Finite Group Databasebeta

← #13140331

Subgroups — #13140331

Order 1728.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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
1D4×C3⋊Dic9C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic
4C3×#63392C211yesmaximal, characteristic
5#11720278C211yesmaximal, characteristic, complemented
6#11720273C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10591479—31nomaximal
9#10588443—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12#88551—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21D4×C3⋊Dic3—31no—
22#63872—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63575—31no—
26#63573—31no—
27#63572—31no—
28D4×Dic9—31no—
29D4×Dic9—61no—
30#63452—31no—
31C9×#102—31no—
32#63392D311yescharacteristic
33#63377—31no—
34Dic9⋊C8—31no—
35#63366—31no—
36C23×C3×C9—21no—
37C3×D4×C9—21no—
38C3×D4×C9Q812yes—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9—42no—
42C2×C3⋊Dic9D411yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C9×C3⋊C8—21no—
47C3×C9⋊C8—21no—
48#4766—91no—
49#4763—91no—
50C2×C32×D4C2×D311yescharacteristic
51C22×C3⋊Dic3—61no—
52#3757—61no—
53C4⋊(C3⋊Dic3)—31no—
54C4×C3⋊Dic3—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8C2×D311yescharacteristic
59Dic3⋊Dic3—61no—
60Dic3⋊Dic3—61no—
61C2×D4×C9—21no—
62C2×D4×C9C2×D311yescharacteristic
63C22×Dic9—61no—
64C22×Dic9—62no—
65C2×C8×C9—31no—
66C9×C4⋊C4—31no—
67#3676—61no—
68#3676—62no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9—61no—
72C4⋊Dic9C2×D311yescharacteristic
73Dic9⋊C4—61no—
74C4×Dic9—31no—
75C4×Dic9—61no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9—22no—
78C22×C3×C9—41no—
79C22×C3×C9—41no—
80C22×C3×C9C2×D411yescharacteristic
81C3×C4×C9C2×Q811yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C3⋊Dic9—21no—
84C3⋊Dic9—21no—
85C3⋊Dic9—41no—
86C9×Dic3—41no—
87C3×Dic9—41no—
88D4×Dic3—91no—
89D4×Dic3—91no—
90D4×Dic3—181no—
91#890—91no—
92#890—91no—
93C3×#102—31no—
94C3×#102—91no—
95#832—31no—
96#832D911yescharacteristic
97#818—91no—
98#817—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#808—91no—
102#807—91no—
103C23×C32—21no—
104C32×D4—21no—
105C32×D4Dic612yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—122no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—61no—
113C23×C9—22no—
114C23×C9—21no—
115C3×C3⋊C8—21no—
116C3×C3⋊C8—61no—
117D4×C9—22no—
118D4×C9—21no—
119D4×C9—22no—
120D4×C9Dic612yes—
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—
131C2×Dic9—124no—
132C2×Dic9—122no—
133C8×C9—61no—
134C9⋊C8—21no—
135#478—271noSylow
136C2×C3×C9—42no—
137C2×C3×C9C22⋊Q811yescharacteristic
138C2×C3×C9#13511yescharacteristic
139C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
140C2×C3×D4—21no—
141C2×C3×D4C2×D911yescharacteristic
142C2×C3×D4D3211yescharacteristic
143C22×Dic3—181no—
144C22×Dic3—181no—
145C22×Dic3—182no—
146C2×C3×C8—31no—
147C2×C3×C8—91no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—91no—
150#217—181no—
151#217—181no—
152#217—182no—
153C4⋊Dic3—31no—
154C4⋊Dic3—91no—
155C4⋊Dic3—91no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158C4⋊Dic3—181no—
159C4⋊Dic3C2×D911yescharacteristic
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162C4×Dic3—91no—
163C4×Dic3—91no—
164C4×Dic3—181no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8C2×D911yescharacteristic
167C22×C32—22no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32D3×D411yescharacteristic
171C32×C4C2×Dic611yescharacteristic
172C32×C4#23711yescharacteristic
173C3⋊Dic3—61no—
174C3⋊Dic3—61no—
175C3⋊Dic3—121no—
176C3×Dic3—41no—
177C3×Dic3—121no—
178C22×C9—24no—
179C22×C9—22no—
180C22×C9—21no—
181C22×C9—42no—
182C22×C9—42no—
183C22×C9—41no—
184C22×C9—41no—
185C22×C9D3×D411yescharacteristic
186C4×C9—21no—
187C4×C9—21no—
188C4×C9—121no—
189C4×C9C2×Dic611yescharacteristic
190C4×C9#23711yescharacteristic
191Dic9—41no—
192Dic9—62no—
193Dic9—61no—
194Dic9—61no—
195Dic9—121no—
196Dic9—122no—
197Dic9—121no—
198#125—271no—
199C4×D4—271no—
200#107—271no—
201#106—271no—
202C4⋊C8—271no—
203#102—31no—
204#102—91no—
205C3×C9#47811yescharacteristic, complemented, Sylow
206C23×C3—22no—
207C23×C3—21no—
208C23×C3—21no—
209C3×D4—22no—
210C3×D4—21no—
211C3×D4—21no—
212C3×D4—22no—
213C3×D4Dic1812yes—
214C3×D4#62712yes—
215C2×C3×C4—21no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—61no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—91no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—182no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—362no—
232C2×Dic3—364no—
233C2×Dic3—362no—
234C3×C8—61no—
235C3×C8—181no—
236C3⋊C8—21no—
237C3⋊C8—61no—
238C2×C32—42no—
239C2×C32#87811yescharacteristic
240C2×C32#91111yescharacteristic
241C2×C32#91611yescharacteristic
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9—44no—
246C2×C9—42no—
247C2×C9#87811yescharacteristic
248C2×C9#91211yescharacteristic
249C2×C9#91411yescharacteristic
250C2×D4D3×D911yescharacteristic
251C22×C4—541no—
252C2×C8—31no—
253C2×C8—91no—
254C4⋊C4—31no—
255C4⋊C4—91no—
256C4⋊C4—271no—
257C4⋊C4—541no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C22×C3—24no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3D4×D911yescharacteristic
272C22×C3#381111yescharacteristic
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—121no—
276C3×C4—361no—
277C3×C4C2×Dic1811yescharacteristic
278C3×C4#369911yescharacteristic
279C3×C4#380511yescharacteristic
280C3×C4C2×#62711yescharacteristic
281Dic3—41no—
282Dic3—121no—
283Dic3—182no—
284Dic3—181no—
285Dic3—181no—
286Dic3—181no—
287Dic3—181no—
288Dic3—361no—
289Dic3—362no—
290Dic3—361no—
291Dic3—361no—
292C32#476311yescharacteristic
293C9—21no—
294C9#476611yescharacteristic, complemented
295C23—21no—
296D4—21no—
297D4#618912yes—
298C2×C4—61no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4—541no—
303C2×C4—1082no—
304C2×C4C2×D3×D911yescharacteristic
305C8—61no—
306C8—181no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—44no—
311C2×C3—42no—
312C2×C3—42no—
313C2×C3#6344011yescharacteristic
314C2×C3#6347311yescharacteristic
315C2×C3#6347811yescharacteristic
316C2×C3#6393611yescharacteristic
317C2×C3#6393711yescharacteristic
318C2×C3#6398211yescharacteristic
319C22—22no—
320C22—41no—
321C22—41no—
322C22#8866811yescharacteristic
323C4—121no—
324C4—361no—
325C4—541no—
326C4—541no—
327C4—1081no—
328C4C2×#618911yescharacteristic
329C4#8866311yescharacteristic
330C3—21no—
331C3#1058844311yescharacteristic, complemented
332C3#1059147911yescharacteristic
333C2—42no—
334C2#1172117911yescharacteristic
335C2#1172118611yescharacteristic
336C2#1172126711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#478
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720937C2semidirect
5. #11720278C2semidirect
6. #11720273C2semidirect
7. #11720262C2semidirect
205. C3×C9#478semidirect
294. C9#4766semidirect
331. C3#10588443semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
#117209372
C9×#8322
C3×#633922
#117202782
#117202732
#117202622
#105914793
#105884433