Finite Group Databasebeta

← #13180643

Subgroups — #13180643

Order 1728.

482 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 112 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#11724369C211yesmaximal, characteristic
2#11724368C212yesmaximal
3C3×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic
4C2×Dic3⋊#157C211yesmaximal, characteristic
5#11724261C212yesmaximal
6#11724154C212yesmaximal, complemented
7#11724153C212yesmaximal, complemented
8#11724152C212yesmaximal, complemented
9#11724152C212yesmaximal, complemented
10#10596543—31nomaximal
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C2×C3⋊PSU(3,𝔽2)C2212yes—
14C2×C3⋊PSU(3,𝔽2)C2211yescharacteristic
15C2×C3×PSU(3,𝔽2)C2211yescharacteristic
16C2×Dic3×C3⋊D3C2211yescharacteristic
17#88994C2212yes—
18C3×C22⋊#157C2212yes—
19#88946C2212yes—
20#88946C2211yescharacteristic
21#88945C2212yes—
22#88945C2212yes—
23#88945C2212yes—
24C3×#3777C2211yescharacteristic
25C3×#3777C2212yes—
26Dic3⋊#157C2212yes—
27Dic3⋊#157C2212yes—
28Dic3⋊#157C2212yes—
29Dic3⋊#157C2212yes—
30Dic3×#157C2212yes—
31Dic3×#157C2212yes—
32#64286—32no—
33C2×C4⋊#157—31no—
34C22⋊PSU(3,𝔽2)—32no—
35C22⋊PSU(3,𝔽2)—31no—
36C22⋊PSU(3,𝔽2)D311yescharacteristic
37C4⋊PSU(3,𝔽2)—32no—
38C4⋊PSU(3,𝔽2)—32no—
39#64243—32no—
40#64243—32no—
41C22×C3×C3⋊D3C2311yescharacteristic
42C2×C3⋊#157—41no—
43C2×C3⋊#157C2312yes—
44C2×C3⋊#157C2312yes—
45C2×C3⋊#157C2312yes—
46C2×C3×#157—41no—
47C2×C3×#157C2312yes—
48C2×C3×#157C2311yescharacteristic
49C2×C3×#157C2311yescharacteristic
50C2×C3×#157C2312yes—
51C3⋊PSU(3,𝔽2)—42no—
52C3⋊PSU(3,𝔽2)—41no—
53C3×PSU(3,𝔽2)—41no—
54C2×C3⋊(C3⋊Dic3)Q811yescharacteristic, complemented
55C2×C32×Dic3Q811yescharacteristic, complemented
56Dic3×C3⋊D3Q812yescomplemented
57Dic3×C3⋊D3C2312yes—
58#5623—91nomaximal
59C22×#157—31no—
60C22×#157C2×D311yescharacteristic
61C2×PSU(3,𝔽2)—31no—
62C2×PSU(3,𝔽2)—32no—
63C2×PSU(3,𝔽2)C2×D311yescharacteristic
64C2×C4×C3⋊D3—31no—
65C2×D3×Dic3—121no—
66C22⋊#157—32no—
67C22⋊#157C2×D312yes—
68C4⋊#157—32no—
69C4⋊#157—32no—
70C4⋊#157—32no—
71C4⋊#157—32no—
72C4×#157—32no—
73C4×#157—32no—
74#3777—32no—
75#3777—32no—
76#3777—32no—
77#3777—31no—
78#3777—32no—
79#3777C2×D311yescharacteristic
80#3777C2×D312yes—
81C22×C33C2×Q811yescharacteristic
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3—21no—
84C2×C3×C3⋊D3C2×Q811yescharacteristic
85C2×C3×C3⋊D3C2411yescharacteristic
86C3⋊#157—22no—
87C3⋊#157—22no—
88C3⋊#157—22no—
89C3×#157—22no—
90C3×#157—21no—
91C3×#157—21no—
92C3×#157—22no—
93C3⋊(C3⋊Dic3)C2×Q812yes—
94C32×Dic3C2×Q812yes—
95C3×C22⋊Q8—91no—
96#944—92no—
97#924—91no—
98C2×Dic3⋊C4—91no—
99#889—92no—
100#889—92no—
101#888—92no—
102#878—92no—
103#877—92no—
104C22×C3⋊D3C22×D311yescharacteristic
105C22×C3×D3—121no—
106C2×#157—32no—
107C2×#157—32no—
108C2×#157—32no—
109C2×#157—41no—
110C2×#157—121no—
111C2×#157C22×D312yes—
112C2×#157C22×D311yescharacteristic
113C2×#157C22×D311yescharacteristic
114C2×#157C22×D312yes—
115PSU(3,𝔽2)—41no—
116PSU(3,𝔽2)—122no—
117PSU(3,𝔽2)—121no—
118C2×C32×C4—31no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—121no—
121C2×C3⋊Dic3—241no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C2×C3×Dic3—41no—
125D3×Dic3—122no—
126D3×Dic3—122no—
127#544—271noSylow
128C2×C33—21no—
129C2×C33C22×Q811yescharacteristic
130C3×C3⋊D3—21no—
131C3×C3⋊D3H(5,𝔽2)11yescharacteristic
132C3×C3⋊D32−1+411yescharacteristic
133C2×C3×Q8—91no—
134C22×C3×C4—91no—
135C22×Dic3—91no—
136C22×Dic3—91no—
137C2×C4×D3—361no—
138C2×Dic6—92no—
139C2×Dic6—91no—
140C3×C4⋊C4—92no—
141C3×C4⋊C4—91no—
142C3×C22⋊C4—92no—
143#217—92no—
144C4⋊Dic3—92no—
145C4⋊Dic3—91no—
146Dic3⋊C4—92no—
147Dic3⋊C4—92no—
148Dic3⋊C4—92no—
149Dic3⋊C4—92no—
150Dic3⋊C4—92no—
151Dic3⋊C4—92no—
152Dic3⋊C4—92no—
153C4×Dic3—92no—
154C4×Dic3—92no—
155C22×C32—41no—
156C22×C32—81no—
157C22×C32D3×Q811yescharacteristic
158C2×C3⋊D3—21no—
159C2×C3⋊D3—21no—
160C2×C3⋊D3D3×Q811yescharacteristic
161C2×C3⋊D3C23×D311yescharacteristic
162C2×C3×D3—121no—
163C2×C3×D3—121no—
164C2×C3×D3—122no—
165C2×C3×D3—122no—
166#157—22no—
167#157—21no—
168#157—21no—
169#157—22no—
170#157—62no—
171#157—62no—
172#157—62no—
173C32×C4—32no—
174C3⋊Dic3—32no—
175C3⋊Dic3—122no—
176C3⋊Dic3—242no—
177C3×Dic3—42no—
178C4⋊Q8—272no—
179C4⋊Q8—272no—
180#125—272no—
181#125—272no—
182C22⋊Q8—91no—
183C22⋊Q8—272no—
184C22⋊Q8—271no—
185#117—272no—
186C2×C4⋊C4—271no—
187C33#54411yescharacteristic, complemented, Sylow
188C23×C3—91no—
189C22×D3—121no—
190C3×Q8—361no—
191C2×C3×C4—92no—
192C2×C3×C4—92no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—121no—
196C2×C3×C4—361no—
197C2×Dic3—91no—
198C2×Dic3—92no—
199C2×Dic3—92no—
200C2×Dic3—92no—
201C2×Dic3—92no—
202C2×Dic3—92no—
203C2×Dic3—361no—
204C2×Dic3—361no—
205C2×Dic3—721no—
206C2×Dic3PSU(3,𝔽2)11yescharacteristic, complemented
207C4×D3—362no—
208C4×D3—362no—
209Dic6—361no—
210Dic6—362no—
211C2×C32—21no—
212C2×C32—42no—
213C2×C32—41no—
214C2×C32—82no—
215C2×C32—81no—
216C2×C32C2×D3×Q811yescharacteristic
217C3⋊D3—21no—
218C3⋊D3#100411yescharacteristic
219C3⋊D3#101011yescharacteristic
220C3×D3—121no—
221C3×D3—121no—
222C3×D3—122no—
223C2×Q8—91no—
224C2×Q8—272no—
225C2×Q8—271no—
226C22×C4—91no—
227C22×C4—271no—
228C22×C4—271no—
229C4⋊C4—91no—
230C4⋊C4—92no—
231C4⋊C4—272no—
232C4⋊C4—272no—
233C4⋊C4—272no—
234C4⋊C4—272no—
235C4⋊C4—272no—
236C4⋊C4—271no—
237C4⋊C4—272no—
238C4⋊C4—272no—
239C4⋊C4—272no—
240C22⋊C4—92no—
241C22⋊C4—272no—
242C42—272no—
243C42—272no—
244C22×C3—41no—
245C22×C3—81no—
246C22×C3—91no—
247C22×C3—91no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C2×PSU(3,𝔽2)11yescharacteristic
251C2×D3—121no—
252C2×D3—121no—
253C2×D3—122no—
254C2×D3—122no—
255C3×C4—122no—
256C3×C4—181no—
257C3×C4—182no—
258C3×C4—181no—
259C3×C4—182no—
260Dic3—92no—
261Dic3—182no—
262Dic3—182no—
263Dic3—182no—
264Dic3—362no—
265Dic3—722no—
266Dic3C2×PSU(3,𝔽2)12yes—
267C32—41no—
268C32—81no—
269C32#562311yescharacteristic, complemented
270C23—91no—
271Q8—361no—
272Q8—1081no—
273Q8—1082no—
274C2×C4—31no—
275C2×C4—92no—
276C2×C4—92no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—272no—
280C2×C4—272no—
281C2×C4—271no—
282C2×C4—272no—
283C2×C4—272no—
284C2×C4—272no—
285C2×C4—361no—
286C2×C4—1081no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—41no—
290C2×C3—82no—
291C2×C3—81no—
292C2×C3—91no—
293C2×C3—91no—
294C2×C3—181no—
295C2×C3C22×PSU(3,𝔽2)11yescharacteristic
296D3—121no—
297D3—121no—
298D3—122no—
299C22—91no—
300C22—91no—
301C22—181no—
302C22—181no—
303C22D3×PSU(3,𝔽2)11yescharacteristic
304C4—32no—
305C4—181no—
306C4—181no—
307C4—182no—
308C4—182no—
309C4—272no—
310C4—542no—
311C4—542no—
312C4—542no—
313C3—41no—
314C3—81no—
315C3#1059654311yescharacteristic, complemented
316C2—21no—
317C2—91no—
318C2—91no—
319C2—181no—
320C2C2×D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×PSU(3,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FittingC22×C33C2×Q8
socleC2×C33C22×Q8
prime radicalC22×C3×C3⋊D3C23
hypercenterC22D3×PSU(3,𝔽2)
hypocenterC33#544
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
6. #11724154C2semidirect
7. #11724153C2semidirect
8. #11724152C2semidirect
9. #11724152C2semidirect
54. C2×C3⋊(C3⋊Dic3)Q8semidirect
55. C2×C32×Dic3Q8semidirect
56. Dic3×C3⋊D3Q8semidirect
187. C33#544semidirect
206. C2×Dic3PSU(3,𝔽2)semidirect
269. C32#5623semidirect
315. C3#10596543semidirect

Maximal subgroups

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

subgroupindex
#117243692
#117243682
C3×C22⋊PSU(3,𝔽2)2
C2×Dic3⋊#1572
#117242612
#117241542
#117241532
#117241522
#105965433
#56239