Finite Group Databasebeta

← #13140704

Subgroups — #13140704

Order 1728.

330 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C3×#63509C211yesmaximal, characteristic
2#11720935C211yesmaximal, characteristic, complemented
3#11720582C211yesmaximal, characteristic, complemented
4C9×#835C211yesmaximal, characteristic, complemented
5#11720283C211yesmaximal, characteristic
6C3⋊C8×Dic9C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic
8#10591614—31nomaximal
9#10588480—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C3×(C2×D9)⋊C4—21no—
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20#88442—21no—
21C3×#947—31no—
22#63869—31no—
23#63662—31no—
24C3×#835D311yescharacteristic
25#63563—31no—
26#63562—31no—
27Dic3×C3⋊C8—31no—
28#63509D311yescharacteristic
29#63452—31no—
30C9×#103—31no—
31#63395—31no—
32#63395—61no—
33#63380—31no—
34C8×Dic9—31no—
35#63369—31no—
36C22×C3×D9—21no—
37C3×Q8×C9—21no—
38C3×Q8×C9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45C3×D36—21no—
46C3×D36—21no—
47C3⋊(C9⋊C8)—21no—
48C9×C3⋊C8—21no—
49#5056—91no—
50#4800—91no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C3×D12—31no—
53C4⋊(C3⋊Dic3)—31no—
54C2×C3⋊(C3⋊C8)—31no—
55C3×(C2×D3)⋊C4—61no—
56C3×(C2×D3)⋊C4—61no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60#3724—61no—
61Dic3⋊Dic3—61no—
62C2×Q8×C9—21no—
63C2×Q8×C9C2×D311yescharacteristic
64C2×D36C2×D311yescharacteristic
65C2×C8×C9—31no—
66C9×C4⋊C4—31no—
67(C2×D9)⋊C4—21no—
68(C2×D9)⋊C4—21no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9—61no—
72Dic9⋊C4—61no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C2×C3×D9—41no—
78C2×C3×D9—41no—
79C2×C3×D9—41no—
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—41no—
84C9×Dic3—41no—
85C3×Dic9—21no—
86C3×Dic9—21no—
87C3×#124—91no—
88#947—31no—
89#890—91no—
90#870—91no—
91C3×#103—31no—
92#835—91no—
93#835—91no—
94#835—181no—
95#835D911yescharacteristic, complemented
96#832—91no—
97#832—91no—
98#821—91no—
99C8×Dic3—91no—
100#809—91no—
101C4×C3⋊C8—91no—
102C22×C3×D3—61no—
103C32×Q8—21no—
104C32×Q8—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C3×D12—61no—
112C3×D12—61no—
113C22×D9—21no—
114C3⋊(C3⋊C8)—61no—
115C3×C3⋊C8—21no—
116Q8×C9—22no—
117Q8×C9—21no—
118Q8×C9—21no—
119Q8×C9—41no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—41no—
123C2×C4×C9—61no—
124C2×C4×C9C22×D311yescharacteristic
125C2×Dic9—61no—
126C2×Dic9—121no—
127C2×Dic9#6711yescharacteristic
128C2×Dic9#6711yescharacteristic
129D36—21no—
130D36—21no—
131C8×C9—61no—
132C9⋊C8—61no—
133C9⋊C8—121no—
134#488—271noSylow
135C2×C3×C9#12411yescharacteristic
136C2×C3×C9#13511yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C3×D9—81no—
139C2×C3×Q8—21no—
140C2×C3×Q8C2×D911yescharacteristic
141C2×C3×Q8D3211yescharacteristic
142C2×C3×D4—91no—
143C2×D12—31no—
144C2×C3×C8—31no—
145C3×C4⋊C4—31no—
146C3×C22⋊C4—181no—
147C3×C22⋊C4—181no—
148C3×C42—91no—
149(C2×D3)⋊C4—61no—
150(C2×D3)⋊C4—61no—
151C4⋊Dic3—91no—
152C4⋊Dic3—91no—
153C4⋊Dic3—181no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×D911yescharacteristic
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3—31no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—181no—
163C2×C3⋊C8C2×D911yescharacteristic
164C22×C32D3×D411yescharacteristic
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C32×C4—41no—
169C32×C4#23511yescharacteristic
170C32×C4#23711yescharacteristic
171C3⋊Dic3—121no—
172C3×Dic3—41no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C22×C9—21no—
176C22×C9C2×#6711yescharacteristic
177C2×D9—41no—
178C2×D9—41no—
179C2×D9—41no—
180C4×C9—21no—
181C4×C9—21no—
182C4×C9—42no—
183C4×C9—41no—
184C4×C9—121no—
185C4×C9#23711yescharacteristic
186C4×C9#23711yescharacteristic
187Dic9—21no—
188Dic9—21no—
189Dic9—121no—
190Dic9—241no—
191#125—271no—
192#124—91no—
193#103—31no—
194#103—271no—
195#102—271no—
196#102—271no—
197C4×C8—271no—
198C3×C9#48811yescharacteristic, complemented, Sylow
199C23×C3—181no—
200C22×D3—61no—
201C3×Q8—22no—
202C3×Q8—21no—
203C3×Q8—21no—
204C3×Q8—21no—
205C3×Q8—21no—
206C3×Q8—41no—
207C3×D4—181no—
208C3×D4—181no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—61no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4C22×D911yescharacteristic
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—181no—
222C2×Dic3—181no—
223C2×Dic3—361no—
224D12—61no—
225D12—61no—
226C3×C8—61no—
227C3⋊C8—21no—
228C3⋊C8—181no—
229C3⋊C8—181no—
230C3⋊C8—361no—
231C2×C32#88511yescharacteristic
232C2×C32#91611yescharacteristic
233C2×C32#91911yescharacteristic
234C3×D3—241no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9#93611yescharacteristic
239C2×C9#95011yescharacteristic
240C2×C9#95011yescharacteristic
241D9—81no—
242C2×Q8D3×D911yescharacteristic
243C2×D4—91no—
244C2×C8—31no—
245C2×C8—271no—
246C4⋊C4—31no—
247C4⋊C4—271no—
248C4⋊C4—541no—
249C4⋊C4—541no—
250C22⋊C4—181no—
251C22⋊C4—181no—
252C42—91no—
253C22×C3—21no—
254C22×C3—361no—
255C22×C3—361no—
256C22×C3—361no—
257C22×C3D4×D911yescharacteristic
258C22×C3D3×#6711yescharacteristic
259C2×D3—121no—
260C2×D3—121no—
261C2×D3—121no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—42no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—121no—
268C3×C4—181no—
269C3×C4—181no—
270C3×C4#369711yescharacteristic
271C3×C4#369911yescharacteristic
272C3×C4#380411yescharacteristic
273C3×C4#380511yescharacteristic
274Dic3—41no—
275Dic3—61no—
276Dic3—61no—
277Dic3—361no—
278Dic3—361no—
279Dic3—721no—
280C32#480011yescharacteristic
281C9—21no—
282C9#505611yescharacteristic, complemented
283C23—181no—
284Q8—21no—
285Q8—21no—
286D4—181no—
287D4—181no—
288C2×C4—21no—
289C2×C4—61no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—541no—
293C2×C4C2×D3×D911yescharacteristic
294C8—61no—
295C8—541no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—721no—
300C2×C3#6344711yescharacteristic
301C2×C3#6347811yescharacteristic
302C2×C3#6348111yescharacteristic
303C2×C3#6394711yescharacteristic
304C2×C3#6394911yescharacteristic
305C2×C3#6395911yescharacteristic
306D3—241no—
307C22—361no—
308C22—361no—
309C22—361no—
310C22D9×#6711yescharacteristic
311C4—41no—
312C4—121no—
313C4—181no—
314C4—181no—
315C4—1081no—
316C4#8865811yescharacteristic
317C4#8866311yescharacteristic
318C3—21no—
319C3#1058848011yescharacteristic, complemented
320C3#1059161411yescharacteristic
321C2—721no—
322C2#1172121511yescharacteristic
323C2#1172122311yescharacteristic
324C2#1172125711yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
hypercenterC2×Q8D3×D9
hypocenterC3×C9#488
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
2. #11720935C2semidirect
3. #11720582C2semidirect
4. C9×#835C2semidirect
6. C3⋊C8×Dic9C2semidirect
95. #835D9semidirect
198. C3×C9#488semidirect
282. C9#5056semidirect
319. C3#10588480semidirect

Maximal subgroups

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

subgroupindex
C3×#635092
#117209352
#117205822
C9×#8352
#117202832
C3⋊C8×Dic92
#117202662
#105916143
#105884803