Finite Group Databasebeta

← C3⋊C8×#627 #13155759

Subgroups — C3⋊C8×#627 #13155759

Order 1728.

550 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C3×C4×#627C211yesmaximal, characteristic
2#11722078C211yesmaximal, characteristic
3Dic3⋊(C3⋊(C3⋊C8))C212yesmaximal
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
5C3×Dic3×C3⋊C8C212yesmaximal
6C8×#627—31nomaximal
7C3⋊C8×Dic6—32nomaximal
8C2×C3×#627C411yescharacteristic
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4×Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14C3×#3724C411yescharacteristic
15C3×Dic3⋊Dic3C412yes—
16C3×C4×Dic6—32no—
17C4×#627D311yescharacteristic
18C8×C3⋊Dic3—31no—
19C4⋊(C3⋊(C3⋊C8))—32no—
20C3×C8×Dic3—32no—
21C3×C4×C3⋊C8—32no—
22#63572—31no—
23Dic3⋊(C3⋊C8)—32no—
24Dic3⋊(C3⋊C8)—32no—
25Dic3⋊(C3⋊C8)—32no—
26Dic3⋊(C3⋊C8)—62no—
27Dic3×C3⋊C8—32no—
28Dic3×C3⋊C8—61no—
29Dic3×C3⋊C8D312yes—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C32×Dic3C2×C412yes—
34C2×C32×Dic3C2×C412yes—
35C3×#627C814yescomplemented
36C3⋊(C3⋊(C3⋊C8))—21no—
37C3×C3⋊(C3⋊C8)—22no—
38C32×C3⋊C8Q812yescomplemented
39Q8×C3⋊C8—91no—
40C8×Dic6—92no—
41C2×C3×Dic6—32no—
42C2×#627Dic311yescharacteristic
43C2×C32×C8—31no—
44C32×C42—32no—
45C4×C3⋊Dic3C2×D311yescharacteristic
46C2×C3⋊(C3⋊C8)—32no—
47C2×C3⋊(C3⋊C8)—31no—
48C2×C3⋊(C3⋊C8)—61no—
49C2×C3⋊(C3⋊C8)—62no—
50C2×C3⋊(C3⋊C8)—121no—
51C2×C3⋊(C3⋊C8)C2×D312yes—
52C3×C4⋊Dic3—32no—
53C3×Dic3⋊C4—32no—
54C3×Dic3⋊C4—32no—
55C3×C4×Dic3—22no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C3×C4×Dic3C2×D312yes—
59C3×C4×Dic3C2×D312yes—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—62no—
64C2×C3×C3⋊C8C2×D312yes—
65#3724Dic311yescharacteristic
66Dic3⋊Dic3Dic312yes—
67C22×C33C22×C411yescharacteristic
68C33×C4C2×Q811yescharacteristic
69C33×C4#4111yescharacteristic
70C3×C3⋊Dic3—21no—
71C3×C3⋊Dic3C2×C812yes—
72C32×Dic3—22no—
73C32×Dic3C2×C814yes—
74C3×C4×Q8—91no—
75C4×Dic6—32no—
76C3×C4×C8—92no—
77Dic3⋊C8—92no—
78Dic3⋊C8—92no—
79C8×Dic3—32no—
80C8×Dic3—92no—
81C8×Dic3—181no—
82C4⋊(C3⋊C8)—91no—
83C4⋊(C3⋊C8)—92no—
84C4⋊(C3⋊C8)—92no—
85C4⋊(C3⋊C8)—182no—
86C4×C3⋊C8—32no—
87C4×C3⋊C8—91no—
88C2×C32×C4—21no—
89C2×C32×C4—22no—
90C2×C32×C4—32no—
91C2×C32×C4—32no—
92C2×C32×C4—41no—
93C2×C32×C4C22×D312yes—
94C2×C32×C4C22×D311yescharacteristic
95C2×C3⋊Dic3C2×Dic311yescharacteristic
96C2×C3⋊Dic3C2×Dic311yescharacteristic
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—32no—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3C4×D312yes—
104C2×C3×Dic3C4×D312yes—
105C2×C3×Dic3C2×Dic312yes—
106C2×C3×Dic3C2×Dic312yes—
107C3×Dic6—38no—
108#627C3⋊C814yesdirect product factor
109C32×C8—32no—
110C3⋊(C3⋊C8)—22no—
111C3⋊(C3⋊C8)—61no—
112C3⋊(C3⋊C8)—62no—
113C3⋊(C3⋊C8)—62no—
114C3⋊(C3⋊C8)—122no—
115C3⋊(C3⋊C8)—122no—
116C3×C3⋊C8—22no—
117C3×C3⋊C8—62no—
118C3×C3⋊C8—62no—
119C3×C3⋊C8—122no—
120C3×C3⋊C8Dic614yescomplemented
121C8×Q8—271noSylow
122C2×C33C4×Q811yescharacteristic
123C2×C33C22×C811yescharacteristic
124C2×C33#13111yescharacteristic
125C2×C3×Q8—91no—
126C2×Dic6—32no—
127C2×C3×C8—32no—
128C2×C3×C8—61no—
129C2×C3×C8—92no—
130C3×C4⋊C4—91no—
131C3×C4⋊C4—92no—
132C3×C42—32no—
133C3×C42—32no—
134C3×C42—62no—
135C3×C42—91no—
136C4⋊Dic3—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139C4×Dic3—32no—
140C4×Dic3—61no—
141C4×Dic3D3212yes—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—32no—
144C2×C3⋊C8—62no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—92no—
147C2×C3⋊C8—181no—
148C2×C3⋊C8—182no—
149C2×C3⋊C8—361no—
150C2×C3⋊C8D3211yescharacteristic
151C22×C32—21no—
152C22×C32—22no—
153C22×C32—41no—
154C22×C32C2×C4×D312yes—
155C22×C32C22×Dic311yescharacteristic
156C32×C4—22no—
157C32×C4—21no—
158C32×C4—22no—
159C32×C4—21no—
160C32×C4—34no—
161C32×C4—41no—
162C32×C4—41no—
163C32×C4—62no—
164C32×C4C2×Dic612yes—
165C32×C4#23512yes—
166C32×C4D3×Q811yescharacteristic
167C32×C4#23911yescharacteristic
168C3⋊Dic3—21no—
169C3⋊Dic3C2×C3⋊C812yes—
170C3×Dic3—22no—
171C3×Dic3—22no—
172C3×Dic3—24no—
173C3×Dic3—34no—
174C3×Dic3—42no—
175C3×Dic3—62no—
176C3×Dic3—62no—
177C3×Dic3—62no—
178C3×Dic3D3×C814yes—
179C3×Dic3C2×C3⋊C814yes—
180C4×Q8—91no—
181C4⋊C8—272no—
182C4⋊C8—271no—
183C4×C8—92no—
184C4×C8—271no—
185C33C8×Q811yescharacteristic, complemented, Sylow
186C3×Q8—94no—
187C2×C3×C4—22no—
188C2×C3×C4—21no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4C2×D3212yes—
199C2×C3×C4C2×D3211yescharacteristic
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3D3×Dic312yes—
205C2×Dic3D3×Dic312yes—
206Dic6—38no—
207C3×C8—34no—
208C3×C8—62no—
209C3×C8—182no—
210C3⋊C8—62no—
211C3⋊C8—62no—
212C3⋊C8—122no—
213C3⋊C8—182no—
214C3⋊C8—181no—
215C3⋊C8—182no—
216C3⋊C8—362no—
217C3⋊C8—362no—
218C3⋊C8#62712yesdirect product factor
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—22no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32C4×Dic612yes—
229C2×C32C2×D3×C812yes—
230C2×C32#90112yes—
231C2×C32C22×C3⋊C811yescharacteristic
232C2×C32Q8×Dic311yescharacteristic
233C2×C32#94811yescharacteristic
234C2×Q8—91no—
235C2×C8—31no—
236C2×C8—92no—
237C2×C8—271no—
238C4⋊C4—91no—
239C4⋊C4—92no—
240C42—32no—
241C42—91no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3C4×D3211yescharacteristic
246C22×C3C2×D3×Dic312yes—
247C3×C4—21no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—34no—
252C3×C4—34no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—64no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—92no—
259C3×C4—122no—
260C3×C4—181no—
261C3×C4D3×Dic612yes—
262C3×C4#379811yescharacteristic
263C3×C4#379912yes—
264C3×C4C2×#62711yescharacteristic
265Dic3—22no—
266Dic3—34no—
267Dic3—62no—
268Dic3—62no—
269Dic3—62no—
270Dic3D3×C3⋊C814yes—
271C32—22no—
272C32—21no—
273C32—41no—
274C32C8×Dic612yescomplemented
275C32Q8×C3⋊C811yescharacteristic, complemented
276Q8—94no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4D3311yescharacteristic
282C8—32no—
283C8—182no—
284C8—541no—
285C2×C3—21no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3D32×C811yescharacteristic
295C2×C3#6380311yescharacteristic
296C2×C3C2×D3×C3⋊C812yes—
297C2×C3#6381412yes—
298C2×C3Dic3×Dic612yes—
299C2×C3C4×#62711yescharacteristic
300C22D32×Dic311yescharacteristic
301C4—34no—
302C4—62no—
303C4—92no—
304C4—181no—
305C4D3×#62711yescharacteristic
306C4#8896511yescharacteristic
307C3—22no—
308C3—21no—
309C3—41no—
310C3C3⋊C8×Dic612yescomplemented
311C3C8×#62711yescharacteristic, complemented
312C2D32×C3⋊C811yescharacteristic
313C2#1172282111yescharacteristic
314C2Dic3×#62711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutatorC2×C33C22×C8
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C22×C4
hypocenterC33C8×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
35. C3×#627C8semidirect
38. C32×C3⋊C8Q8semidirect
108. #627C3⋊C8direct (N × H)
120. C3×C3⋊C8Dic6semidirect
185. C33C8×Q8semidirect
218. C3⋊C8#627direct (N × H)
274. C32C8×Dic6semidirect
275. C32Q8×C3⋊C8semidirect
310. C3C3⋊C8×Dic6semidirect
311. C3C8×#627semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6272
#117220782
Dic3⋊(C3⋊(C3⋊C8))2
C3⋊C8×C3⋊Dic32
C3×Dic3×C3⋊C82
C8×#6273
C3⋊C8×Dic63