Finite Group Databasebeta

← #13167062

Subgroups — #13167062

Order 1728.

428 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 79 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⋊#157C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4#11722763C214yesmaximal, complemented
5#10593328—31nomaximal
6C22×C3⋊#157C412yes—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C4×C3⋊#157C411yescharacteristic
9C4×C3⋊#157C411yescharacteristic
10C4×C3⋊#157C2212yes—
11C3×#3748—21no—
12C32×#208—21no—
13#88809—21no—
14#88808C2212yes—
15C3×#3712—21no—
16C3×#3710C2212yes—
17C2×C4×#157—31no—
18C3×D3×M16—61no—
19#63818—31no—
20#63818D311yescharacteristic
21D3×#208—61no—
22#63732—34no—
23C22×C3×C3⋊D3C2×C411yescharacteristic
24C2×C3⋊#157—24no—
25C2×C3⋊#157C2×C412yes—
26C2×C3⋊#157C2×C412yes—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3Q811yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3D411yescharacteristic
31C3×C4×C3⋊D3Q811yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C3×C3⋊(C3⋊C8)—22no—
34C32×C3⋊C8—22no—
35#5118—91nomaximal
36C22×#157—32no—
37C2×C4×C3⋊D3C2×D311yescharacteristic
38C2×C3×C4×D3—61no—
39C2×C3×C4×D3—61no—
40C4×#157—31no—
41C4×#157—31no—
42C4×#157—32no—
43C32×M16—61no—
44#3748—21no—
45C3×#208—21no—
46C3×#208—21no—
47C3×#208—41no—
48C3×#208—61no—
49C3×#208—61no—
50C3×#208—121no—
51C3×ΓL(1,𝔽25)—62no—
52C3×D3×C8—62no—
53#3712—21no—
54#3712—61no—
55#3711—62no—
56#3710—32no—
57#3710C2×D312yes—
58D3×C3⋊C8—62no—
59C22×C33C4⋊C411yescharacteristic
60C2×C3×C3⋊D3—22no—
61C2×C3×C3⋊D3C4⋊C411yescharacteristic
62C2×C3×C3⋊D3C22×C411yescharacteristic
63C3⋊#157—22no—
64C3⋊#157M1614yescomplemented
65C33×C4C4⋊C411yescharacteristic
66C33×C4C2×D411yescharacteristic
67C3×C3⋊Dic3C4⋊C411yescharacteristic
68C3×C3⋊Dic3C2×Q811yescharacteristic
69C2×C3×M16—91no—
70C2×C4×Dic3—91no—
71C2×#208—91no—
72D3×M16—61no—
73D3×M16—181no—
74Dic3⋊C8—94no—
75C22×C3⋊D3C4×D311yescharacteristic
76C22×C3×D3—61no—
77C22×C3×D3—61no—
78C2×#157—32no—
79C2×#157—32no—
80C2×#157—64no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4#6711yescharacteristic
86C2×C3⋊Dic3Dic611yescharacteristic
87C4×C3⋊D3Dic611yescharacteristic
88C4×C3⋊D3C4×D311yescharacteristic
89C4×C3⋊D3#6711yescharacteristic
90C4×C3⋊D3C22×D311yescharacteristic
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C3×C4×D3—61no—
94C3×C4×D3—61no—
95C3×C4×D3—61no—
96C3×C4×D3—61no—
97C3×C4×D3—61no—
98C3×C4×D3—61no—
99C3×C4×D3—61no—
100C3×C4×D3—61no—
101C32×C8—62no—
102C3⋊(C3⋊C8)—22no—
103C3×C3⋊C8—22no—
104C3×C3⋊C8—22no—
105C3×C3⋊C8—42no—
106C3×C3⋊C8—62no—
107C3×C3⋊C8—62no—
108C3×C3⋊C8—122no—
109#423—271noSylow
110C2×C33—21no—
111C2×C33C2×C4⋊C411yescharacteristic
112C3×C3⋊D3—21no—
113C3×C3⋊D3C2×M1612yes—
114C22×C3×C4—91no—
115C22×Dic3—92no—
116C2×C4×D3—61no—
117C2×C4×D3—61no—
118C3×M16—61no—
119C3×M16—61no—
120C3×M16—121no—
121C3×M16—181no—
122C3×M16—181no—
123C2×C3×C8—92no—
124C4×Dic3—92no—
125C4×Dic3—91no—
126C4×Dic3—91no—
127#208—21no—
128#208—61no—
129#208—61no—
130#208—121no—
131#208—181no—
132C2×C3⋊C8—92no—
133ΓL(1,𝔽25)—62no—
134ΓL(1,𝔽25)—182no—
135D3×C8—62no—
136D3×C8—182no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—41no—
140C22×C32—41no—
141C22×C32Dic3⋊C411yescharacteristic
142C2×C3⋊D3—22no—
143C2×C3⋊D3Dic3⋊C411yescharacteristic
144C2×C3⋊D3C2×C4×D311yescharacteristic
145C2×C3×D3—61no—
146C2×C3×D3—61no—
147C2×C3×D3—61no—
148C2×C3×D3—61no—
149C2×C3×D3—122no—
150C2×C3×D3—122no—
151#157—34no—
152#157—62no—
153C32×C4—21no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—41no—
158C32×C4—41no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4Dic3⋊C411yescharacteristic
162C32×C4C2×#6711yescharacteristic
163C3⋊Dic3Dic3⋊C411yescharacteristic
164C3⋊Dic3C2×Dic611yescharacteristic
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C2×M16—91no—
170C2×M16—271no—
171C2×C42—271no—
172C4⋊C8—274no—
173C33#42311yescharacteristic, complemented, Sylow
174C23×C3—91no—
175C22×D3—61no—
176C22×D3—61no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—91no—
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—91no—
185C2×C3×C4—91no—
186C2×C3×C4GO+(4,𝔽2)11yescharacteristic
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3—92no—
190C2×Dic3—92no—
191C2×Dic3—184no—
192C4×D3—61no—
193C4×D3—61no—
194C4×D3—61no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—61no—
198C4×D3—61no—
199C4×D3—61no—
200C3×C8—62no—
201C3×C8—62no—
202C3×C8—122no—
203C3×C8—182no—
204C3⋊C8—22no—
205C3⋊C8—62no—
206C3⋊C8—62no—
207C3⋊C8—122no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—81no—
216C2×C32—81no—
217C2×C32C2×Dic3⋊C411yescharacteristic
218C3⋊D3—21no—
219C3⋊D3D3×M1612yes—
220C3×D3—62no—
221C3×D3—62no—
222C3×D3—121no—
223C3×D3—121no—
224C22×C4—91no—
225C22×C4—272no—
226M16—61no—
227M16—181no—
228M16—181no—
229M16—541no—
230C2×C8—92no—
231C2×C8—272no—
232C42—272no—
233C42—271no—
234C42—271no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—91no—
240C22×C3—91no—
241C22×C3—182no—
242C22×C3#377311yescharacteristic
243C2×D3—61no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—61no—
247C2×D3—122no—
248C2×D3—122no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—91no—
258C3×C4—91no—
259C3×C4#377311yescharacteristic
260C3×C4C2×GO+(4,𝔽2)11yescharacteristic
261Dic3—61no—
262Dic3—61no—
263Dic3—61no—
264Dic3—61no—
265Dic3—94no—
266Dic3—182no—
267C32—21no—
268C32—21no—
269C32—41no—
270C32—41no—
271C32#511811yescharacteristic, complemented
272C23—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—272no—
279C2×C4—272no—
280C2×C4—544no—
281C2×C4D32⋊D311yescharacteristic
282C8—62no—
283C8—182no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—81no—
292C2×C3—81no—
293C2×C3—92no—
294C2×C3—181no—
295C2×C3C2×#377311yescharacteristic
296D3—62no—
297D3—62no—
298D3—121no—
299D3—121no—
300C22—91no—
301C22—91no—
302C22—182no—
303C22#8893411yescharacteristic
304C4—91no—
305C4—91no—
306C4—274no—
307C4—542no—
308C4#8893411yescharacteristic
309C4C2×D32⋊D311yescharacteristic
310C3—21no—
311C3—21no—
312C3—41no—
313C3—41no—
314C3#1059332811yescharacteristic, complemented
315C2—21no—
316C2—92no—
317C2—181no—
318C2C2×#8893411yescharacteristic
subgroupG/H
center, FrattiniC4C2×D32⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#423
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722763C2semidirect
64. C3⋊#157M16semidirect
173. C33#423semidirect
271. C32#5118semidirect
314. C3#10593328semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C3⋊#1572
#117236562
C3×#638182
#117227632
#105933283
#51189