Finite Group Databasebeta

← C3×#63971 #11723296

Subgroups — C3×#63971 #11723296

Order 864.

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

normal subgroups only

This group has 550 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×C3×#638C211yesmaximal, characteristic, complemented
2C2×C32×#67C211yesmaximal, characteristic, complemented
3C2×C3×#626C211yesmaximal, characteristic, complemented
4C2×C3×D3×Dic3C211yesmaximal, characteristic, complemented
5C32×#217C211yesmaximal, characteristic, complemented
6C3×#3724C211yesmaximal, characteristic, complemented
7C3×#3722C211yesmaximal, characteristic, complemented
8C3×#939—31nomaximal
9C3×#884—31nomaximal
10#63971C311yesmaximal, characteristic, direct product factor
11C23×C33C2211yescharacteristic
12C22×C3×C3⋊D3C2211yescharacteristic
13C22×C32×D3C2211yescharacteristic
14C3×#638—22no—
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C32×#67—22no—
17C2×C32×Dic3C2211yescharacteristic
18C2×C32×Dic3C2211yescharacteristic
19C2×C32×Dic3C2211yescharacteristic
20C3×#626—22no—
21C3×D3×Dic3—22no—
22C2×C32×D4—31no—
23C2×#638C2×C311yescharacteristic, complemented
24C2×C3×#67—21no—
25C2×C3×#67—31no—
26C2×C3×#67—31no—
27C2×C3×#67—31no—
28C2×C3×#67—61no—
29C2×C3×#67D311yescharacteristic, complemented
30C2×C3×#67C2×C311yescharacteristic, complemented
31C22×C3×Dic3—31no—
32C2×C3×D12—31no—
33C2×C3×C4×D3—31no—
34C2×#626C2×C311yescharacteristic, complemented
35C2×D3×Dic3C2×C311yescharacteristic, complemented
36C32×C22⋊C4—31no—
37C3×#217—21no—
38C3×#217D311yescharacteristic, complemented
39C3×#217C2×C311yescharacteristic, complemented
40C3×(C2×D3)⋊C4—31no—
41C3×(C2×D3)⋊C4—31no—
42C3×Dic3⋊C4—31no—
43C3×Dic3⋊C4—31no—
44#3724C2×C311yescharacteristic, complemented
45#3722C2×C311yescharacteristic, complemented
46C22×C33—21no—
47C22×C33—21no—
48C22×C33—21no—
49C22×C33C2311yescharacteristic
50C2×C3×C3⋊D3—21no—
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—21no—
53C2×C32×D3—21no—
54C2×C32×D3—21no—
55C2×C32×D3D412yescomplemented
56C3×C3⋊Dic3D412yescomplemented
57C32×Dic3—21no—
58C32×Dic3—21no—
59C32×Dic3—21no—
60C3×#121—91no—
61#939—31no—
62#884—31no—
63C23×C32—21no—
64C23×C32—21no—
65C23×C32—21no—
66C23×C32—31no—
67C23×C32—41no—
68C23×C32C2×D311yescharacteristic
69C23×C32C2×D311yescharacteristic
70C23×C32C22×C311yescharacteristic
71C22×C3⋊D3C22×C311yescharacteristic
72C22×C3×D3—21no—
73C22×C3×D3—31no—
74C22×C3×D3—31no—
75C22×C3×D3—61no—
76C22×C3×D3C2×D311yescharacteristic
77C22×C3×D3C22×C311yescharacteristic
78C32×D4—62no—
79C2×C32×C4—31no—
80C2×C32×C4—31no—
81C2×C32×C4—31no—
82#638—22no—
83C2×C3⋊Dic3C22×C311yescharacteristic
84C3×#67—24no—
85C3×#67—22no—
86C3×#67—22no—
87C3×#67—62no—
88C3×#67—62no—
89C3×#67—62no—
90C3×#67—64no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—31no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3C2×D311yescharacteristic
99C2×C3×Dic3C2×D311yescharacteristic
100C2×C3×Dic3C2×D311yescharacteristic
101C2×C3×Dic3C22×C311yescharacteristic
102C2×C3×Dic3C22×C311yescharacteristic
103C2×C3×Dic3C22×C311yescharacteristic
104C3×D12—62no—
105C3×C4×D3—62no—
106#626—22no—
107D3×Dic3—22no—
108C2×C33—41no—
109C2×C33C2×D411yescharacteristic
110C2×C33C2×D411yescharacteristic
111C2×C33#4111yescharacteristic
112C3×C3⋊D3—41no—
113C32×D3—22no—
114C2×C3×D4—31no—
115C2×C3×D4—31no—
116C2×C3×D4—61no—
117C2×C3×D4—91no—
118C2×C3×D4—91no—
119C22×C3×C4—91no—
120C2×#67—31no—
121C2×#67—31no—
122C2×#67—31no—
123C2×#67—61no—
124C2×#67C3×D311yescharacteristic, complemented
125C22×Dic3—31no—
126C2×D12—31no—
127C2×C4×D3—31no—
128C3×C4⋊C4—91no—
129C3×C22⋊C4—31no—
130C3×C22⋊C4—31no—
131C3×C22⋊C4—61no—
132C3×C22⋊C4—91no—
133#217C3×D311yescharacteristic, complemented
134(C2×D3)⋊C4—31no—
135(C2×D3)⋊C4—31no—
136Dic3⋊C4—31no—
137Dic3⋊C4—31no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—21no—
141C22×C32—22no—
142C22×C32—22no—
143C22×C32—22no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—32no—
154C22×C32—41no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—42no—
162C22×C32—42no—
163C22×C32—42no—
164C22×C32—61no—
165C22×C32—61no—
166C22×C32C22×D311yescharacteristic
167C22×C32C22×D311yescharacteristic
168C22×C32C23×C311yescharacteristic
169C2×C3⋊D3—21no—
170C2×C3⋊D3—21no—
171C2×C3⋊D3—21no—
172C2×C3×D3—21no—
173C2×C3×D3—21no—
174C2×C3×D3—21no—
175C2×C3×D3—21no—
176C2×C3×D3—22no—
177C2×C3×D3—41no—
178C2×C3×D3—41no—
179C2×C3×D3—62no—
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3#6712yescomplemented
189C2×C3×D3C3×D412yescomplemented
190C32×C4—61no—
191C32×C4—61no—
192C32×C4—61no—
193C3⋊Dic3C3×D412yescomplemented
194C3×Dic3—22no—
195C3×Dic3—21no—
196C3×Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—32no—
202C3×Dic3—32no—
203C3×Dic3—41no—
204C3×Dic3—41no—
205C3×Dic3—62no—
206#121—91noSylow
207C33#12111yescharacteristic, complemented, Sylow
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—31no—
212C23×C3—31no—
213C23×C3—41no—
214C23×C3—61no—
215C23×C3—91no—
216C23×C3D3211yescharacteristic
217C23×C3C2×C3×D311yescharacteristic
218C23×C3C2×C3×D311yescharacteristic
219C22×D3—31no—
220C22×D3—31no—
221C22×D3—61no—
222C22×D3C2×C3×D311yescharacteristic
223C3×D4—62no—
224C3×D4—62no—
225C3×D4—64no—
226C3×D4—182no—
227C3×D4—182no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—182no—
239#67—22no—
240#67—62no—
241#67—62no—
242#67—62no—
243#67—64no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—62no—
248C2×Dic3C2×C3×D311yescharacteristic
249C2×Dic3C2×C3×D311yescharacteristic
250C2×Dic3C2×C3×D311yescharacteristic
251D12—62no—
252C4×D3—62no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—42no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32—44no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—62no—
273C2×C32#23511yescharacteristic
274C2×C32D3×D411yescharacteristic
275C2×C32D3×D411yescharacteristic
276C2×C32D3×D411yescharacteristic
277C2×C32#23711yescharacteristic
278C2×C32C2×#6711yescharacteristic
279C2×C32C2×C3×D411yescharacteristic
280C2×C32C2×C3×D411yescharacteristic
281C2×C32C3×#4111yescharacteristic
282C3⋊D3—41no—
283C3×D3—22no—
284C3×D3—22no—
285C3×D3—42no—
286C3×D3—122no—
287C3×D3—121no—
288C3×D3—121no—
289C2×D4—31no—
290C2×D4—91no—
291C2×D4—91no—
292C22×C4—91no—
293C4⋊C4—91no—
294C22⋊C4—31no—
295C22⋊C4—91no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—32no—
312C22×C3—32no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—42no—
320C22×C3—42no—
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—62no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3—61no—
327C22×C3—61no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3C2×D3211yescharacteristic
334C22×C3C22×C3×D311yescharacteristic
335C22×C3C22×C3×D311yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—62no—
339C2×D3—61no—
340C2×D3—61no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3C3×#6712yescomplemented
348C3×C4—62no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—92no—
356C3×C4—121no—
357C3×C4—121no—
358Dic3—21no—
359Dic3—21no—
360Dic3—21no—
361Dic3—32no—
362Dic3—32no—
363Dic3—62no—
364C32—21no—
365C32—21no—
366C32—21no—
367C32—41no—
368C32#88411yescharacteristic, complemented
369C32#93911yescharacteristic, complemented
370C32C3×#12111yescharacteristic, complemented
371C23—31no—
372C23—91no—
373C23C3×D3211yescharacteristic
374D4—62no—
375D4—182no—
376D4—182no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—91no—
381C2×C4—182no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—42no—
393C2×C3—42no—
394C2×C3—44no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—62no—
402C2×C3—62no—
403C2×C3—122no—
404C2×C3—361no—
405C2×C3#380411yescharacteristic
406C2×C3D3×#6711yescharacteristic
407C2×C3#381111yescharacteristic
408C2×C3C3×#23511yescharacteristic
409C2×C3C3×D3×D411yescharacteristic
410C2×C3C3×D3×D411yescharacteristic
411C2×C3C3×D3×D411yescharacteristic
412C2×C3C3×#23711yescharacteristic
413C2×C3C2×C3×#6711yescharacteristic
414D3—22no—
415D3—122no—
416D3—121no—
417D3—121no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—32no—
422C22—61no—
423C22—61no—
424C22—181no—
425C22—181no—
426C22—181no—
427C22C2×C3×D3211yescharacteristic
428C4—61no—
429C4—61no—
430C4—61no—
431C4—92no—
432C3—21no—
433C3—21no—
434C3—21no—
435C3—41no—
436C3#6397111yescharacteristic, direct product factor
437C3C3×#88411yescharacteristic, complemented
438C3C3×#93911yescharacteristic, complemented
439C2—41no—
440C2—62no—
441C2—361no—
442C2C3×#380411yescharacteristic
443C2C3×D3×#6711yescharacteristic
444C2C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC23×C33C22
socleC22×C33C23
hypercenterC23×C3D32
hypocenterC32C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×#638C2semidirect
2. C2×C32×#67C2semidirect
3. C2×C3×#626C2semidirect
4. C2×C3×D3×Dic3C2semidirect
5. C32×#217C2semidirect
6. C3×#3724C2semidirect
7. C3×#3722C2semidirect
10. #63971C3direct (N × H)
23. C2×#638C2×C3semidirect
29. C2×C3×#67D3semidirect
30. C2×C3×#67C2×C3semidirect
34. C2×#626C2×C3semidirect
35. C2×D3×Dic3C2×C3semidirect
38. C3×#217D3semidirect
39. C3×#217C2×C3semidirect
44. #3724C2×C3semidirect
45. #3722C2×C3semidirect
55. C2×C32×D3D4semidirect
56. C3×C3⋊Dic3D4semidirect
124. C2×#67C3×D3semidirect
133. #217C3×D3semidirect
188. C2×C3×D3#67semidirect
189. C2×C3×D3C3×D4semidirect
193. C3⋊Dic3C3×D4semidirect
207. C33#121semidirect
347. C2×D3C3×#67semidirect
368. C32#884semidirect
369. C32#939semidirect
370. C32C3×#121semidirect
436. C3#63971direct (N × H)
437. C3C3×#884semidirect
438. C3C3×#939semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6382
C2×C32×#672
C2×C3×#6262
C2×C3×D3×Dic32
C32×#2172
C3×#37242
C3×#37222
C3×#9393
C3×#8843
#639713