Finite Group Databasebeta

← C3×#63956 #11723281

Subgroups — C3×#63956 #11723281

Order 864.

374 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 88 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×#3757C211yesmaximal, characteristic
2C32×#217C211yesmaximal, characteristic
3C32×#217C211yesmaximal, characteristic
4C3×#3724C211yesmaximal, characteristic, complemented
5C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×Dic32C211yesmaximal, characteristic, complemented
8C3×#879—31nomaximal
9C3×#879—31nomaximal
10#63956C311yesmaximal, characteristic, direct product factor
11C23×C33C2211yescharacteristic
12C2×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C32×Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C32×C22⋊C4—31no—
19C32×C22⋊C4—31no—
20#3757C2×C311yescharacteristic
21C3×#217—21no—
22C3×#217—21no—
23C3×#217—31no—
24C3×#217—31no—
25C3×#217—61no—
26C3×#217D311yescharacteristic
27C3×#217D311yescharacteristic
28C3×#217C2×C311yescharacteristic
29C3×#217C2×C311yescharacteristic
30C3×C4⋊Dic3—31no—
31C3×C4⋊Dic3—31no—
32C3×Dic3⋊C4—31no—
33C3×Dic3⋊C4—31no—
34C3×Dic3⋊C4—31no—
35C3×Dic3⋊C4—31no—
36C3×C4×Dic3—31no—
37C3×C4×Dic3—31no—
38#3724C2×C311yescharacteristic, complemented
39Dic3⋊Dic3C2×C311yescharacteristic, complemented
40Dic3⋊Dic3C2×C311yescharacteristic, complemented
41Dic32C2×C311yescharacteristic, complemented
42C22×C33—21no—
43C22×C33—21no—
44C22×C33—21no—
45C22×C33C2311yescharacteristic
46C3×C3⋊Dic3—21no—
47C3×C3⋊Dic3—21no—
48C32×Dic3—21no—
49C32×Dic3—21no—
50C32×Dic3—21no—
51C32×Dic3—21no—
52C3×#126—91no—
53#879—31no—
54#879—31no—
55C23×C32—21no—
56C23×C32—21no—
57C23×C32—21no—
58C23×C32—41no—
59C23×C32C2×D311yescharacteristic
60C23×C32C2×D311yescharacteristic
61C23×C32C22×C311yescharacteristic
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C3⋊Dic3C22×C311yescharacteristic
67C2×C3⋊Dic3C22×C311yescharacteristic
68C2×C3×Dic3—21no—
69C2×C3×Dic3—21no—
70C2×C3×Dic3—21no—
71C2×C3×Dic3—21no—
72C2×C3×Dic3—31no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3C2×D311yescharacteristic
79C2×C3×Dic3C2×D311yescharacteristic
80C2×C3×Dic3C2×D311yescharacteristic
81C2×C3×Dic3C2×D311yescharacteristic
82C2×C3×Dic3C22×C311yescharacteristic
83C2×C3×Dic3C22×C311yescharacteristic
84C2×C3×Dic3C22×C311yescharacteristic
85C2×C3×Dic3C22×C311yescharacteristic
86C2×C33—41no—
87C2×C33#4111yescharacteristic
88C2×C33#4111yescharacteristic
89C2×C33#4111yescharacteristic
90C3×C4⋊C4—91no—
91C3×C4⋊C4—91no—
92C3×C4⋊C4—91no—
93C3×C22⋊C4—31no—
94C3×C22⋊C4—31no—
95C3×C22⋊C4—31no—
96C3×C22⋊C4—31no—
97C3×C22⋊C4—61no—
98C3×C22⋊C4—61no—
99C3×C22⋊C4—91no—
100C3×C42—91no—
101#217—31no—
102#217—31no—
103#217—61no—
104#217C3×D311yescharacteristic
105#217C3×D311yescharacteristic
106C4⋊Dic3—31no—
107C4⋊Dic3—31no—
108Dic3⋊C4—31no—
109Dic3⋊C4—31no—
110Dic3⋊C4—31no—
111Dic3⋊C4—31no—
112C4×Dic3—31no—
113C4×Dic3—31no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—22no—
118C22×C32—22no—
119C22×C32—22no—
120C22×C32—21no—
121C22×C32—21no—
122C22×C32—21no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—41no—
130C22×C32—41no—
131C22×C32—41no—
132C22×C32—41no—
133C22×C32—41no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32C22×D311yescharacteristic
140C22×C32C22×D311yescharacteristic
141C22×C32C23×C311yescharacteristic
142C32×C4—61no—
143C32×C4—61no—
144C32×C4—61no—
145C32×C4—61no—
146C3⋊Dic3—21no—
147C3⋊Dic3—21no—
148C3×Dic3—21no—
149C3×Dic3—21no—
150C3×Dic3—21no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3—21no—
154C3×Dic3—21no—
155C3×Dic3—21no—
156C3×Dic3—41no—
157C3×Dic3—41no—
158C3×Dic3—41no—
159C3×Dic3—41no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—61no—
163C3×Dic3—61no—
164C3×Dic3—121no—
165C3×Dic3—121no—
166#126—91noSylow
167C33#12611yescharacteristic, complemented, Sylow
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—41no—
172C23×C3D3211yescharacteristic
173C23×C3C2×C3×D311yescharacteristic
174C23×C3C2×C3×D311yescharacteristic
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—91no—
188C2×C3×C4—91no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3C2×C3×D311yescharacteristic
196C2×Dic3C2×C3×D311yescharacteristic
197C2×Dic3C2×C3×D311yescharacteristic
198C2×Dic3C2×C3×D311yescharacteristic
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—42no—
212C2×C32—42no—
213C2×C32—42no—
214C2×C32—44no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32#23511yescharacteristic
219C2×C32#23511yescharacteristic
220C2×C32#23711yescharacteristic
221C2×C32#23711yescharacteristic
222C2×C32#23711yescharacteristic
223C2×C32#23711yescharacteristic
224C2×C32C3×#4111yescharacteristic
225C2×C32C3×#4111yescharacteristic
226C2×C32C3×#4111yescharacteristic
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4—91no—
230C22⋊C4—31no—
231C22⋊C4—31no—
232C22⋊C4—91no—
233C42—91no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—42no—
258C22×C3—41no—
259C22×C3C2×D3211yescharacteristic
260C22×C3C22×C3×D311yescharacteristic
261C22×C3C22×C3×D311yescharacteristic
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—121no—
271C3×C4—121no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—181no—
275C3×C4—181no—
276Dic3—21no—
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—61no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—121no—
285Dic3—121no—
286C32—21no—
287C32—21no—
288C32—21no—
289C32—41no—
290C32#87911yescharacteristic, complemented
291C32#87911yescharacteristic, complemented
292C32C3×#12611yescharacteristic, complemented
293C23C3×D3211yescharacteristic
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C3—21no—
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—42no—
310C2×C3—42no—
311C2×C3—42no—
312C2×C3—44no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3#380411yescharacteristic
320C2×C3#380411yescharacteristic
321C2×C3#380511yescharacteristic
322C2×C3C3×#23511yescharacteristic
323C2×C3C3×#23511yescharacteristic
324C2×C3C3×#23711yescharacteristic
325C2×C3C3×#23711yescharacteristic
326C2×C3C3×#23711yescharacteristic
327C2×C3C3×#23711yescharacteristic
328C22—21no—
329C22—21no—
330C22—21no—
331C22C2×C3×D3211yescharacteristic
332C4—61no—
333C4—61no—
334C4—61no—
335C4—61no—
336C4—181no—
337C4—181no—
338C3—21no—
339C3—21no—
340C3—21no—
341C3—41no—
342C3#6395611yescharacteristic, direct product factor
343C3C3×#87911yescharacteristic, complemented
344C3C3×#87911yescharacteristic, complemented
345C2—41no—
346C2C3×#380411yescharacteristic
347C2C3×#380411yescharacteristic
348C2C3×#380511yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
Fitting, prime radicalC23×C33C22
socleC22×C33C23
hypercenterC23×C3D32
hypocenterC32C3×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
4. C3×#3724C2semidirect
5. C3×Dic3⋊Dic3C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
7. C3×Dic32C2semidirect
10. #63956C3direct (N × H)
38. #3724C2×C3semidirect
39. Dic3⋊Dic3C2×C3semidirect
40. Dic3⋊Dic3C2×C3semidirect
41. Dic32C2×C3semidirect
167. C33#126semidirect
290. C32#879semidirect
291. C32#879semidirect
292. C32C3×#126semidirect
342. C3#63956direct (N × H)
343. C3C3×#879semidirect
344. C3C3×#879semidirect

Maximal subgroups

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

subgroupindex
C3×#37572
C32×#2172
C3×#37242
C3×Dic3⋊Dic32
C3×Dic322
C3×#8793
#639563