C9#23711yescharacteristic
191Dic9—32no—
192Dic9—62no—
193Dic9—62no—
194Dic9—61no—
195Dic9—122no—
196Dic9—121no—
197C4⋊Q8—271no—
198C4×D4—271no—
199#106—31no—
200#106—271no—
201C4⋊C8—271no—
202#102—91no—
203#102—271no—
204C3×C9#47611yescharacteristic, complemented, Sylow
205C23×C3—181no—
206C22×D3—61no—
207C3×D4—92no—
208C3×D4—181no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—61no—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C2×D3211yescharacteristic
216C2×Dic3—21no—
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—
226C2×Dic3—362no—
227D12—32no—
228D12—61no—
229C4×D3—362no—
230Dic6—362no—
231Dic6—364no—
232Dic6—362no—
233C3×C8—61no—
234C3×C8—181no—
235C3⋊C8—21no—
236C3⋊C8—61no—
237C2×C32#89611yescharacteristic
238C2×C32D3×SD1611yescharacteristic
239C2×C32#91411yescharacteristic
240C3×D3—122no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9#87811yescharacteristic
245C2×C9#91111yescharacteristic
246C2×C9D3×SD1611yescharacteristic
247D9—42no—
248C2×Q8—541no—
249C2×D4—91no—
250C22×C4—541no—
251C2×C8—31no—
252C2×C8—91no—
253C4⋊C4—31no—
254C4⋊C4—271no—
255C4⋊C4—541no—
256C4⋊C4D3×D911yescharacteristic
257C22⋊C4—541no—
258C42—271no—
259C22×C3—21no—
260C22×C3—182no—
261C22×C3—361no—
262C22×C3—361no—
263C22×C3D4×D911yescharacteristic
264C22×C3#381111yescharacteristic
265C2×D3—62no—
266C2×D3—121no—
267C2×D3—121no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—121no—
274C3×C4#369711yescharacteristic
275C3×C4Q8×D911yescharacteristic
276C3×C4D3×Dic611yescharacteristic
277C3×C4#379511yescharacteristic
278Dic3—41no—
279Dic3—92no—
280Dic3—92no—
281Dic3—182no—
282Dic3—182no—
283Dic3—181no—
284Dic3—181no—
285Dic3—362no—
286Dic3—361no—
287Dic3—361no—
288C32#487011yescharacteristic
289C9—21no—
290C9#476111yescharacteristic, complemented
291C23—181no—
292Q8—1082no—
293D4—92no—
294D4—181no—
295C2×C4—21no—
296C2×C4—61no—
297C2×C4—271no—
298C2×C4—271no—
299C2×C4—541no—
300C2×C4—1082no—
301C2×C4C2×D3×D911yescharacteristic
302C8—61no—
303C8—181no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—362no—
308C2×C3#6345811yescharacteristic
309C2×C3SD16×D911yescharacteristic
310C2×C3#6347611yescharacteristic
311C2×C3#6389611yescharacteristic
312C2×C3#6393711yescharacteristic
313C2×C3#6394011yescharacteristic
314D3—122no—
315C22—182no—
316C22—361no—
317C22—361no—
318C22#8866811yescharacteristic
319C4—41no—
320C4—121no—
321C4—272no—
322C4—541no—
323C4—1081no—
324C4D9×Dic611yescharacteristic
325C4#8864111yescharacteristic
326C3—21no—
327C3#1058855011yescharacteristic, complemented
328C3#1059047211yescharacteristic
329C2—362no—
330C2#1172099611yescharacteristic
331C2#1172118711yescharacteristic
332C2#1172121211yescharacteristic
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#476
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. #11721744C2semidirect
4. C9×#808C2semidirect
6. #11720273C2semidirect
7. #11720264C2semidirect
45. C3×D36Q8semidirect
103. #808D9semidirect
130. D36Dic6semidirect
204. C3×C9#476semidirect
290. C9#4761semidirect
327. C3#10588550semidirect

Maximal subgroups

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

subgroupindex
#117217442
#117209982
C3×#633692
C9×#8082
#117202832
#117202732
#117202642
#105904723
#105885503