Finite Group Databasebeta

← C3×#63888 #11723213

Subgroups — C3×#63888 #11723213

Order 864.

410 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 100 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×C32×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×#624C211yesmaximal, characteristic
3C3×#3752C211yesmaximal, characteristic
4C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×#3722C211yesmaximal, characteristic
8C3×#946—31nomaximal
9C3×#896—31nomaximal
10#63888C311yesmaximal, 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×Dic6—22no—
19C3×#624—22no—
20C2×C32×Q8—31no—
21C2×C3×C4×D3—31no—
22C2×C3×C4×D3—31no—
23C2×C3×Dic6—21no—
24C2×C3×Dic6D311yescharacteristic, complemented
25C2×C3×Dic6C2×C311yescharacteristic, complemented
26C2×#624C2×C311yescharacteristic
27C32×C4⋊C4—31no—
28#3752C2×C311yescharacteristic
29C3×(C2×D3)⋊C4—31no—
30C3×(C2×D3)⋊C4—31no—
31C3×(C2×D3)⋊C4—31no—
32C3×(C2×D3)⋊C4—31no—
33C3×(C2×D3)⋊C4—61no—
34C3×C4⋊Dic3—31no—
35C3×Dic3⋊C4—21no—
36C3×Dic3⋊C4—31no—
37C3×Dic3⋊C4—31no—
38C3×Dic3⋊C4—31no—
39C3×Dic3⋊C4D311yescharacteristic, complemented
40C3×Dic3⋊C4C2×C311yescharacteristic, complemented
41#3724C2×C311yescharacteristic, complemented
42Dic3⋊Dic3C2×C311yescharacteristic, complemented
43#3722C2×C311yescharacteristic
44C22×C33C2311yescharacteristic
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3Q812yescomplemented
48C33×C4—21no—
49C3×C3⋊Dic3—21no—
50C32×Dic3—21no—
51C32×Dic3—21no—
52C32×Dic3—21no—
53C32×Dic3D412yes—
54C3×C22⋊Q8—91no—
55#946—31no—
56#896—31no—
57C22×C3⋊D3C22×C311yescharacteristic
58C22×C3×D3—31no—
59C22×C3×D3—31no—
60C22×C3×D3—61no—
61C32×Q8—62no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69C2×C32×C4—41no—
70C2×C32×C4C2×D311yescharacteristic
71C2×C32×C4C2×D311yescharacteristic
72C2×C32×C4C22×C311yescharacteristic
73C2×C3⋊Dic3C22×C311yescharacteristic
74C2×C3×Dic3—21no—
75C2×C3×Dic3—21no—
76C2×C3×Dic3—21no—
77C2×C3×Dic3—21no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—31no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3C2×D311yescharacteristic
82C2×C3×Dic3C2×D311yescharacteristic
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C3×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C22×C311yescharacteristic
86C2×C3×Dic3C22×C311yescharacteristic
87C2×C3×Dic3C22×C311yescharacteristic
88C2×C3×Dic3C22×C311yescharacteristic
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91C3×Dic6—24no—
92C3×Dic6—22no—
93C3×Dic6—22no—
94#624—22no—
95C2×C33C2×D411yescharacteristic
96C2×C33C2×Q811yescharacteristic
97C2×C33#4111yescharacteristic
98C3×C3⋊D3—22no—
99C2×C3×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—61no—
102C22×C3×C4—91no—
103C2×C4×D3—31no—
104C2×C4×D3—31no—
105C2×Dic6C3×D311yescharacteristic, complemented
106C3×C4⋊C4—31no—
107C3×C4⋊C4—31no—
108C3×C4⋊C4—61no—
109C3×C4⋊C4—91no—
110C3×C4⋊C4—91no—
111C3×C22⋊C4—91no—
112C3×C22⋊C4—91no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4—31no—
115(C2×D3)⋊C4—31no—
116(C2×D3)⋊C4—31no—
117(C2×D3)⋊C4—61no—
118C4⋊Dic3—31no—
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—31no—
122Dic3⋊C4C3×D311yescharacteristic, complemented
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—41no—
127C22×C32C22×D311yescharacteristic
128C22×C32C22×D311yescharacteristic
129C22×C32C23×C311yescharacteristic
130C2×C3⋊D3—21no—
131C2×C3⋊D3—21no—
132C2×C3⋊D3C3×Q812yescomplemented
133C2×C3×D3—32no—
134C2×C3×D3—32no—
135C2×C3×D3—62no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—121no—
141C2×C3×D3—121no—
142C32×C4—21no—
143C32×C4—22no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—32no—
147C32×C4—41no—
148C32×C4—41no—
149C32×C4—42no—
150C32×C4—61no—
151C32×C4—61no—
152C32×C4—61no—
153C3⋊Dic3—21no—
154C3×Dic3—22no—
155C3×Dic3—22no—
156C3×Dic3—21no—
157C3×Dic3—21no—
158C3×Dic3—21no—
159C3×Dic3—21no—
160C3×Dic3—21no—
161C3×Dic3—21no—
162C3×Dic3—22no—
163C3×Dic3—41no—
164C3×Dic3—61no—
165C3×Dic3—61no—
166C3×Dic3—121no—
167C3×Dic3#6712yes—
168C3×Dic3C3×D412yes—
169C22⋊Q8—91noSylow
170C33C22⋊Q811yescharacteristic, complemented, Sylow
171C23×C3—91no—
172C22×D3—31no—
173C22×D3—31no—
174C22×D3—61no—
175C3×Q8—62no—
176C3×Q8—62no—
177C3×Q8—64no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—41no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—91no—
195C2×C3×C4—182no—
196C2×C3×C4D3211yescharacteristic
197C2×C3×C4C2×C3×D311yescharacteristic
198C2×C3×C4C2×C3×D311yescharacteristic
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—61no—
202C2×Dic3C2×C3×D311yescharacteristic
203C2×Dic3C2×C3×D311yescharacteristic
204C2×Dic3C2×C3×D311yescharacteristic
205C2×Dic3C2×C3×D311yescharacteristic
206C4×D3—62no—
207C4×D3—62no—
208Dic6—22no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32#23511yescharacteristic
222C2×C32D3×D411yescharacteristic
223C2×C32D3×Q811yescharacteristic
224C2×C32D3×Q811yescharacteristic
225C2×C32#23911yescharacteristic
226C2×C32C2×#6711yescharacteristic
227C2×C32C2×C3×D411yescharacteristic
228C2×C32C2×C3×Q811yescharacteristic
229C2×C32C3×#4111yescharacteristic
230C3⋊D3—22no—
231C3×D3—62no—
232C3×D3—62no—
233C3×D3—122no—
234C2×Q8—31no—
235C22×C4—91no—
236C4⋊C4—31no—
237C4⋊C4—91no—
238C4⋊C4—91no—
239C22⋊C4—91no—
240C22⋊C4—91no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—41no—
245C22×C3—92no—
246C22×C3—181no—
247C22×C3—181no—
248C22×C3C2×D3211yescharacteristic
249C22×C3C22×C3×D311yescharacteristic
250C22×C3C22×C3×D311yescharacteristic
251C2×D3—32no—
252C2×D3—32no—
253C2×D3—62no—
254C2×D3—61no—
255C2×D3—61no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—121no—
259C2×D3—121no—
260C3×C4—22no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—32no—
265C3×C4—32no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—42no—
269C3×C4—62no—
270C3×C4—62no—
271C3×C4—62no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—121no—
279C3×C4—181no—
280Dic3—21no—
281Dic3—21no—
282Dic3—21no—
283Dic3—61no—
284Dic3—61no—
285Dic3—121no—
286Dic3C3×#6712yes—
287C32—21no—
288C32—21no—
289C32—21no—
290C32—41no—
291C32#89611yescharacteristic, complemented
292C32#94611yescharacteristic, complemented
293C32C3×C22⋊Q811yescharacteristic, complemented
294C23—91no—
295Q8—62no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—91no—
301C2×C4—182no—
302C2×C4C3×D3211yescharacteristic
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—182no—
316C2×C3#379711yescharacteristic
317C2×C3#379911yescharacteristic
318C2×C3D3×#6711yescharacteristic
319C2×C3C3×#23511yescharacteristic
320C2×C3C3×D3×D411yescharacteristic
321C2×C3C3×D3×Q811yescharacteristic
322C2×C3C3×D3×Q811yescharacteristic
323C2×C3C3×#23911yescharacteristic
324C2×C3C2×C3×#6711yescharacteristic
325D3—62no—
326D3—62no—
327D3—122no—
328C22—92no—
329C22—181no—
330C22—181no—
331C22C2×C3×D3211yescharacteristic
332C4—21no—
333C4—32no—
334C4—61no—
335C4—61no—
336C4—61no—
337C4—181no—
338C3—21no—
339C3—21no—
340C3—21no—
341C3—41no—
342C3#6388811yescharacteristic, direct product factor
343C3C3×#89611yescharacteristic, complemented
344C3C3×#94611yescharacteristic, complemented
345C2—182no—
346C2C3×#379711yescharacteristic
347C2C3×#379911yescharacteristic
348C2C3×D3×#6711yescharacteristic
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×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×C32×Dic6C2semidirect
4. C32×Dic3⋊C4C2semidirect
5. C3×#3724C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
10. #63888C3direct (N × H)
24. C2×C3×Dic6D3semidirect
25. C2×C3×Dic6C2×C3semidirect
39. C3×Dic3⋊C4D3semidirect
40. C3×Dic3⋊C4C2×C3semidirect
41. #3724C2×C3semidirect
42. Dic3⋊Dic3C2×C3semidirect
47. C2×C3×C3⋊D3Q8semidirect
105. C2×Dic6C3×D3semidirect
122. Dic3⋊C4C3×D3semidirect
132. C2×C3⋊D3C3×Q8semidirect
170. C33C22⋊Q8semidirect
291. C32#896semidirect
292. C32#946semidirect
293. C32C3×C22⋊Q8semidirect
342. C3#63888direct (N × H)
343. C3C3×#896semidirect
344. C3C3×#946semidirect

Maximal subgroups

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

subgroupindex
C2×C32×Dic62
C2×C3×#6242
C3×#37522
C32×Dic3⋊C42
C3×#37242
C3×Dic3⋊Dic32
C3×#37222
C3×#9463
C3×#8963
#638883