Finite Group Databasebeta

← C3×#63893 #11723218

Subgroups — C3×#63893 #11723218

Order 864.

398 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 104 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×C3×#627C211yesmaximal, characteristic, complemented
2C2×C3×D3×Dic3C211yesmaximal, characteristic
3C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C32×(C2×D3)⋊C4C211yesmaximal, characteristic
5C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic
8C3×#897—31nomaximal
9C3×#878—31nomaximal
10#63893C311yesmaximal, characteristic, direct product factor
11C22×C32×D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C3×#627—22no—
19C3×D3×Dic3—22no—
20C22×C3×Dic3—31no—
21C2×C3×C4×D3—31no—
22C2×C3×Dic6—31no—
23C2×C3×Dic6—31no—
24C2×#627C2×C311yescharacteristic, complemented
25C2×D3×Dic3C2×C311yescharacteristic
26C32×C4⋊C4—31no—
27C32×C22⋊C4—31no—
28C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
29C3×#217—31no—
30C3×(C2×D3)⋊C4—21no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4D311yescharacteristic
33C3×(C2×D3)⋊C4C2×C311yescharacteristic
34C3×C4⋊Dic3—31no—
35C3×C4⋊Dic3—31no—
36C3×C4⋊Dic3—31no—
37C3×C4⋊Dic3—61no—
38C3×Dic3⋊C4—21no—
39C3×Dic3⋊C4—31no—
40C3×Dic3⋊C4D311yescharacteristic, complemented
41C3×Dic3⋊C4C2×C311yescharacteristic, complemented
42Dic3⋊Dic3C2×C311yescharacteristic, complemented
43(C2×D3)⋊Dic3C2×C311yescharacteristic
44C22×C33C2311yescharacteristic
45C2×C32×D3—21no—
46C2×C32×D3—21no—
47C2×C32×D3Q812yescomplemented
48C33×C4—21no—
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3—21no—
51C32×Dic3—21no—
52C32×Dic3—21no—
53C32×Dic3D412yes—
54C3×C22⋊Q8—91no—
55#897—31no—
56#878—31no—
57C23×C32—31no—
58C22×C3×D3—21no—
59C22×C3×D3C2×D311yescharacteristic
60C22×C3×D3C22×C311yescharacteristic
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—41no—
68C2×C32×C4C2×D311yescharacteristic
69C2×C32×C4C2×D311yescharacteristic
70C2×C32×C4C22×C311yescharacteristic
71C2×C3⋊Dic3C22×C311yescharacteristic
72C2×C3⋊Dic3C22×C311yescharacteristic
73C2×C3×Dic3—21no—
74C2×C3×Dic3—21no—
75C2×C3×Dic3—21no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—31no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—31no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C3×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C2×D311yescharacteristic
86C2×C3×Dic3C22×C311yescharacteristic
87C2×C3×Dic3C22×C311yescharacteristic
88C2×C3×Dic3C22×C311yescharacteristic
89C3×C4×D3—62no—
90C3×Dic6—62no—
91C3×Dic6—62no—
92#627—22no—
93D3×Dic3—22no—
94C2×C33C2×D411yescharacteristic
95C2×C33C2×Q811yescharacteristic
96C2×C33#4111yescharacteristic
97C32×D3—22no—
98C2×C3×Q8—91no—
99C22×C3×C4—91no—
100C22×Dic3—31no—
101C2×C4×D3—31no—
102C2×Dic6—31no—
103C2×Dic6—31no—
104C3×C4⋊C4—31no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4—61no—
107C3×C4⋊C4—91no—
108C3×C4⋊C4—91no—
109C3×C22⋊C4—31no—
110C3×C22⋊C4—31no—
111C3×C22⋊C4—61no—
112C3×C22⋊C4—91no—
113#217—31no—
114(C2×D3)⋊C4—31no—
115(C2×D3)⋊C4C3×D311yescharacteristic
116C4⋊Dic3—31no—
117C4⋊Dic3—31no—
118C4⋊Dic3—31no—
119C4⋊Dic3—61no—
120Dic3⋊C4—31no—
121Dic3⋊C4C3×D311yescharacteristic, complemented
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—32no—
126C22×C32—41no—
127C22×C32—61no—
128C22×C32—61no—
129C22×C32C22×D311yescharacteristic
130C22×C32C22×D311yescharacteristic
131C22×C32C23×C311yescharacteristic
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3—21no—
136C2×C3×D3—22no—
137C2×C3×D3—41no—
138C2×C3×D3—41no—
139C2×C3×D3Dic612yescomplemented
140C2×C3×D3C3×Q812yescomplemented
141C32×C4—21no—
142C32×C4—22no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—32no—
146C32×C4—41no—
147C32×C4—41no—
148C32×C4—42no—
149C32×C4—61no—
150C32×C4—61no—
151C3⋊Dic3—21no—
152C3⋊Dic3—21no—
153C3×Dic3—21no—
154C3×Dic3—21no—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3—22no—
158C3×Dic3—41no—
159C3×Dic3—41no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—61no—
164C3×Dic3—121no—
165C3×Dic3—121no—
166C3×Dic3D1212yes—
167C3×Dic3C3×D412yes—
168C22⋊Q8—91noSylow
169C33C22⋊Q811yescharacteristic, complemented, Sylow
170C23×C3—31no—
171C23×C3—31no—
172C23×C3—61no—
173C22×D3C2×C3×D311yescharacteristic
174C3×Q8—182no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—41no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—91no—
189C2×C3×C4—91no—
190C2×C3×C4—182no—
191C2×C3×C4D3211yescharacteristic
192C2×C3×C4C2×C3×D311yescharacteristic
193C2×C3×C4C2×C3×D311yescharacteristic
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—31no—
197C2×Dic3—31no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—62no—
201C2×Dic3C2×C3×D311yescharacteristic
202C2×Dic3C2×C3×D311yescharacteristic
203C2×Dic3C2×C3×D311yescharacteristic
204C4×D3—62no—
205Dic6—62no—
206Dic6—62no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—62no—
220C2×C32C2×Dic611yescharacteristic
221C2×C32C2×D1211yescharacteristic
222C2×C32D3×D411yescharacteristic
223C2×C32#23711yescharacteristic
224C2×C32#23711yescharacteristic
225C2×C32D3×Q811yescharacteristic
226C2×C32C2×C3×D411yescharacteristic
227C2×C32C2×C3×Q811yescharacteristic
228C2×C32C3×#4111yescharacteristic
229C3×D3—22no—
230C3×D3—22no—
231C3×D3—42no—
232C2×Q8—91no—
233C22×C4—91no—
234C4⋊C4—31no—
235C4⋊C4—91no—
236C4⋊C4—91no—
237C22⋊C4—31no—
238C22⋊C4—91no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—41no—
245C22×C3—62no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—121no—
251C22×C3—121no—
252C22×C3C2×D3211yescharacteristic
253C22×C3C22×C3×D311yescharacteristic
254C22×C3C22×C3×D311yescharacteristic
255C2×D3—21no—
256C2×D3—21no—
257C2×D3C3×Dic612yescomplemented
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—32no—
263C3×C4—32no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—42no—
267C3×C4—62no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—181no—
275C3×C4—181no—
276Dic3—21no—
277Dic3—21no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—61no—
282Dic3—121no—
283Dic3—121no—
284Dic3C3×D1212yes—
285C32—21no—
286C32—21no—
287C32—21no—
288C32—41no—
289C32#87811yescharacteristic, complemented
290C32#89711yescharacteristic, complemented
291C32C3×C22⋊Q811yescharacteristic, complemented
292C23—31no—
293Q8—182no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—182no—
300C2×C4C3×D3211yescharacteristic
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—62no—
314C2×C3—62no—
315C2×C3—122no—
316C2×C3D3×Dic611yescharacteristic
317C2×C3D3×D1211yescharacteristic
318C2×C3#380511yescharacteristic
319C2×C3C2×C3×Dic611yescharacteristic
320C2×C3C2×C3×D1211yescharacteristic
321C2×C3C3×D3×D411yescharacteristic
322C2×C3C3×#23711yescharacteristic
323C2×C3C3×#23711yescharacteristic
324C2×C3C3×D3×Q811yescharacteristic
325D3—22no—
326C22—32no—
327C22—61no—
328C22—61no—
329C22C2×C3×D3211yescharacteristic
330C4—21no—
331C4—32no—
332C4—61no—
333C4—61no—
334C4—181no—
335C4—181no—
336C3—21no—
337C3—21no—
338C3—21no—
339C3—41no—
340C3#6389311yescharacteristic, direct product factor
341C3C3×#87811yescharacteristic, complemented
342C3C3×#89711yescharacteristic, complemented
343C2—62no—
344C2C3×D3×Dic611yescharacteristic
345C2C3×D3×D1211yescharacteristic
346C2C3×#380511yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC22×C32×D3C22
hypercenterC2×C3×C4D32
hypocenterC32C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#627C2semidirect
3. C3×C4⋊(C3⋊Dic3)C2semidirect
5. C32×Dic3⋊C4C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
10. #63893C3direct (N × H)
24. C2×#627C2×C3semidirect
28. C4⋊(C3⋊Dic3)C2×C3semidirect
40. C3×Dic3⋊C4D3semidirect
41. C3×Dic3⋊C4C2×C3semidirect
42. Dic3⋊Dic3C2×C3semidirect
47. C2×C32×D3Q8semidirect
121. Dic3⋊C4C3×D3semidirect
139. C2×C3×D3Dic6semidirect
140. C2×C3×D3C3×Q8semidirect
169. C33C22⋊Q8semidirect
257. C2×D3C3×Dic6semidirect
289. C32#878semidirect
290. C32#897semidirect
291. C32C3×C22⋊Q8semidirect
340. C3#63893direct (N × H)
341. C3C3×#878semidirect
342. C3C3×#897semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6272
C2×C3×D3×Dic32
C3×C4⋊(C3⋊Dic3)2
C32×(C2×D3)⋊C42
C32×Dic3⋊C42
C3×Dic3⋊Dic32
C3×(C2×D3)⋊Dic32
C3×#8973
C3×#8783
#638933