Finite Group Databasebeta

← #13155103

Subgroups — #13155103

Order 1728.

296 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 29 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#11723670C211yesmaximal, characteristic
2C3×#63780C211yesmaximal, characteristic, complemented
3#11722074C211yesmaximal, characteristic, complemented
4#10590059—31nomaximal
5C2×C3×#634C2211yescharacteristic
6C2×#6295C411yescharacteristic
7C3×#3792—21no—
8C3×#3752—21no—
9C3×#3748—21no—
10#88814—21no—
11#88813C411yescharacteristic
12#63913—31no—
13#63907—61no—
14#63780D311yescharacteristic, complemented
15#63561—31no—
16#63560—61no—
17C22×C3×C3⋊D3—21no—
18C2×C33×C4D411yescharacteristic
19C2×C3×C3⋊Dic3—21no—
20C2×C3×C3⋊Dic3C2×C411yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C3×#634—41no—
23#6295—41no—
24C3×#622—41no—
25C3×C3⋊(C3⋊C8)—41no—
26#4477—91nomaximal
27C2×#634C2×D311yescharacteristic
28C2×C3×Dic6—61no—
29C2×C3×Dic6—61no—
30C2×#626—61no—
31C2×#625—31no—
32#3792—21no—
33#3752—21no—
34#3748—21no—
35C3×(C2×D3)⋊C4—61no—
36C3×(C2×D3)⋊C4—61no—
37C3×(C2×D3)⋊C4—121no—
38C3×#208—61no—
39C3×#208—61no—
40C3×#208—121no—
41#3722—61no—
42(C2×D3)⋊Dic3—61no—
43(C2×D3)⋊Dic3—61no—
44Dic32—31no—
45Dic32—61no—
46C22×C33C22⋊C411yescharacteristic
47C2×C3×C3⋊D3—41no—
48C2×C3×C3⋊D3—41no—
49C2×C3×C3⋊D3—41no—
50C33×C4—21no—
51C3×C3⋊Dic3—21no—
52C3×C3⋊Dic3—21no—
53C3×C3⋊Dic3—41no—
54#947—91no—
55#885—181no—
56C3×#101—91no—
57#836—91no—
58#824—181no—
59C22×C3⋊D3—21no—
60C22×C3×D3—61no—
61C22×C3×D3—61no—
62C22×C3×D3—121no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4—41no—
66C2×C32×C4—41no—
67C2×C32×C4D1211yescharacteristic
68C2×C3⋊Dic3—21no—
69C2×C3⋊Dic3C4×D311yescharacteristic
70C2×C3⋊Dic3#6711yescharacteristic
71#634—41no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—121no—
79C3×Dic6—122no—
80C3×Dic6—121no—
81C3×Dic6—121no—
82#626—122no—
83#625—121no—
84#622—41no—
85C3⋊(C3⋊C8)—41no—
86C3×C3⋊C8—121no—
87C3×C3⋊C8—121no—
88C3×C3⋊C8—122no—
89#355—271noSylow
90C2×C33—21no—
91C2×C33#9911yescharacteristic
92C3×C3⋊D3—81no—
93C2×C3×Q8—91no—
94C2×#67—181no—
95C2×D12—91no—
96C2×Dic6—61no—
97C2×Dic6—61no—
98C3×M16—181no—
99C3×M16—181no—
100C3×C22⋊C4—181no—
101#217—181no—
102(C2×D3)⋊C4—61no—
103(C2×D3)⋊C4—61no—
104(C2×D3)⋊C4—121no—
105(C2×D3)⋊C4—181no—
106(C2×D3)⋊C4—181no—
107C4×Dic3—91no—
108C4×Dic3—181no—
109#208—61no—
110#208—61no—
111#208—121no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—41no—
115C22×C32—41no—
116C22×C32(C2×D3)⋊C411yescharacteristic
117C2×C3⋊D3—41no—
118C2×C3⋊D3—41no—
119C2×C3⋊D3—41no—
120C2×C3×D3—121no—
121C2×C3×D3—121no—
122C2×C3×D3—121no—
123C2×C3×D3—121no—
124C2×C3×D3—121no—
125C2×C3×D3—122no—
126C2×C3×D3—121no—
127C2×C3×D3—241no—
128C2×C3×D3—241no—
129C32×C4—21no—
130C32×C4—21no—
131C32×C4—21no—
132C32×C4—41no—
133C32×C4—41no—
134C32×C4—41no—
135C32×C4—81no—
136C3⋊Dic3—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3—41no—
139C3×Dic3—62no—
140C3×Dic3—121no—
141C3×Dic3—121no—
142C3×Dic3—121no—
143C3×Dic3—121no—
144C3×Dic3—121no—
145C3×Dic3—241no—
146#124—271no—
147#101—91no—
148#101—271no—
149C33#35511yescharacteristic, complemented, Sylow
150C23×C3—181no—
151C22×D3—61no—
152C22×D3—61no—
153C22×D3—121no—
154C3×Q8—361no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—181no—
162C2×C3×C4GO+(4,𝔽2)11yescharacteristic
163#67—362no—
164C2×Dic3—61no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—121no—
171D12—361no—
172Dic6—122no—
173Dic6—121no—
174Dic6—121no—
175C3×C8—361no—
176C3×C8—361no—
177C3⋊C8—121no—
178C3⋊C8—121no—
179C3⋊C8—122no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—41no—
186C2×C32—41no—
187C2×C32—41no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—81no—
191C2×C32#80611yescharacteristic
192C3⋊D3—81no—
193C3×D3—241no—
194C3×D3—242no—
195C3×D3—241no—
196C2×Q8—91no—
197C2×D4—271no—
198M16—181no—
199M16—181no—
200C22⋊C4—181no—
201C22⋊C4—541no—
202C42—271no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—361no—
208C22×C3—361no—
209C22×C3—361no—
210C22×C3D32⋊C411yescharacteristic
211C2×D3—121no—
212C2×D3—121no—
213C2×D3—121no—
214C2×D3—121no—
215C2×D3—121no—
216C2×D3—122no—
217C2×D3—121no—
218C2×D3—241no—
219C2×D3—241no—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—81no—
227C3×C4—181no—
228C3×C4—181no—
229C3×C4—361no—
230Dic3—62no—
231Dic3—121no—
232Dic3—121no—
233Dic3—121no—
234Dic3—121no—
235Dic3—121no—
236Dic3—241no—
237C32—21no—
238C32—21no—
239C32—41no—
240C32—41no—
241C32#447711yescharacteristic, complemented
242C23—181no—
243Q8—361no—
244D4—1081no—
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—181no—
248C2×C4#157⋊D311yescharacteristic
249C8—361no—
250C8—361no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—41no—
257C2×C3—41no—
258C2×C3—41no—
259C2×C3—41no—
260C2×C3—41no—
261C2×C3—81no—
262C2×C3—721no—
263C2×C3#6373811yescharacteristic
264D3—241no—
265D3—242no—
266D3—241no—
267C22—361no—
268C22—361no—
269C22—361no—
270C22#8893911yescharacteristic
271C4—21no—
272C4—181no—
273C4—181no—
274C4—361no—
275C3—21no—
276C3—21no—
277C3—41no—
278C3—41no—
279C3#1059005911yescharacteristic, complemented
280C2—21no—
281C2—721no—
282C2#1172279211yescharacteristic
subgroupG/H
centerC2#11722792
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC2×C4#157⋊D3
FittingC2×C33×C4D4
socleC2×C33#99
prime radicalC2×#6295C4
cosocleC2×C3×#634C22
hypocenterC33#355
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63780C2semidirect
3. #11722074C2semidirect
14. #63780D3semidirect
149. C33#355semidirect
241. C32#4477semidirect
279. C3#10590059semidirect

Maximal subgroups

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

subgroupindex
#117236702
C3×#637802
#117220742
#105900593
#44779