Finite Group Databasebeta

← C2×#5667 #86015

Subgroups — C2×#5667 #86015

Order 384.

1152 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#938C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C2×#924C211yesmaximal, characteristic, complemented
6#5667C2116yesmaximal, direct product factor
7C2×#897C211yesmaximal, characteristic, complemented
8C2×#896C211yesmaximal, characteristic, complemented
9C2×#894C211yesmaximal, characteristic, complemented
10C2×#890C211yesmaximal, characteristic, complemented
11C2×#886C211yesmaximal, characteristic, complemented
12C2×#885C211yesmaximal, characteristic, complemented
13C2×#885C211yesmaximal, characteristic, complemented
14C2×#884C211yesmaximal, characteristic, complemented
15C2×#883C211yesmaximal, characteristic, complemented
16C2×#878C211yesmaximal, characteristic, complemented
17C2×#562—31nomaximal, Sylow
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C22×D12C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×Dic6C2211yescharacteristic
29C22×Dic6C2211yescharacteristic
30C3×#123C2218yescomplemented
31C2×C3×C4⋊C4C2211yescharacteristic
32C2×C3×C4⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39#939C2218yescomplemented
40#938C2218yescomplemented
41#930C2218yescomplemented
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×(C2×D3)⋊C4C2211yescharacteristic
48C2×C4⋊Dic3C2211yescharacteristic
49C2×C4⋊Dic3C2211yescharacteristic
50C2×C4⋊Dic3C2211yescharacteristic
51C2×C4⋊Dic3C2211yescharacteristic
52#924C2218yescomplemented
53C2×Dic3⋊C4C2211yescharacteristic
54C2×Dic3⋊C4C2211yescharacteristic
55C2×Dic3⋊C4C2211yescharacteristic
56C2×Dic3⋊C4C2211yescharacteristic
57C2×C4×Dic3C2211yescharacteristic
58#897C2218yescomplemented
59#896C2218yescomplemented
60#894C2218yescomplemented
61#890C2218yescomplemented
62#886C2218yescomplemented
63#885C2218yescomplemented
64#885C2218yescomplemented
65#884C2218yescomplemented
66#883C2218yescomplemented
67#878C2218yescomplemented
68#562—316no—
69C2×#125—31no—
70C2×#124—31no—
71C2×#124—31no—
72C2×#123—31no—
73C2×#123—31no—
74C2×#123—31no—
75C2×#123D311yescharacteristic, complemented
76C2×C22⋊Q8—31no—
77C2×C22⋊Q8—31no—
78C2×C22⋊Q8—31no—
79C2×C22⋊Q8—31no—
80C2×#121—31no—
81C2×#121—31no—
82C2×#121—31no—
83C2×#121—31no—
84C24×C3C2311yescharacteristic
85C24×C3C2311yescharacteristic
86C23×D3C2311yescharacteristic
87C23×D3C2311yescharacteristic
88C2×C3×D4—42no—
89C2×C3×D4C2314yes—
90C22×C3×C4—42no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2314yes—
97C2×#67—42no—
98C2×#67—42no—
99C2×#67—42no—
100C2×#67—42no—
101C2×#67C2314yes—
102C2×#67C2314yes—
103C2×#67C2314yes—
104C2×#67C2314yes—
105C22×Dic3—42no—
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2314yes—
113C2×D12—42no—
114C2×D12C2314yes—
115C2×C4×D3—42no—
116C2×C4×D3—42no—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yes—
119C2×Dic6—42no—
120C2×Dic6—42no—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C3×C4⋊C4C2314yes—
124C3×C4⋊C4C2314yes—
125C3×C22⋊C4C2314yes—
126C3×C22⋊C4C2314yes—
127C3×C22⋊C4C2314yes—
128#217C2314yes—
129#217C2314yes—
130#217C2314yes—
131(C2×D3)⋊C4C2314yes—
132(C2×D3)⋊C4C2314yes—
133(C2×D3)⋊C4C2314yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2314yes—
136(C2×D3)⋊C4C2314yes—
137C4⋊Dic3C2314yes—
138C4⋊Dic3C2314yes—
139C4⋊Dic3C2314yes—
140C4⋊Dic3C2314yes—
141Dic3⋊C4C2314yes—
142Dic3⋊C4C2314yes—
143Dic3⋊C4C2314yes—
144Dic3⋊C4C2314yes—
145C4×Dic3C2314yes—
146C22×Q8—31no—
147C22×Q8—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—31no—
151C22×D4—31no—
152C22×D4—31no—
153C22×D4C2×D311yescharacteristic
154C23×C4—31no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4C2×D311yescharacteristic
158#125—38no—
159#124—38no—
160#124—38no—
161#123—38no—
162#123—38no—
163#123—38no—
164#123C2×D318yescomplemented
165C22⋊Q8—38no—
166C22⋊Q8—38no—
167C22⋊Q8—38no—
168C22⋊Q8—38no—
169#121—38no—
170#121—38no—
171#121—38no—
172#121—38no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4C2×D311yescharacteristic
182C2×C4⋊C4C2×D311yescharacteristic
183C2×C22⋊C4—31no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—31no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—31no—
190C2×C22⋊C4—31no—
191C2×C22⋊C4—31no—
192C2×C22⋊C4C2×D311yescharacteristic
193C2×C22⋊C4C2×D311yescharacteristic
194C2×C22⋊C4C2×D311yescharacteristic
195C2×C42—31no—
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3C2412yes—
203C23×C3C2412yes—
204C23×C3C2411yescharacteristic
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—22no—
210C22×D3—22no—
211C22×D3C2412yes—
212C22×D3C2412yes—
213C3×D4—44no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—44no—
220C2×C3×C4C2412yes—
221C2×C3×C4C2412yes—
222C2×C3×C4C2412yes—
223C2×C3×C4C2412yes—
224C2×C3×C4C2412yes—
225#67—44no—
226#67—44no—
227#67—44no—
228#67—44no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—44no—
236C2×Dic3C2412yes—
237C2×Dic3C2412yes—
238C2×Dic3C2412yes—
239C2×Dic3C2412yes—
240C2×Dic3C2412yes—
241C2×Dic3C2412yes—
242D12—44no—
243C4×D3—44no—
244C4×D3—44no—
245Dic6—44no—
246Dic6—44no—
247C24—31no—
248C24—31no—
249C24C22×D311yescharacteristic
250C24C22×D311yescharacteristic
251C2×Q8—34no—
252C2×Q8—34no—
253C2×Q8—122no—
254C2×Q8—122no—
255C2×D4—34no—
256C2×D4—34no—
257C2×D4—34no—
258C2×D4—34no—
259C2×D4—34no—
260C2×D4—42no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—122no—
264C2×D4—122no—
265C2×D4—122no—
266C2×D4C22×D314yes—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—34no—
274C22×C4—34no—
275C22×C4—34no—
276C22×C4—42no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4—122no—
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D314yes—
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—34no—
294C4⋊C4C22×D314yes—
295C4⋊C4C22×D314yes—
296C22⋊C4—34no—
297C22⋊C4—34no—
298C22⋊C4—34no—
299C22⋊C4—34no—
300C22⋊C4—34no—
301C22⋊C4—34no—
302C22⋊C4—34no—
303C22⋊C4—34no—
304C22⋊C4—34no—
305C22⋊C4C22×D314yes—
306C22⋊C4C22×D314yes—
307C22⋊C4C22×D314yes—
308C42—34no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3H(5,𝔽2)12yes—
318C22×C3H(5,𝔽2)12yes—
319C22×C32−1+412yes—
320C22×C3C2511yescharacteristic
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—44no—
328C2×D3—44no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334Dic3—22no—
335Dic3—22no—
336Dic3—22no—
337Dic3—22no—
338Dic3—22no—
339Dic3—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—32no—
347C23—32no—
348C23—62no—
349C23—62no—
350C23—62no—
351C23—62no—
352C23—62no—
353C23—62no—
354C23C23×D312yes—
355C23C23×D312yes—
356C23C23×D311yescharacteristic
357Q8—124no—
358Q8—124no—
359D4—44no—
360D4—124no—
361D4—124no—
362D4—124no—
363D4—124no—
364D4—124no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—44no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4—124no—
386C2×C4C23×D312yes—
387C2×C4C23×D312yes—
388C2×C4C23×D312yes—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C3—42no—
392C2×C3—42no—
393C2×C3#56214yescomplemented
394C2×C3C2×H(5,𝔽2)11yescharacteristic
395C2×C3C2×H(5,𝔽2)11yescharacteristic
396C2×C3C2×2−1+411yescharacteristic
397D3—42no—
398D3—42no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—44no—
406C22—44no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22—124no—
414C22—124no—
415C22#100412yes—
416C22#100912yes—
417C22#101012yes—
418C22C24×D311yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—62no—
425C4—62no—
426C4—62no—
427C4—62no—
428C4—62no—
429C4—62no—
430C3C2×#56211yescharacteristic, complemented, Sylow
431C2—42no—
432C2—42no—
433C2—122no—
434C2—122no—
435C2#566714yesdirect product factor
436C2C2×#100411yescharacteristic
437C2C2×#100911yescharacteristic
438C2C2×#101011yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#123C2
socleC23×C3C24
hypercenterC2×#123D3
hypocenterC3C2×#562
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×#939C2semidirect
3. C2×#938C2semidirect
4. C2×#930C2semidirect
5. C2×#924C2semidirect
6. #5667C2direct (N × H)
7. C2×#897C2semidirect
8. C2×#896C2semidirect
9. C2×#894C2semidirect
10. C2×#890C2semidirect
11. C2×#886C2semidirect
12. C2×#885C2semidirect
13. C2×#885C2semidirect
14. C2×#884C2semidirect
15. C2×#883C2semidirect
16. C2×#878C2semidirect
30. C3×#123C22semidirect
39. #939C22semidirect
40. #938C22semidirect
41. #930C22semidirect
52. #924C22semidirect
58. #897C22semidirect
59. #896C22semidirect
60. #894C22semidirect
61. #890C22semidirect
62. #886C22semidirect
63. #885C22semidirect
64. #885C22semidirect
65. #884C22semidirect
66. #883C22semidirect
67. #878C22semidirect
75. C2×#123D3semidirect
164. #123C2×D3semidirect
393. C2×C3#562semidirect
430. C3C2×#562semidirect
435. C2#5667direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9392
C2×#9382
C2×#9302
C2×#9242
#56672
C2×#8972
C2×#8962
C2×#8942
C2×#8902
C2×#8862
C2×#8852
C2×#8842
C2×#8832
C2×#8782
C2×#5623