Finite Group Databasebeta

← #13155005

Subgroups — #13155005

Order 1728.

372 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 63 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×C4⋊(C3⋊#157)C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4#10590053—31nomaximal
5C22×C3⋊#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C4⋊(C3⋊#157)—21no—
8C4⋊(C3⋊#157)—21no—
9C3×#3748—21no—
10C32×#208—21no—
11#88809C412yescomplemented
12#88808—21no—
13C3×#3712C412yescomplemented
14C3×#3710—21no—
15C2×C4⋊#157—31no—
16C3×D3×M16—61no—
17#63818—31no—
18#63818D311yescharacteristic
19D3×#208—61no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3⋊#157—22no—
22C2×C3⋊#157—42no—
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3Q811yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3C2×C411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3×C4×C3⋊D3D411yescharacteristic
29C3×C3⋊(C3⋊C8)—22no—
30C32×C3⋊C8—22no—
31#4554—91nomaximal
32C22×#157—32no—
33C2×C4×C3⋊D3C2×D311yescharacteristic
34C2×C3×C4×D3—61no—
35C2×C3×C4×D3—61no—
36C4⋊#157—61no—
37C4⋊#157—61no—
38C32×M16—61no—
39#3748—21no—
40C3×#208—21no—
41C3×#208—21no—
42C3×#208—41no—
43C3×#208—61no—
44C3×#208—61no—
45C3×#208—121no—
46C3×ΓL(1,𝔽25)—62no—
47C3×D3×C8—62no—
48#3712—32no—
49#3712Dic312yescomplemented
50#3711—62no—
51#3710—21no—
52#3710—61no—
53D3×C3⋊C8—62no—
54C22×C33C4⋊C411yescharacteristic
55C2×C3×C3⋊D3—21no—
56C2×C3×C3⋊D3—21no—
57C2×C3×C3⋊D3C4211yescharacteristic
58C2×C3×C3⋊D3C22⋊C411yescharacteristic
59C3⋊#157—42no—
60C33×C4C22⋊C411yescharacteristic
61C33×C4C22⋊C411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C3×C3⋊Dic3C4⋊C411yescharacteristic
64C2×C3×M16—91no—
65C2×C4⋊Dic3—91no—
66C2×#208—91no—
67D3×M16—61no—
68D3×M16—181no—
69C22×C3⋊D3C4×D311yescharacteristic
70C22×C3×D3—61no—
71C22×C3×D3—61no—
72C2×#157—62no—
73C2×#157—122no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—41no—
77C2×C32×C4—41no—
78C2×C32×C4#6711yescharacteristic
79C2×C3⋊Dic3Dic611yescharacteristic
80C4×C3⋊D3C4×D311yescharacteristic
81C4×C3⋊D3D1211yescharacteristic
82C4×C3⋊D3C2×Dic311yescharacteristic
83C4×C3⋊D3#6711yescharacteristic
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94C32×C8—62no—
95C3⋊(C3⋊C8)—22no—
96C3×C3⋊C8—22no—
97C3×C3⋊C8—22no—
98C3×C3⋊C8—42no—
99C3×C3⋊C8—62no—
100C3×C3⋊C8—62no—
101C3×C3⋊C8—122no—
102#343—271noSylow
103C2×C33—21no—
104C2×C33#9511yescharacteristic
105C3×C3⋊D3—21no—
106C3×C3⋊D3#10011yescharacteristic
107C3×C3⋊D3#10111yescharacteristic
108C22×C3×C4—91no—
109C22×Dic3—92no—
110C2×C4×D3—61no—
111C2×C4×D3—61no—
112C3×M16—61no—
113C3×M16—61no—
114C3×M16—92no—
115C3×M16—121no—
116C3×M16—181no—
117C2×C3×C8—181no—
118C4⋊Dic3—181no—
119C4⋊Dic3—181no—
120#208—21no—
121#208—61no—
122#208—61no—
123#208—92no—
124#208—121no—
125C2×C3⋊C8—181no—
126ΓL(1,𝔽25)—62no—
127ΓL(1,𝔽25)—182no—
128D3×C8—62no—
129D3×C8—182no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—41no—
133C22×C32—41no—
134C22×C32Dic3⋊C411yescharacteristic
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3C4×Dic311yescharacteristic
138C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
139C2×C3×D3—61no—
140C2×C3×D3—61no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C2×C3×D3—121no—
144C2×C3×D3—121no—
145C2×C3×D3—121no—
146C2×C3×D3—121no—
147#157—122no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—41no—
153C32×C4—41no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4(C2×D3)⋊C411yescharacteristic
157C32×C4#21711yescharacteristic
158C3⋊Dic3Dic3⋊C411yescharacteristic
159C3⋊Dic3C4⋊Dic311yescharacteristic
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—61no—
164C2×M16—91no—
165C2×M16—271no—
166C2×C4⋊C4—271no—
167C33#34311yescharacteristic, complemented, Sylow
168C23×C3—91no—
169C22×D3—61no—
170C22×D3—61no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—91no—
180C2×C3×C4GO+(4,𝔽2)11yescharacteristic
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—182no—
184C2×Dic3—362no—
185C4×D3—61no—
186C4×D3—61no—
187C4×D3—61no—
188C4×D3—61no—
189C4×D3—61no—
190C4×D3—61no—
191C4×D3—61no—
192C4×D3—61no—
193C3×C8—62no—
194C3×C8—62no—
195C3×C8—122no—
196C3×C8—182no—
197C3⋊C8—22no—
198C3⋊C8—62no—
199C3⋊C8—62no—
200C3⋊C8—122no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—81no—
209C2×C32—81no—
210C2×C32C3⋊#9511yescharacteristic
211C3⋊D3—21no—
212C3⋊D3#82311yescharacteristic
213C3⋊D3#82411yescharacteristic
214C3×D3—61no—
215C3×D3—61no—
216C3×D3—61no—
217C3×D3—61no—
218C3×D3—121no—
219C3×D3—121no—
220C22×C4—91no—
221C22×C4—272no—
222M16—61no—
223M16—92no—
224M16—181no—
225M16—272no—
226C2×C8—181no—
227C2×C8—541no—
228C4⋊C4—541no—
229C4⋊C4—541no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—91no—
235C22×C3—91no—
236C22×C3—181no—
237C22×C3—181no—
238C22×C3#377311yescharacteristic
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—61no—
242C2×D3—61no—
243C2×D3—121no—
244C2×D3—121no—
245C2×D3—121no—
246C2×D3—121no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—91no—
256C3×C4—91no—
257C3×C4D32⋊C411yescharacteristic
258C3×C4D32⋊C411yescharacteristic
259Dic3—61no—
260Dic3—61no—
261Dic3—61no—
262Dic3—61no—
263Dic3—362no—
264C32—21no—
265C32—21no—
266C32—41no—
267C32—41no—
268C32#455411yescharacteristic, complemented
269C23—91no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—542no—
276C2×C4—1082no—
277C2×C4D32⋊D311yescharacteristic
278C8—62no—
279C8—182no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—81no—
288C2×C3—81no—
289C2×C3—91no—
290C2×C3—91no—
291C2×C3—181no—
292C2×C3#6373711yescharacteristic
293D3—61no—
294D3—61no—
295D3—61no—
296D3—61no—
297D3—121no—
298D3—121no—
299C22—91no—
300C22—91no—
301C22—181no—
302C22—181no—
303C22#8893411yescharacteristic
304C4—91no—
305C4—91no—
306C4—1082no—
307C4#8893211yescharacteristic
308C4D32⋊Dic311yescharacteristic
309C3—21no—
310C3—21no—
311C3—41no—
312C3—41no—
313C3#1059005311yescharacteristic, complemented
314C2—21no—
315C2—91no—
316C2—91no—
317C2—181no—
318C2#1172276911yescharacteristic
subgroupG/H
centerC2#11722769
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC2×C4D32⋊D3
FittingC2×C33×C4D4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#343
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
11. #88809C4semidirect
13. C3×#3712C4semidirect
49. #3712Dic3semidirect
167. C33#343semidirect
268. C32#4554semidirect
313. C3#10590053semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
#117236562
C3×#638182
#105900533
#45549