Finite Group Databasebeta

← #13155295

Subgroups — #13155295

Order 1728.

706 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 164 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×#88809C211yesmaximal, characteristic
2C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
3C3×#63874C211yesmaximal, characteristic
4Dic3×C3⋊(C3⋊C8)C212yesmaximal, complemented
5C3×#63554C212yesmaximal, complemented
6#10590703—31nomaximal
7#10590319—32nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2212yes—
11C32×C4×Dic3C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88809C414yescomplemented
14C3⋊D3×C3⋊C8C414yescomplemented
15C3×#3724C412yes—
16C3×#3722C412yes—
17C2×C8×C3⋊D3—31no—
18C3×#872—32no—
19#63874D311yescharacteristic
20C2×#3712—31no—
21C2×#3711—32no—
22C2×D3×C3⋊C8—32no—
23C2×D3×C3⋊C8—61no—
24C4×C3⋊(C3⋊C8)—32no—
25C3×#815—32no—
26C3×#803—32no—
27#63554D312yescomplemented
28Dic3×C3⋊C8—32no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—62no—
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3C2×C412yes—
35C3×C4×C3⋊D3C2×C412yes—
36C2×C32×Dic3C2×C414yes—
37C3⋊(C3⋊(C3⋊C8))C2×C412yes—
38C3×C3⋊(C3⋊C8)C2×C414yes—
39C32×C3⋊C8C2×C412yes—
40#4997—91no—
41#4708—92no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×C3×C4×D3—32no—
44C2×C3×C4×D3—61no—
45C2×C32×C8—31no—
46C32×C42—32no—
47C2×C3⋊(C3⋊C8)—31no—
48C2×C3⋊(C3⋊C8)—32no—
49C2×C3⋊(C3⋊C8)—62no—
50C2×C3⋊(C3⋊C8)—61no—
51C2×C3⋊(C3⋊C8)—121no—
52C2×C3⋊(C3⋊C8)C2×D312yes—
53C8×C3⋊D3—34no—
54C3×(C2×D3)⋊C4—34no—
55C3×Dic3⋊C4—34no—
56C3×C4×Dic3—22no—
57C3×C4×Dic3C2×D312yes—
58C3×C4×Dic3C2×D312yes—
59C2×C3×C3⋊C8—21no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—62no—
63C2×C3×C3⋊C8C2×D312yes—
64#3724Dic312yes—
65#3722Dic312yes—
66#3712—34no—
67#3711—38no—
68D3×C3⋊C8—38no—
69D3×C3⋊C8—64no—
70C22×C33C22×C411yescharacteristic
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3C4212yes—
73C33×C4C22×C411yescharacteristic
74C33×C4C22×C411yescharacteristic
75C3×C3⋊Dic3C4212yes—
76C32×Dic3—24no—
77C3×#117—91no—
78C2×#208—91no—
79C22×C3⋊C8—91no—
80C2×ΓL(1,𝔽25)—92no—
81C2×D3×C8—92no—
82C2×D3×C8—181no—
83#872—32no—
84C3×#97—92no—
85#815—32no—
86C8×Dic3—92no—
87#803—32no—
88C4×C3⋊C8—92no—
89C4×C3⋊C8—92no—
90C4×C3⋊C8—182no—
91C22×C3⋊D3C2×Dic311yescharacteristic
92C22×C3×D3—32no—
93C22×C3×D3—61no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—34no—
97C2×C32×C4—41no—
98C2×C32×C4C22×D312yes—
99C2×C32×C4C22×D311yescharacteristic
100C2×C3⋊Dic3C2×Dic311yescharacteristic
101C4×C3⋊D3C4×D312yes—
102C4×C3⋊D3C4×D312yes—
103C2×C3×Dic3—24no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3C4×D314yes—
107C2×C3×Dic3C2×Dic314yes—
108C3×C4×D3—34no—
109C3×C4×D3—34no—
110C3×C4×D3—62no—
111C3×C4×D3—62no—
112C32×C8—32no—
113C3⋊(C3⋊C8)—34no—
114C3⋊(C3⋊C8)—32no—
115C3⋊(C3⋊C8)—62no—
116C3⋊(C3⋊C8)—64no—
117C3⋊(C3⋊C8)—122no—
118C3⋊(C3⋊C8)C4×D314yes—
119C3×C3⋊C8—22no—
120C3×C3⋊C8—34no—
121C3×C3⋊C8—34no—
122C3×C3⋊C8—64no—
123C3×C3⋊C8C4×D314yes—
124#405—271noSylow
125C2×C33C2×C4211yescharacteristic
126C2×C33#13112yes—
127C3×C3⋊D3—22no—
128C22×C3×C4—91no—
129C2×C4×D3—32no—
130C2×C4×D3—61no—
131C2×C3×C8—32no—
132C2×C3×C8—61no—
133C2×C3×C8—92no—
134C3×C4⋊C4—92no—
135C3×C22⋊C4—92no—
136C3×C42—32no—
137C3×C42—32no—
138C3×C42—62no—
139(C2×D3)⋊C4—34no—
140Dic3⋊C4—34no—
141C4×Dic3D3212yes—
142#208—94no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—32no—
145C2×C3⋊C8—62no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—92no—
148C2×C3⋊C8—94no—
149C2×C3⋊C8—182no—
150C2×C3⋊C8—181no—
151C2×C3⋊C8—361no—
152C2×C3⋊C8D3211yescharacteristic
153ΓL(1,𝔽25)—98no—
154D3×C8—98no—
155D3×C8—184no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—41no—
159C22×C32C2×C4×D312yes—
160C22×C32C22×Dic311yescharacteristic
161C2×C3⋊D3—22no—
162C2×C3⋊D3C4×Dic312yes—
163C2×C3×D3—34no—
164C2×C3×D3—62no—
165C2×C3×D3—64no—
166C2×C3×D3—64no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—41no—
173C32×C4—64no—
174C32×C4C2×C4×D312yes—
175C32×C4C2×C4×D311yescharacteristic
176C32×C4C2×C4×D311yescharacteristic
177C32×C4C2×C4×D312yes—
178C3⋊Dic3C4×Dic312yes—
179C3×Dic3—28no—
180C3×Dic3—24no—
181C3×Dic3—24no—
182C3×Dic3—34no—
183C3×Dic3—62no—
184C2×M16—271no—
185C22×C8—271no—
186#117—91no—
187#97—92no—
188C4×C8—272no—
189C33#40511yescharacteristic, complemented, Sylow
190C23×C3—91no—
191C22×D3—32no—
192C22×D3—61no—
193C2×C3×C4—22no—
194C2×C3×C4—21no—
195C2×C3×C4—34no—
196C2×C3×C4—34no—
197C2×C3×C4—41no—
198C2×C3×C4—64no—
199C2×C3×C4—91no—
200C2×C3×C4—92no—
201C2×C3×C4—92no—
202C2×C3×C4C2×D3212yes—
203C2×C3×C4C2×D3211yescharacteristic
204C2×Dic3—32no—
205C2×Dic3—61no—
206C2×Dic3D3×Dic314yes—
207C4×D3—34no—
208C4×D3—34no—
209C4×D3—62no—
210C4×D3—62no—
211C3×C8—34no—
212C3×C8—62no—
213C3×C8—94no—
214C3⋊C8—34no—
215C3⋊C8—34no—
216C3⋊C8—64no—
217C3⋊C8—94no—
218C3⋊C8—92no—
219C3⋊C8—182no—
220C3⋊C8—184no—
221C3⋊C8—362no—
222C3⋊C8#62412yes—
223C2×C32—22no—
224C2×C32—24no—
225C2×C32—22no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—42no—
229C2×C32C42×D312yes—
230C2×C32#90714yes—
231C2×C32C2×C4×Dic311yescharacteristic
232C2×C32#94812yes—
233C3⋊D3—22no—
234C3×D3—64no—
235C3×D3—64no—
236C22×C4—91no—
237M16—274no—
238C2×C8—31no—
239C2×C8—92no—
240C2×C8—274no—
241C2×C8—271no—
242C4⋊C4—92no—
243C22⋊C4—92no—
244C42—32no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—41no—
248C22×C3—92no—
249C22×C3—182no—
250C22×C3C4×D3211yescharacteristic
251C22×C3C2×D3×Dic312yes—
252C2×D3—34no—
253C2×D3—62no—
254C2×D3—64no—
255C2×D3—64no—
256C3×C4—21no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—68no—
263C3×C4—64no—
264C3×C4—64no—
265C3×C4—92no—
266C3×C4C4×D3212yes—
267C3×C4C4×D3212yes—
268C3×C4C4×D3211yescharacteristic
269C3×C4C2×#62411yescharacteristic
270Dic3—24no—
271Dic3—34no—
272Dic3—62no—
273C32—22no—
274C32—21no—
275C32—41no—
276C32#470812yescomplemented
277C32#499711yescharacteristic, complemented
278C23—91no—
279C2×C4—34no—
280C2×C4—91no—
281C2×C4—92no—
282C2×C4—92no—
283C2×C4D3311yescharacteristic
284C8—32no—
285C8—94no—
286C8—272no—
287C2×C3—24no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—42no—
292C2×C3—41no—
293C2×C3—182no—
294C2×C3#6380512yes—
295C2×C3#6381514yes—
296C2×C3C4×D3×Dic312yes—
297C2×C3C4×#62411yescharacteristic
298D3—64no—
299D3—64no—
300C22—92no—
301C22—182no—
302C22D32×Dic311yescharacteristic
303C4—64no—
304C4—92no—
305C4D3×#62411yescharacteristic
306C4#8894911yescharacteristic
307C3—21no—
308C3—22no—
309C3—41no—
310C3#1059031912yescomplemented
311C3#1059070311yescharacteristic, complemented
312C2—182no—
313C2#1172282312yes—
314C2Dic3×#62411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C2×C42
Fitting, cosocleC2×C33×C4C23
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. Dic3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63554C2semidirect
13. #88809C4semidirect
14. C3⋊D3×C3⋊C8C4semidirect
27. #63554D3semidirect
189. C33#405semidirect
276. C32#4708semidirect
277. C32#4997semidirect
310. C3#10590319semidirect
311. C3#10590703semidirect

Maximal subgroups

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

subgroupindex
C2×#888092
C2×C3⋊D3×C3⋊C82
C3×#638742
Dic3×C3⋊(C3⋊C8)2
C3×#635542
#105907033
#105903193