Finite Group Databasebeta

← C3×#63906 #11723231

Subgroups — C3×#63906 #11723231

Order 864.

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

normal subgroups only

This group has 514 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×C32×C4×D3C211yesmaximal, characteristic, complemented
2C2×C3×#626C211yesmaximal, characteristic, complemented
3C2×C3×#626C211yesmaximal, characteristic, complemented
4C2×C3×#625C211yesmaximal, characteristic, complemented
5C3×#3752C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
8C3×#930—31nomaximal
9C3×#884—31nomaximal
10#63906C311yesmaximal, characteristic, direct product factor
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C22×C32×D3C2211yescharacteristic
14C2×C33×C4C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C32×C4×D3—22no—
19C3×#626—22no—
20C3×#626—22no—
21C3×#625—22no—
22C22×C32×C4—31no—
23C2×C3×#67—31no—
24C2×C3×#67—31no—
25C2×C3×#67—31no—
26C2×C3×#67—31no—
27C2×C3×D12—31no—
28C2×C3×D12—31no—
29C2×C3×C4×D3—21no—
30C2×C3×C4×D3D311yescharacteristic, complemented
31C2×C3×C4×D3C2×C311yescharacteristic, complemented
32C2×#626C2×C311yescharacteristic, complemented
33C2×#626C2×C311yescharacteristic, complemented
34C2×#625C2×C311yescharacteristic, complemented
35C32×C22⋊C4—31no—
36#3752C2×C311yescharacteristic, complemented
37C3×(C2×D3)⋊C4—21no—
38C3×(C2×D3)⋊C4—31no—
39C3×(C2×D3)⋊C4—31no—
40C3×(C2×D3)⋊C4—61no—
41C3×(C2×D3)⋊C4D311yescharacteristic, complemented
42C3×(C2×D3)⋊C4C2×C311yescharacteristic, complemented
43C3×C4⋊Dic3—31no—
44C3×Dic3⋊C4—31no—
45Dic3⋊Dic3C2×C311yescharacteristic, complemented
46C22×C33C2311yescharacteristic
47C2×C3×C3⋊D3—21no—
48C2×C3×C3⋊D3—21no—
49C2×C3×C3⋊D3—21no—
50C2×C32×D3—21no—
51C2×C32×D3—21no—
52C2×C32×D3—21no—
53C2×C32×D3—21no—
54C2×C32×D3—21no—
55C2×C32×D3D412yescomplemented
56C33×C4—21no—
57C3×C3⋊Dic3—21no—
58C32×Dic3—21no—
59C32×Dic3D412yescomplemented
60C3×#121—91no—
61#930—31no—
62#884—31no—
63C23×C32—31no—
64C23×C32—31no—
65C22×C3⋊D3C22×C311yescharacteristic
66C22×C3×D3—21no—
67C22×C3×D3—21no—
68C22×C3×D3—31no—
69C22×C3×D3—31no—
70C22×C3×D3—61no—
71C22×C3×D3C2×D311yescharacteristic
72C22×C3×D3C2×D311yescharacteristic
73C22×C3×D3C22×C311yescharacteristic
74C22×C3×D3C22×C311yescharacteristic
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4—21no—
78C2×C32×C4—31no—
79C2×C32×C4—31no—
80C2×C32×C4—41no—
81C2×C32×C4—62no—
82C2×C32×C4C2×D311yescharacteristic
83C2×C32×C4C2×D311yescharacteristic
84C2×C32×C4C22×C311yescharacteristic
85C2×C3⋊Dic3C22×C311yescharacteristic
86C3×#67—62no—
87C3×#67—62no—
88C3×#67—62no—
89C3×#67—62no—
90C2×C3×Dic3—21no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3C2×D311yescharacteristic
96C2×C3×Dic3C2×D311yescharacteristic
97C2×C3×Dic3C22×C311yescharacteristic
98C2×C3×Dic3C22×C311yescharacteristic
99C3×D12—62no—
100C3×D12—62no—
101C3×C4×D3—24no—
102C3×C4×D3—22no—
103C3×C4×D3—22no—
104#626—22no—
105#626—22no—
106#625—22no—
107C2×C33C2×D411yescharacteristic
108C2×C33C2×D411yescharacteristic
109C2×C33#4111yescharacteristic
110C3×C3⋊D3—41no—
111C32×D3—22no—
112C32×D3—41no—
113C2×C3×D4—91no—
114C2×C3×D4—91no—
115C2×C3×D4—91no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C2×#67—31no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×D12—31no—
124C2×D12—31no—
125C2×C4×D3C3×D311yescharacteristic, complemented
126C3×C4⋊C4—91no—
127C3×C22⋊C4—31no—
128C3×C22⋊C4—31no—
129C3×C22⋊C4—61no—
130C3×C22⋊C4—91no—
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4—31no—
133(C2×D3)⋊C4—61no—
134(C2×D3)⋊C4C3×D311yescharacteristic, complemented
135C4⋊Dic3—31no—
136Dic3⋊C4—31no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—32no—
141C22×C32—41no—
142C22×C32—61no—
143C22×C32—61no—
144C22×C32—61no—
145C22×C32—61no—
146C22×C32—61no—
147C22×C32C22×D311yescharacteristic
148C22×C32C22×D311yescharacteristic
149C22×C32C23×C311yescharacteristic
150C2×C3⋊D3—21no—
151C2×C3⋊D3—21no—
152C2×C3⋊D3—21no—
153C2×C3×D3—22no—
154C2×C3×D3—22no—
155C2×C3×D3—21no—
156C2×C3×D3—21no—
157C2×C3×D3—21no—
158C2×C3×D3—21no—
159C2×C3×D3—22no—
160C2×C3×D3—21no—
161C2×C3×D3—21no—
162C2×C3×D3—21no—
163C2×C3×D3—21no—
164C2×C3×D3—21no—
165C2×C3×D3—21no—
166C2×C3×D3—22no—
167C2×C3×D3—41no—
168C2×C3×D3—41no—
169C2×C3×D3—62no—
170C2×C3×D3—61no—
171C2×C3×D3—61no—
172C2×C3×D3—61no—
173C2×C3×D3—61no—
174C2×C3×D3—61no—
175C2×C3×D3—61no—
176C2×C3×D3—121no—
177C2×C3×D3—121no—
178C2×C3×D3D1212yescomplemented
179C2×C3×D3C3×D412yescomplemented
180C32×C4—21no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—32no—
185C32×C4—41no—
186C32×C4—41no—
187C32×C4—42no—
188C32×C4—61no—
189C3⋊Dic3—21no—
190C3×Dic3—21no—
191C3×Dic3—21no—
192C3×Dic3—22no—
193C3×Dic3—41no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—121no—
197C3×Dic3#6712yescomplemented
198C3×Dic3C3×D412yescomplemented
199#121—91noSylow
200C33#12111yescharacteristic, complemented, Sylow
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—31no—
204C23×C3—31no—
205C23×C3—61no—
206C23×C3—61no—
207C23×C3—91no—
208C22×D3—31no—
209C22×D3—31no—
210C22×D3—61no—
211C22×D3C2×C3×D311yescharacteristic
212C22×D3C2×C3×D311yescharacteristic
213C3×D4—182no—
214C3×D4—182no—
215C3×D4—182no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—41no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—64no—
229C2×C3×C4—91no—
230C2×C3×C4D3211yescharacteristic
231C2×C3×C4C2×C3×D311yescharacteristic
232C2×C3×C4C2×C3×D311yescharacteristic
233#67—62no—
234#67—62no—
235#67—62no—
236#67—62no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—61no—
240C2×Dic3C2×C3×D311yescharacteristic
241C2×Dic3C2×C3×D311yescharacteristic
242D12—62no—
243D12—62no—
244C4×D3—22no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—62no—
258C2×C32—121no—
259C2×C32C2×D1211yescharacteristic
260C2×C32#23511yescharacteristic
261C2×C32#23511yescharacteristic
262C2×C32D3×D411yescharacteristic
263C2×C32D3×D411yescharacteristic
264C2×C32C2×#6711yescharacteristic
265C2×C32C2×C3×D411yescharacteristic
266C2×C32C2×C3×D411yescharacteristic
267C2×C32C3×#4111yescharacteristic
268C3⋊D3—41no—
269C3×D3—24no—
270C3×D3—22no—
271C3×D3—22no—
272C3×D3—42no—
273C3×D3—41no—
274C3×D3—41no—
275C3×D3—122no—
276C3×D3—121no—
277C3×D3—121no—
278C2×D4—91no—
279C2×D4—91no—
280C2×D4—91no—
281C22×C4—31no—
282C4⋊C4—91no—
283C22⋊C4—31no—
284C22⋊C4—91no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—32no—
289C22×C3—32no—
290C22×C3—41no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—62no—
294C22×C3—62no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—61no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—61no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3C2×D3211yescharacteristic
311C22×C3C22×C3×D311yescharacteristic
312C22×C3C22×C3×D311yescharacteristic
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—62no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3C3×D1212yescomplemented
328C3×C4—22no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—32no—
333C3×C4—32no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—42no—
337C3×C4—62no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—121no—
341C3×C4—181no—
342Dic3—21no—
343Dic3—61no—
344Dic3—61no—
345Dic3—121no—
346Dic3C3×#6712yescomplemented
347C32—21no—
348C32—21no—
349C32—21no—
350C32—41no—
351C32#88411yescharacteristic, complemented
352C32#93011yescharacteristic, complemented
353C32C3×#12111yescharacteristic, complemented
354C23—31no—
355C23—31no—
356C23—91no—
357D4—182no—
358D4—182no—
359D4—182no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—62no—
363C2×C4—91no—
364C2×C4C3×D3211yescharacteristic
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—64no—
378C2×C3—62no—
379C2×C3—62no—
380C2×C3—122no—
381C2×C3—121no—
382C2×C3—121no—
383C2×C3—361no—
384C2×C3#379811yescharacteristic
385C2×C3D3×D1211yescharacteristic
386C2×C3D3×#6711yescharacteristic
387C2×C3C2×C3×D1211yescharacteristic
388C2×C3C3×#23511yescharacteristic
389C2×C3C3×#23511yescharacteristic
390C2×C3C3×D3×D411yescharacteristic
391C2×C3C3×D3×D411yescharacteristic
392C2×C3C2×C3×#6711yescharacteristic
393D3—22no—
394D3—41no—
395D3—122no—
396D3—121no—
397D3—121no—
398C22—32no—
399C22—61no—
400C22—61no—
401C22—61no—
402C22—61no—
403C22—61no—
404C22—181no—
405C22—181no—
406C22—181no—
407C22C2×C3×D3211yescharacteristic
408C4—21no—
409C4—32no—
410C4—61no—
411C4—181no—
412C3—21no—
413C3—21no—
414C3—21no—
415C3—41no—
416C3#6390611yescharacteristic, direct product factor
417C3C3×#88411yescharacteristic, complemented
418C3C3×#93011yescharacteristic, complemented
419C2—62no—
420C2—121no—
421C2—361no—
422C2C3×#379811yescharacteristic
423C2C3×D3×D1211yescharacteristic
424C2C3×D3×#6711yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
hypercenterC2×C3×C4D32
hypocenterC32C3×#121
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×C4×D3C2semidirect
2. C2×C3×#626C2semidirect
3. C2×C3×#626C2semidirect
4. C2×C3×#625C2semidirect
5. C3×#3752C2semidirect
6. C32×(C2×D3)⋊C4C2semidirect
7. C3×Dic3⋊Dic3C2semidirect
10. #63906C3direct (N × H)
30. C2×C3×C4×D3D3semidirect
31. C2×C3×C4×D3C2×C3semidirect
32. C2×#626C2×C3semidirect
33. C2×#626C2×C3semidirect
34. C2×#625C2×C3semidirect
36. #3752C2×C3semidirect
41. C3×(C2×D3)⋊C4D3semidirect
42. C3×(C2×D3)⋊C4C2×C3semidirect
45. Dic3⋊Dic3C2×C3semidirect
55. C2×C32×D3D4semidirect
59. C32×Dic3D4semidirect
125. C2×C4×D3C3×D3semidirect
134. (C2×D3)⋊C4C3×D3semidirect
178. C2×C3×D3D12semidirect
179. C2×C3×D3C3×D4semidirect
197. C3×Dic3#67semidirect
198. C3×Dic3C3×D4semidirect
200. C33#121semidirect
327. C2×D3C3×D12semidirect
346. Dic3C3×#67semidirect
351. C32#884semidirect
352. C32#930semidirect
353. C32C3×#121semidirect
416. C3#63906direct (N × H)
417. C3C3×#884semidirect
418. C3C3×#930semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D32
C2×C3×#6262
C2×C3×#6252
C3×#37522
C32×(C2×D3)⋊C42
C3×Dic3⋊Dic32
C3×#9303
C3×#8843
#639063