Finite Group Databasebeta

← C3×#63868 #11723193

Subgroups — C3×#63868 #11723193

Order 864.

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

normal subgroups only

This group has 346 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×#3752C211yesmaximal, characteristic
2C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C32×C4×Dic3C211yesmaximal, characteristic, complemented
4C3×#3724C211yesmaximal, characteristic, complemented
5C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#3722C211yesmaximal, characteristic
7C3×#3722C211yesmaximal, characteristic
8C3×#898—31nomaximal
9C3×#876—31nomaximal
10#63868C311yesmaximal, characteristic, direct product factor
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C32×Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C32×C4⋊C4—31no—
19C32×C42—31no—
20#3752C2×C311yescharacteristic
21C3×(C2×D3)⋊C4—31no—
22C3×(C2×D3)⋊C4—31no—
23C3×(C2×D3)⋊C4—31no—
24C3×(C2×D3)⋊C4—31no—
25C3×(C2×D3)⋊C4—31no—
26C3×(C2×D3)⋊C4—31no—
27C3×(C2×D3)⋊C4—61no—
28C3×C4⋊Dic3—31no—
29C3×Dic3⋊C4—21no—
30C3×Dic3⋊C4—31no—
31C3×Dic3⋊C4—31no—
32C3×Dic3⋊C4—31no—
33C3×Dic3⋊C4D311yescharacteristic, complemented
34C3×Dic3⋊C4C2×C311yescharacteristic, complemented
35C3×C4×Dic3—21no—
36C3×C4×Dic3D311yescharacteristic, complemented
37C3×C4×Dic3C2×C311yescharacteristic, complemented
38#3724C2×C311yescharacteristic, complemented
39Dic3⋊Dic3C2×C311yescharacteristic, complemented
40#3722C2×C311yescharacteristic
41#3722C2×C311yescharacteristic
42C22×C33C2311yescharacteristic
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3—21no—
45C2×C3×C3⋊D3—21no—
46C33×C4—21no—
47C3×C3⋊Dic3—21no—
48C32×Dic3—21no—
49C32×Dic3—21no—
50C32×Dic3—21no—
51C32×Dic3—21no—
52C3×#126—91no—
53#898—31no—
54#876—31no—
55C22×C3⋊D3C22×C311yescharacteristic
56C22×C3×D3—31no—
57C22×C3×D3—31no—
58C22×C3×D3—61no—
59C2×C32×C4—21no—
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—41no—
67C2×C32×C4C2×D311yescharacteristic
68C2×C32×C4C2×D311yescharacteristic
69C2×C32×C4C22×C311yescharacteristic
70C2×C3⋊Dic3C22×C311yescharacteristic
71C2×C3×Dic3—21no—
72C2×C3×Dic3—21no—
73C2×C3×Dic3—21no—
74C2×C3×Dic3—21no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
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#4111yescharacteristic
87C2×C33#4111yescharacteristic
88C2×C33#4111yescharacteristic
89C3×C3⋊D3—41no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—31no—
92C3×C4⋊C4—61no—
93C3×C4⋊C4—91no—
94C3×C4⋊C4—91no—
95C3×C22⋊C4—91no—
96C3×C22⋊C4—91no—
97C3×C22⋊C4—91no—
98C3×C42—31no—
99C3×C42—31no—
100C3×C42—61no—
101(C2×D3)⋊C4—31no—
102(C2×D3)⋊C4—31no—
103(C2×D3)⋊C4—31no—
104(C2×D3)⋊C4—31no—
105(C2×D3)⋊C4—31no—
106(C2×D3)⋊C4—31no—
107(C2×D3)⋊C4—61no—
108C4⋊Dic3—31no—
109Dic3⋊C4—31no—
110Dic3⋊C4—31no—
111Dic3⋊C4—31no—
112Dic3⋊C4C3×D311yescharacteristic, complemented
113C4×Dic3C3×D311yescharacteristic, complemented
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—41no—
118C22×C32C22×D311yescharacteristic
119C22×C32C22×D311yescharacteristic
120C22×C32C23×C311yescharacteristic
121C2×C3⋊D3—21no—
122C2×C3⋊D3—21no—
123C2×C3⋊D3—21no—
124C2×C3×D3—62no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—61no—
128C2×C3×D3—61no—
129C2×C3×D3—61no—
130C2×C3×D3—61no—
131C2×C3×D3—121no—
132C2×C3×D3—121no—
133C32×C4—21no—
134C32×C4—22no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—41no—
138C32×C4—41no—
139C32×C4—42no—
140C32×C4—61no—
141C32×C4—61no—
142C32×C4—61no—
143C32×C4—61no—
144C3⋊Dic3—21no—
145C3×Dic3—22no—
146C3×Dic3—22no—
147C3×Dic3—22no—
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—61no—
158C3×Dic3—61no—
159C3×Dic3—121no—
160#126—91noSylow
161C33#12611yescharacteristic, complemented, Sylow
162C23×C3—91no—
163C22×D3—31no—
164C22×D3—31no—
165C22×D3—61no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—41no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—91no—
183C2×C3×C4D3211yescharacteristic
184C2×C3×C4C2×C3×D311yescharacteristic
185C2×C3×C4C2×C3×D311yescharacteristic
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—61no—
189C2×Dic3C2×C3×D311yescharacteristic
190C2×Dic3C2×C3×D311yescharacteristic
191C2×Dic3C2×C3×D311yescharacteristic
192C2×Dic3C2×C3×D311yescharacteristic
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32#23511yescharacteristic
206C2×C32#23511yescharacteristic
207C2×C32#23511yescharacteristic
208C2×C32#23511yescharacteristic
209C2×C32#23711yescharacteristic
210C2×C32#23911yescharacteristic
211C2×C32C3×#4111yescharacteristic
212C2×C32C3×#4111yescharacteristic
213C2×C32C3×#4111yescharacteristic
214C3⋊D3—41no—
215C3×D3—122no—
216C3×D3—121no—
217C3×D3—121no—
218C4⋊C4—31no—
219C4⋊C4—91no—
220C4⋊C4—91no—
221C22⋊C4—91no—
222C22⋊C4—91no—
223C22⋊C4—91no—
224C42—31no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—41no—
229C22×C3—181no—
230C22×C3—181no—
231C22×C3—181no—
232C22×C3C2×D3211yescharacteristic
233C22×C3C22×C3×D311yescharacteristic
234C22×C3C22×C3×D311yescharacteristic
235C2×D3—62no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—61no—
242C2×D3—121no—
243C2×D3—121no—
244C3×C4—22no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4—42no—
251C3×C4—62no—
252C3×C4—62no—
253C3×C4—62no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—121no—
263C3×C4—181no—
264Dic3—21no—
265Dic3—21no—
266Dic3—21no—
267Dic3—21no—
268Dic3—61no—
269Dic3—61no—
270Dic3—121no—
271C32—21no—
272C32—21no—
273C32—21no—
274C32—41no—
275C32#87611yescharacteristic, complemented
276C32#89811yescharacteristic, complemented
277C32C3×#12611yescharacteristic, complemented
278C23—91no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—91no—
284C2×C4C3×D3211yescharacteristic
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—361no—
298C2×C3#379811yescharacteristic
299C2×C3#379911yescharacteristic
300C2×C3#380411yescharacteristic
301C2×C3C3×#23511yescharacteristic
302C2×C3C3×#23511yescharacteristic
303C2×C3C3×#23511yescharacteristic
304C2×C3C3×#23511yescharacteristic
305C2×C3C3×#23711yescharacteristic
306C2×C3C3×#23911yescharacteristic
307D3—122no—
308D3—121no—
309D3—121no—
310C22—181no—
311C22—181no—
312C22—181no—
313C22C2×C3×D3211yescharacteristic
314C4—21no—
315C4—61no—
316C4—61no—
317C4—61no—
318C4—61no—
319C4—181no—
320C3—21no—
321C3—21no—
322C3—21no—
323C3—41no—
324C3#6386811yescharacteristic, direct product factor
325C3C3×#87611yescharacteristic, complemented
326C3C3×#89811yescharacteristic, complemented
327C2—361no—
328C2C3×#379811yescharacteristic
329C2C3×#379911yescharacteristic
330C2C3×#380411yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC22×C3×C3⋊D3C22
hypercenterC2×C3×C4D32
hypocenterC32C3×#126
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C32×Dic3⋊C4C2semidirect
3. C32×C4×Dic3C2semidirect
4. C3×#3724C2semidirect
5. C3×Dic3⋊Dic3C2semidirect
10. #63868C3direct (N × H)
33. C3×Dic3⋊C4D3semidirect
34. C3×Dic3⋊C4C2×C3semidirect
36. C3×C4×Dic3D3semidirect
37. C3×C4×Dic3C2×C3semidirect
38. #3724C2×C3semidirect
39. Dic3⋊Dic3C2×C3semidirect
112. Dic3⋊C4C3×D3semidirect
113. C4×Dic3C3×D3semidirect
161. C33#126semidirect
275. C32#876semidirect
276. C32#898semidirect
277. C32C3×#126semidirect
324. C3#63868direct (N × H)
325. C3C3×#876semidirect
326. C3C3×#898semidirect

Maximal subgroups

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

subgroupindex
C3×#37522
C32×Dic3⋊C42
C32×C4×Dic32
C3×#37242
C3×Dic3⋊Dic32
C3×#37222
C3×#8983
C3×#8763
#638683