Finite Group Databasebeta

← C3×#10593550 #13167625

Subgroups — C3×#10593550 #13167625

Order 1728.

378 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×C3×C4×#157C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C2×C3×#3777C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5C3×#63781C211yesmaximal, characteristic
6C3×#63747C211yesmaximal, characteristic
7C3×#63747C211yesmaximal, characteristic
8#10593550C311yesmaximal, characteristic, direct product factor
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C3×C4×#157—22no—
17C3×#3777—22no—
18C3×#3777C414yescomplemented
19C2×C4×#157C2×C311yescharacteristic
20C2×#3777C2×C311yescharacteristic
21C2×#3777C2×C311yescharacteristic
22#63781C2×C311yescharacteristic
23#63781C2×C311yescharacteristic
24#63747C2×C311yescharacteristic
25#63747C2×C311yescharacteristic
26C22×C3×C3⋊D3C2311yescharacteristic
27C2×C3×#157—21no—
28C2×C3×#157—21no—
29C2×C3×#157—21no—
30C2×C3×#157—21no—
31C2×C3×#157—21no—
32C2×C3×#157—21no—
33C2×C3×#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157—21no—
37C2×C3×#157—21no—
38C2×C3×#157—21no—
39C2×C3×#157—21no—
40C2×C3×#157—21no—
41C2×C3×#157C2×C412yes—
42C2×C3×#157C2×C412yes—
43C2×C3×#157C2×C412yes—
44C2×C3×#157D412yes—
45C2×C33×C4Q811yescharacteristic
46C2×C3×C3⋊Dic3Q811yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C3×#387—91nomaximal
50C22×#157C22×C311yescharacteristic
51C22×#157C22×C311yescharacteristic
52C22×#157C22×C311yescharacteristic
53C22×#157C22×C311yescharacteristic
54C22×#157C22×C311yescharacteristic
55C22×#157C22×C311yescharacteristic
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×C4×D3—121no—
58C4×#157—22no—
59#3777—22no—
60#3777C3×C414yescomplemented
61C22×C33C2×Q811yescharacteristic
62C2×C3×C3⋊D3C22×C411yescharacteristic
63C2×C3×C3⋊D3C2×D411yescharacteristic
64C2×C3×C3⋊D3#4111yescharacteristic
65C2×C3×C3⋊D3#4111yescharacteristic
66C2×C3×C3⋊D3#4111yescharacteristic
67C2×C3×C3⋊D3#4111yescharacteristic
68C3×#157—22no—
69C3×#157—22no—
70C3×#157—22no—
71C3×#157—22no—
72C3×#157—41no—
73C3×#157—41no—
74C33×C4—21no—
75C3×C3⋊Dic3—21no—
76C2×C3×C4⋊C4—91no—
77C2×C3×C4⋊C4—91no—
78C2×C3×C42—91no—
79C3×#95—91no—
80C3×#95—91no—
81C3×#95—91no—
82C3×#95—91no—
83C22×C3⋊D3C23×C311yescharacteristic
84C22×C3×D3—121no—
85C2×#157—21no—
86C2×#157—21no—
87C2×#157—21no—
88C2×#157—21no—
89C2×#157—21no—
90C2×#157—21no—
91C2×#157—21no—
92C2×#157—21no—
93C2×#157—21no—
94C2×#157—21no—
95C2×#157—21no—
96C2×#157—21no—
97C2×#157—21no—
98C2×#157—21no—
99C2×#157C2×C3×C412yes—
100C2×#157C2×C3×C412yes—
101C2×#157C2×C3×C412yes—
102C2×#157C3×D412yes—
103C2×C32×C4—41no—
104C2×C32×C4—81no—
105C2×C32×C4C3×Q811yescharacteristic
106C2×C3⋊Dic3C3×Q811yescharacteristic
107C4×C3⋊D3—21no—
108C4×C3⋊D3—21no—
109C2×C3×Dic3—121no—
110C3×C4×D3—122no—
111C3×C4×D3—122no—
112#387—91noSylow
113C2×C33C4×Q811yescharacteristic
114C2×C33C22⋊Q811yescharacteristic
115C2×C33#12511yescharacteristic
116C3×C3⋊D3#11711yescharacteristic
117C3×C3⋊D3C4×D411yescharacteristic
118C3×C3⋊D3#12311yescharacteristic
119C3×C3⋊D3#12611yescharacteristic
120C22×C3×C4—91no—
121C22×C3×C4—91no—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C2×C4×D3—121no—
128C3×C4⋊C4—94no—
129C3×C4⋊C4—182no—
130C3×C42—182no—
131C22×C32—41no—
132C22×C32—81no—
133C22×C32C2×C3×Q811yescharacteristic
134C2×C3⋊D3C22×C3×C411yescharacteristic
135C2×C3⋊D3C2×C3×D411yescharacteristic
136C2×C3⋊D3C3×#4111yescharacteristic
137C2×C3⋊D3C3×#4111yescharacteristic
138C2×C3⋊D3C3×#4111yescharacteristic
139C2×C3⋊D3C3×#4111yescharacteristic
140C2×C3×D3—121no—
141C2×C3×D3—121no—
142C2×C3×D3—121no—
143C2×C3×D3—121no—
144C2×C3×D3—121no—
145C2×C3×D3—121no—
146#157—22no—
147#157—22no—
148#157—22no—
149#157—22no—
150#157—41no—
151#157—41no—
152C32×C4—21no—
153C32×C4—42no—
154C32×C4—82no—
155C3⋊Dic3—21no—
156C3×Dic3—122no—
157C2×C4⋊C4—91no—
158C2×C4⋊C4—91no—
159C2×C42—91no—
160#95—91no—
161#95—91no—
162#95—91no—
163#95—91no—
164C33#38711yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C22×D3—121no—
167C2×C3×C4—41no—
168C2×C3×C4—81no—
169C2×C3×C4—92no—
170C2×C3×C4—92no—
171C2×C3×C4—92no—
172C2×C3×C4—92no—
173C2×C3×C4—91no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4PSU(3,𝔽2)11yescharacteristic
191C2×Dic3—121no—
192C4×D3—122no—
193C4×D3—122no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—81no—
198C2×C32—81no—
199C2×C32—81no—
200C2×C32C3×C4×Q811yescharacteristic
201C2×C32C3×C22⋊Q811yescharacteristic
202C2×C32C3×#12511yescharacteristic
203C3⋊D3C3×#11711yescharacteristic
204C3⋊D3C3×C4×D411yescharacteristic
205C3⋊D3C3×#12311yescharacteristic
206C3⋊D3C3×#12611yescharacteristic
207C3×D3—121no—
208C3×D3—121no—
209C3×D3—121no—
210C3×D3—121no—
211C22×C4—91no—
212C22×C4—91no—
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4—91no—
218C4⋊C4—94no—
219C4⋊C4—182no—
220C42—182no—
221C22×C3—41no—
222C22×C3—81no—
223C22×C3—91no—
224C22×C3—91no—
225C22×C3—91no—
226C22×C3—91no—
227C22×C3—91no—
228C22×C3—91no—
229C22×C3C2×PSU(3,𝔽2)11yescharacteristic
230C2×D3—121no—
231C2×D3—121no—
232C2×D3—121no—
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—121no—
236C3×C4—21no—
237C3×C4—42no—
238C3×C4—82no—
239C3×C4—182no—
240C3×C4—182no—
241C3×C4—182no—
242C3×C4—182no—
243C3×C4—181no—
244C3×C4—361no—
245C3×C4—361no—
246Dic3—122no—
247C32—41no—
248C32—81no—
249C32C3×#38711yescharacteristic, complemented
250C23—91no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4—92no—
254C2×C4—92no—
255C2×C4—91no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—181no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4C3×PSU(3,𝔽2)11yescharacteristic
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—81no—
277C2×C3—81no—
278C2×C3—81no—
279C2×C3—91no—
280C2×C3—91no—
281C2×C3—91no—
282C2×C3—91no—
283C2×C3#6424311yescharacteristic
284C2×C3C4×PSU(3,𝔽2)11yescharacteristic
285C2×C3C22⋊PSU(3,𝔽2)11yescharacteristic
286D3—121no—
287D3—121no—
288D3—121no—
289D3—121no—
290C22—91no—
291C22—91no—
292C22—91no—
293C22—91no—
294C22—91no—
295C22—91no—
296C22C2×C3×PSU(3,𝔽2)11yescharacteristic
297C4—21no—
298C4—182no—
299C4—182no—
300C4—182no—
301C4—182no—
302C4—181no—
303C4—361no—
304C4—361no—
305C3—41no—
306C3—81no—
307C3#1059355011yescharacteristic, direct product factor
308C2—91no—
309C2—91no—
310C2—91no—
311C2—91no—
312C2C3×#6424311yescharacteristic
313C2C3×C4×PSU(3,𝔽2)11yescharacteristic
314C2C3×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×PSU(3,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC22×C33C2×Q8
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4PSU(3,𝔽2)
hypocenterC32C3×#387
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
8. #10593550C3direct (N × H)
18. C3×#3777C4semidirect
60. #3777C3×C4semidirect
164. C33#387semidirect
249. C32C3×#387semidirect
307. C3#10593550direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C2×C3×#37772
C3×#637812
C3×#637472
#105935503
C3×#3879