Finite Group Databasebeta

← #13155078

Subgroups — #13155078

Order 1728.

332 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 41 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#11723671C211yesmaximal, characteristic
2C3×#63781C211yesmaximal, characteristic
3#11722072C211yesmaximal, characteristic
4#10590034—31nomaximal
5C22×C3×#157—21no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×#6296C411yescharacteristic, complemented
8C3×#3750—21no—
9C2×C3×C3⋊(C3⋊C8)—21no—
10#88815C411yescharacteristic, complemented
11#88814—21no—
12#63865—31no—
13#63851—61no—
14#63781D311yescharacteristic
15#63557—31no—
16(C2×D3)⋊(C3⋊C8)—61no—
17C22×C3×C3⋊D3D411yescharacteristic
18C2×C3×#157—22no—
19C2×C3×#157—41no—
20C2×C3×#157—41no—
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C3×C4×C3⋊D3—41no—
26#6296—41no—
27C3×C3⋊(C3⋊C8)—41no—
28#4452—91nomaximal
29C22×#157—21no—
30C2×C4×C3⋊D3C2×D311yescharacteristic
31C2×C3×C4×D3—61no—
32C2×C3×C4×D3—61no—
33C2×#627—31no—
34C2×#627—61no—
35#3750—21no—
36C2×C3⋊(C3⋊C8)—21no—
37C3×Dic3⋊C4—61no—
38C3×Dic3⋊C4—61no—
39C3×Dic3⋊C4—121no—
40C2×C3×C3⋊C8—61no—
41C2×C3×C3⋊C8—61no—
42C2×C3×C3⋊C8—121no—
43#3724—31no—
44Dic3⋊Dic3—61no—
45#3722—61no—
46(C2×D3)⋊Dic3—61no—
47(C2×D3)⋊Dic3—61no—
48C22×C33C22⋊C411yescharacteristic
49C2×C3×C3⋊D3—21no—
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3SD1611yescharacteristic
52C2×C3×C3⋊D3Q1611yescharacteristic
53C3×#157—42no—
54C33×C4—21no—
55C3×C3⋊Dic3—21no—
56C3×C3⋊Dic3—41no—
57C3×C3⋊Dic3—41no—
58#924—91no—
59#896—181no—
60C3×#95—91no—
61#830—91no—
62(C2×D3)⋊C8—181no—
63C22×C3⋊D3#6711yescharacteristic
64C22×C3×D3—61no—
65C22×C3×D3—61no—
66C2×#157—22no—
67C2×#157—41no—
68C2×#157—41no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4D1211yescharacteristic
74C2×C3⋊Dic3—21no—
75C2×C3⋊Dic3—21no—
76C2×C3⋊Dic3C4×D311yescharacteristic
77C4×C3⋊D3—41no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—121no—
85C2×C3×Dic3—121no—
86C3×C4×D3—122no—
87C3×C4×D3—121no—
88C3×C4×D3—121no—
89#627—122no—
90#627—121no—
91C3⋊(C3⋊C8)—41no—
92C3×C3⋊C8—121no—
93C3×C3⋊C8—121no—
94C3×C3⋊C8—122no—
95#328—271noSylow
96C2×C33#9911yescharacteristic
97C2×C33#10311yescharacteristic
98C2×C33#10411yescharacteristic
99C3×C3⋊D3—21no—
100C3×C3⋊D3—21no—
101C22×C3×C4—91no—
102C22×C3×C4—181no—
103C2×C4×D3—61no—
104C2×C4×D3—61no—
105C2×Dic6—91no—
106C2×Dic6—181no—
107C2×C3×C8—181no—
108C3×C4⋊C4—181no—
109#217—181no—
110(C2×D3)⋊C4—181no—
111(C2×D3)⋊C4—181no—
112C4⋊Dic3—91no—
113Dic3⋊C4—61no—
114Dic3⋊C4—61no—
115Dic3⋊C4—121no—
116Dic3⋊C4—181no—
117C2×C3⋊C8—61no—
118C2×C3⋊C8—61no—
119C2×C3⋊C8—121no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32(C2×D3)⋊C411yescharacteristic
125C2×C3⋊D3—21no—
126C2×C3⋊D3—21no—
127C2×C3⋊D3#21411yescharacteristic
128C2×C3⋊D3#21611yescharacteristic
129C2×C3×D3—61no—
130C2×C3×D3—61no—
131C2×C3×D3—61no—
132C2×C3×D3—61no—
133C2×C3×D3—121no—
134C2×C3×D3—121no—
135C2×C3×D3—121no—
136C2×C3×D3—121no—
137#157—42no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—41no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—81no—
145C3⋊Dic3—21no—
146C3⋊Dic3—41no—
147C3⋊Dic3—41no—
148C3×Dic3—62no—
149C3×Dic3—121no—
150C3×Dic3—121no—
151C3×Dic3—121no—
152C3×Dic3—122no—
153C3×Dic3—121no—
154C3×Dic3—121no—
155C3×Dic3—241no—
156C22⋊Q8—271no—
157C22⋊C8—271no—
158#95—91no—
159C33#32811yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C22×D3—61no—
162C22×D3—61no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—91no—
168C2×C3×C4—181no—
169C2×C3×C4—181no—
170C2×C3×C4—182no—
171C2×C3×C4—361no—
172C2×C3×C4—361no—
173C2×C3×C4—361no—
174C2×C3×C4GO+(4,𝔽2)11yescharacteristic
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—61no—
181C2×Dic3—121no—
182C2×Dic3—121no—
183C4×D3—122no—
184C4×D3—121no—
185C4×D3—121no—
186Dic6—362no—
187Dic6—361no—
188C3×C8—361no—
189C3⋊C8—121no—
190C3⋊C8—121no—
191C3⋊C8—122no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32#80511yescharacteristic
205C2×C32#80611yescharacteristic
206C2×C32#81011yescharacteristic
207C3⋊D3—21no—
208C3⋊D3—21no—
209C3×D3—121no—
210C3×D3—121no—
211C3×D3—121no—
212C3×D3—121no—
213C2×Q8—271no—
214C22×C4—91no—
215C22×C4—181no—
216C2×C8—181no—
217C4⋊C4—181no—
218C4⋊C4—271no—
219C22⋊C4—541no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—91no—
225C22×C3—91no—
226C22×C3—181no—
227C22×C3—181no—
228C22×C3D32⋊C411yescharacteristic
229C2×D3—61no—
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—121no—
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—121no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—81no—
244C3×C4—181no—
245C3×C4—361no—
246C3×C4—361no—
247C3×C4—362no—
248Dic3—62no—
249Dic3—121no—
250Dic3—121no—
251Dic3—121no—
252Dic3—122no—
253Dic3—121no—
254Dic3—121no—
255Dic3—241no—
256C32—21no—
257C32—21no—
258C32—41no—
259C32—41no—
260C32#445211yescharacteristic, complemented
261C23—91no—
262Q8—1081no—
263C2×C4—91no—
264C2×C4—182no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—361no—
268C2×C4—361no—
269C2×C4—361no—
270C2×C4#157⋊D311yescharacteristic
271C8—361no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—181no—
285C2×C3—181no—
286C2×C3#6373011yescharacteristic
287C2×C3#6373111yescharacteristic
288C2×C3#6373811yescharacteristic
289D3—121no—
290D3—121no—
291D3—121no—
292D3—121no—
293C22—91no—
294C22—91no—
295C22—181no—
296C22—181no—
297C22#8893911yescharacteristic
298C4—21no—
299C4—181no—
300C4—362no—
301C4—361no—
302C4—361no—
303C3—21no—
304C3—21no—
305C3—41no—
306C3—41no—
307C3#1059003411yescharacteristic, complemented
308C2—181no—
309C2—181no—
310C2#1172278411yescharacteristic
311C2#1172278511yescharacteristic
312C2#1172279211yescharacteristic
subgroupG/H
centerC22#88939
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC2×C4#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#328
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. C2×#6296C4semidirect
10. #88815C4semidirect
159. C33#328semidirect
260. C32#4452semidirect
307. C3#10590034semidirect

Maximal subgroups

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

subgroupindex
#117236712
C3×#637812
#117220722
#105900343
#44529