Finite Group Databasebeta

← #13165803

Subgroups — #13165803

Order 1728.

378 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 80 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×#88945C211yesmaximal, characteristic
2C2×C3×#3777C211yesmaximal, characteristic
3C2×Dic3×#157C211yesmaximal, characteristic
4C3⋊#63747C211yesmaximal, characteristic
5C3⋊#63747C211yesmaximal, characteristic
6#11722722C211yesmaximal, characteristic
7#11722722C211yesmaximal, characteristic
8#10593550—31nomaximal
9C22×C3⋊#157C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2211yescharacteristic
14C22×C3×#157C2211yescharacteristic
15C2×Dic3×C3⋊D3C2211yescharacteristic
16#88945C414yescomplemented
17C3×#3777—22no—
18Dic3×#157—22no—
19C2×C4×#157—31no—
20C2×#3777—31no—
21C2×#3777D311yescharacteristic
22#63781—31no—
23#63781—31no—
24#63747—31no—
25#63747—31no—
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⋊#157C2×C412yes—
35C2×C3⋊#157C2×C412yes—
36C2×C3×#157—21no—
37C2×C3×#157—21no—
38C2×C3×#157—21no—
39C2×C3×#157—21no—
40C2×C3×#157—21no—
41C2×C3×#157—21no—
42C2×C3×#157—21no—
43C2×C3×#157C2×C412yes—
44C2×C3×#157D412yes—
45C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
46C2×C32×Dic3Q811yescharacteristic
47Dic3×C3⋊D3—21no—
48Dic3×C3⋊D3—21no—
49#4978—91nomaximal
50C22×#157—31no—
51C22×#157—31no—
52C22×#157—31no—
53C22×#157C2×D311yescharacteristic
54C22×#157C2×D311yescharacteristic
55C22×#157C2×D311yescharacteristic
56C2×C4×C3⋊D3—31no—
57C2×D3×Dic3—121no—
58C4×#157—62no—
59#3777—22no—
60#3777—34no—
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—41no—
71C3×#157—22no—
72C3×#157—22no—
73C3×#157—41no—
74C3⋊(C3⋊Dic3)—21no—
75C32×Dic3—21no—
76C2×C3×C4⋊C4—91no—
77C2×Dic3⋊C4—91no—
78C2×C4×Dic3—91no—
79C3⋊#95—91no—
80C3⋊#95—91no—
81C3⋊#95—91no—
82C3⋊#95—91no—
83C22×C3⋊D3C22×D311yescharacteristic
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—32no—
93C2×#157—32no—
94C2×#157—61no—
95C2×#157—61no—
96C2×#157—61no—
97C2×#157—61no—
98C2×#157—61no—
99C2×#157—61no—
100C2×#157—61no—
101C2×#157C4×D312yes—
102C2×#157#6712yes—
103C2×C32×C4—31no—
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—121no—
106C2×C3⋊Dic3—241no—
107C4×C3⋊D3—61no—
108C4×C3⋊D3—61no—
109C2×C3×Dic3—41no—
110D3×Dic3—122no—
111D3×Dic3—122no—
112#387—271noSylow
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×Dic3—91no—
124C22×Dic3—91no—
125C22×Dic3—91no—
126C22×Dic3—91no—
127C2×C4×D3—361no—
128C3×C4⋊C4—182no—
129Dic3⋊C4—94no—
130C4×Dic3—182no—
131C22×C32—41no—
132C22×C32—81no—
133C22×C32D3×Q811yescharacteristic
134C2×C3⋊D3C2×C4×D311yescharacteristic
135C2×C3⋊D3#23511yescharacteristic
136C2×C3⋊D3#23711yescharacteristic
137C2×C3⋊D3#23711yescharacteristic
138C2×C3⋊D3#23911yescharacteristic
139C2×C3⋊D3C2×#6711yescharacteristic
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—41no—
149#157—62no—
150#157—62no—
151#157—121no—
152C32×C4—61no—
153C3⋊Dic3—61no—
154C3⋊Dic3—122no—
155C3⋊Dic3—242no—
156C3×Dic3—42no—
157C2×C4⋊C4—91no—
158C2×C4⋊C4—271no—
159C2×C42—271no—
160#95—271no—
161#95—271no—
162#95—271no—
163#95—271no—
164C33#38711yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C22×D3—121no—
167C2×C3×C4—92no—
168C2×C3×C4—92no—
169C2×C3×C4—121no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C2×C3×C4—181no—
173C2×C3×C4—181no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177C2×Dic3—92no—
178C2×Dic3—92no—
179C2×Dic3—91no—
180C2×Dic3—181no—
181C2×Dic3—181no—
182C2×Dic3—181no—
183C2×Dic3—181no—
184C2×Dic3—181no—
185C2×Dic3—181no—
186C2×Dic3—181no—
187C2×Dic3—181no—
188C2×Dic3—181no—
189C2×Dic3—361no—
190C2×Dic3—721no—
191C2×Dic3PSU(3,𝔽2)11yescharacteristic
192C4×D3—362no—
193C4×D3—362no—
194C2×C32—41no—
195C2×C32—41no—
196C2×C32—41no—
197C2×C32—81no—
198C2×C32—81no—
199C2×C32—81no—
200C2×C32#88711yescharacteristic
201C2×C32#88911yescharacteristic
202C2×C32#94611yescharacteristic
203C3⋊D3#89211yescharacteristic
204C3⋊D3#89811yescharacteristic
205C3⋊D3C4×#6711yescharacteristic
206C3⋊D3#93511yescharacteristic
207C3×D3—121no—
208C3×D3—121no—
209C3×D3—121no—
210C3×D3—121no—
211C22×C4—91no—
212C22×C4—91no—
213C22×C4—91no—
214C22×C4—271no—
215C22×C4—271no—
216C22×C4—271no—
217C22×C4—271no—
218C4⋊C4—182no—
219C4⋊C4—274no—
220C42—542no—
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—122no—
237C3×C4—182no—
238C3×C4—182no—
239C3×C4—361no—
240Dic3—21no—
241Dic3—182no—
242Dic3—182no—
243Dic3—181no—
244Dic3—362no—
245Dic3—361no—
246Dic3—722no—
247C32—41no—
248C32—81no—
249C32#497811yescharacteristic, complemented
250C23—91no—
251C2×C4—31no—
252C2×C4—92no—
253C2×C4—92no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—272no—
262C2×C4—272no—
263C2×C4—271no—
264C2×C4—541no—
265C2×C4—541no—
266C2×C4—541no—
267C2×C4—541no—
268C2×C4—541no—
269C2×C4—541no—
270C2×C4—541no—
271C2×C4—541no—
272C2×C4—541no—
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—
296C22D3×PSU(3,𝔽2)11yescharacteristic
297C4—61no—
298C4—182no—
299C4—182no—
300C4—361no—
301C4—542no—
302C4—542no—
303C4—541no—
304C4—1081no—
305C3—41no—
306C3—81no—
307C3#1059355011yescharacteristic, complemented
308C2—91no—
309C2—91no—
310C2—91no—
311C2—91no—
312C2#1172414811yescharacteristic
313C2#1172414911yescharacteristic
314C2#1172415211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×PSU(3,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×Q8
prime radical, cosocleC22×C3×C3⋊D3C23
hypocenterC33#387
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
16. #88945C4semidirect
164. C33#387semidirect
249. C32#4978semidirect
307. C3#10593550semidirect

Maximal subgroups

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

subgroupindex
C2×#889452
C2×C3×#37772
C2×Dic3×#1572
C3⋊#637472
#117227222
#105935503
#49789