Finite Group Databasebeta

← C2×#5655 #86009

Subgroups — C2×#5655 #86009

Order 384.

1632 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#237C211yesmaximal, characteristic, complemented
2C22×#235C211yesmaximal, characteristic, complemented
3C2×C3×#123C211yesmaximal, characteristic, complemented
4C2×#939C211yesmaximal, characteristic, complemented
5C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
6#5655C2116yesmaximal, direct product factor
7C2×#896C211yesmaximal, characteristic, complemented
8C2×#894C211yesmaximal, characteristic, complemented
9C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
10C2×#888C211yesmaximal, characteristic, complemented
11C2×#886C211yesmaximal, characteristic, complemented
12C2×#884C211yesmaximal, characteristic, complemented
13C2×#883C211yesmaximal, characteristic, complemented
14C2×#67⋊C4C211yesmaximal, characteristic, complemented
15C2×#67⋊C4C211yesmaximal, characteristic, complemented
16C2×#878C211yesmaximal, characteristic, complemented
17C2×#547—31nomaximal, Sylow
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C23×Dic3C2211yescharacteristic
26C23×Dic3C2211yescharacteristic
27C2×#237—28no—
28C2×#237C2218yescomplemented
29C2×#235—28no—
30C2×#235C2218yescomplemented
31C22×D12C2211yescharacteristic
32C22×C4×D3C2211yescharacteristic
33C22×C4×D3C2211yescharacteristic
34C22×C4×D3C2211yescharacteristic
35C22×C4×D3C2211yescharacteristic
36C22×Dic6C2211yescharacteristic
37C22×Dic6C2211yescharacteristic
38C3×#123C2218yescomplemented
39C2×C3×C4⋊C4C2211yescharacteristic
40C2×C3×C4⋊C4C2211yescharacteristic
41C2×C3×C22⋊C4C2211yescharacteristic
42C2×C3×C22⋊C4C2211yescharacteristic
43C2×C3×C22⋊C4C2211yescharacteristic
44C2×#217C2211yescharacteristic
45#939C2218yescomplemented
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×(C2×D3)⋊C4C2211yescharacteristic
48C2×(C2×D3)⋊C4C2211yescharacteristic
49C2×(C2×D3)⋊C4C2211yescharacteristic
50C2×C4⋊Dic3C2211yescharacteristic
51C2×C4⋊Dic3C2211yescharacteristic
52C2×Dic3⋊C4C2211yescharacteristic
53C2×Dic3⋊C4C2211yescharacteristic
54C2×Dic3⋊C4C2211yescharacteristic
55C2×Dic3⋊C4C2211yescharacteristic
56C2×Dic3⋊C4C2211yescharacteristic
57C2×Dic3⋊C4C2211yescharacteristic
58C2×Dic3⋊C4C2218yescomplemented
59C2×C4×Dic3C2211yescharacteristic
60#896C2218yescomplemented
61#894C2218yescomplemented
62D3×C4⋊C4C2218yescomplemented
63#888C2218yescomplemented
64#886C2218yescomplemented
65#884C2218yescomplemented
66#883C2218yescomplemented
67#67⋊C4C2218yescomplemented
68#67⋊C4C2218yescomplemented
69#878C2218yescomplemented
70C22×#41—31no—
71C22×#41—31no—
72#547—316no—
73C2×C4⋊Q8—31no—
74C2×#123—31no—
75C2×#123—31no—
76C2×#123—31no—
77C2×#123D311yescharacteristic, complemented
78C2×C22⋊Q8—31no—
79C2×C22⋊Q8—31no—
80C2×#121—31no—
81C2×#121—31no—
82C2×C4×D4—31no—
83C2×C4×D4—31no—
84C22×C4⋊C4—31no—
85C22×C4⋊C4—31no—
86C24×C3C2311yescharacteristic
87C24×C3C2311yescharacteristic
88C23×D3C2311yescharacteristic
89C23×D3C2311yescharacteristic
90C2×C3×D4—24no—
91C2×C3×D4C2314yes—
92C22×C3×C4—24no—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2314yes—
99C2×#67—24no—
100C2×#67—24no—
101C2×#67—24no—
102C2×#67D418yescomplemented
103C2×#67C2314yes—
104C2×#67C2314yes—
105C2×#67C2314yes—
106C2×#67C2314yes—
107C22×Dic3—24no—
108C22×Dic3—24no—
109C22×Dic3—24no—
110C22×Dic3C2314yes—
111C22×Dic3C2314yes—
112C22×Dic3C2314yes—
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119#237—232no—
120#235—232no—
121C2×D12—24no—
122C2×D12C2314yes—
123C2×C4×D3—24no—
124C2×C4×D3—24no—
125C2×C4×D3—24no—
126C2×C4×D3—24no—
127C2×C4×D3C2314yes—
128C2×C4×D3C2314yes—
129C2×C4×D3C2314yes—
130C2×C4×D3C2314yes—
131C2×Dic6—24no—
132C2×Dic6—24no—
133C2×Dic6C2314yes—
134C2×Dic6C2314yes—
135C3×C4⋊C4C2314yes—
136C3×C4⋊C4C2314yes—
137C3×C22⋊C4C2314yes—
138C3×C22⋊C4C2314yes—
139C3×C22⋊C4C2314yes—
140#217C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145C4⋊Dic3C2314yes—
146C4⋊Dic3C2314yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2314yes—
153C4×Dic3C2314yes—
154C2×#41—38no—
155C2×#41—38no—
156C2×#41—68no—
157C2×#41—68no—
158C22×Q8—31no—
159C22×Q8—31no—
160C22×D4—31no—
161C22×D4—31no—
162C22×D4—31no—
163C22×D4—31no—
164C22×D4—31no—
165C22×D4C2×D311yescharacteristic
166C23×C4—31no—
167C23×C4—31no—
168C23×C4—31no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4—31no—
172C23×C4—31no—
173C23×C4C2×D311yescharacteristic
174C4⋊Q8—38no—
175#123—38no—
176#123—38no—
177#123—38no—
178#123C2×D318yescomplemented
179C22⋊Q8—38no—
180C22⋊Q8—38no—
181#121—38no—
182#121—38no—
183C4×D4—38no—
184C4×D4—38no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—38no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—38no—
195C2×C4⋊C4C2×D311yescharacteristic
196C2×C4⋊C4C2×D311yescharacteristic
197C2×C22⋊C4—31no—
198C2×C22⋊C4—31no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4C2×D311yescharacteristic
203C2×C22⋊C4C2×D311yescharacteristic
204C2×C22⋊C4C2×D311yescharacteristic
205C2×C42—31no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—22no—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3C2×D414yes—
212C23×C3C2412yes—
213C23×C3C2412yes—
214C23×C3C2411yescharacteristic
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—22no—
219C22×D3—22no—
220C22×D3C2×D414yes—
221C22×D3C2412yes—
222C22×D3C2412yes—
223C3×D4—28no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—28no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4C2412yes—
231C2×C3×C4C2412yes—
232C2×C3×C4C2412yes—
233C2×C3×C4C2412yes—
234C2×C3×C4C2412yes—
235#67—28no—
236#67—28no—
237#67—28no—
238#67C2×D4116yescomplemented
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—28no—
243C2×Dic3—22no—
244C2×Dic3—28no—
245C2×Dic3—28no—
246C2×Dic3C2×D414yes—
247C2×Dic3#4114yescomplemented
248C2×Dic3C2412yes—
249C2×Dic3C2412yes—
250C2×Dic3C2412yes—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254D12—28no—
255C4×D3—28no—
256C4×D3—28no—
257C4×D3—28no—
258C4×D3—28no—
259Dic6—28no—
260Dic6—28no—
261C24—31no—
262C24—31no—
263C24C22×D311yescharacteristic
264C24C22×D311yescharacteristic
265#41—632no—
266#41—632no—
267C2×Q8—34no—
268C2×Q8—34no—
269C2×Q8—64no—
270C2×Q8—64no—
271C2×D4—24no—
272C2×D4—38no—
273C2×D4—34no—
274C2×D4—34no—
275C2×D4—34no—
276C2×D4—34no—
277C2×D4—34no—
278C2×D4—64no—
279C2×D4—64no—
280C2×D4—64no—
281C2×D4—64no—
282C2×D4C22×D314yes—
283C22×C4—24no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—34no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—31no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—34no—
294C22×C4—34no—
295C22×C4—34no—
296C22×C4—34no—
297C22×C4—64no—
298C22×C4—64no—
299C22×C4—64no—
300C22×C4—64no—
301C22×C4—64no—
302C22×C4—64no—
303C22×C4—64no—
304C22×C4C22×D314yes—
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C22×C4C22×D311yescharacteristic
308C22×C4C22×D311yescharacteristic
309C22×C4C22×D311yescharacteristic
310C4⋊C4—34no—
311C4⋊C4—34no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—34no—
317C4⋊C4—34no—
318C4⋊C4C22×D314yes—
319C4⋊C4C22×D314yes—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4C22×D314yes—
326C22⋊C4C22×D314yes—
327C22⋊C4C22×D314yes—
328C42—34no—
329C22×C3—28no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—44no—
336C22×C3C22×D414yes—
337C22×C3C22×D412yes—
338C22×C3C2×#4112yes—
339C22×C32−1+412yes—
340C22×C3C2511yescharacteristic
341C2×D3—28no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—44no—
348C2×D3C22×D414yes—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—22no—
354Dic3—22no—
355Dic3—22no—
356Dic3—22no—
357Dic3—22no—
358Dic3C22×D414yes—
359Dic3C2×#4114yescomplemented
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—34no—
366C23—32no—
367C23—32no—
368C23—62no—
369C23—62no—
370C23—62no—
371C23—62no—
372C23—62no—
373C23D3×D414yes—
374C23C23×D312yes—
375C23C23×D312yes—
376C23C23×D311yescharacteristic
377Q8—68no—
378Q8—68no—
379D4—28no—
380D4—316no—
381D4—68no—
382D4—68no—
383D4—68no—
384D4—68no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—34no—
392C2×C4—34no—
393C2×C4—32no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—32no—
399C2×C4—68no—
400C2×C4—68no—
401C2×C4—68no—
402C2×C4—68no—
403C2×C4—68no—
404C2×C4—68no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—68no—
409C2×C4—62no—
410C2×C4C23×D312yes—
411C2×C4C23×D312yes—
412C2×C4C23×D312yes—
413C2×C4C23×D312yes—
414C2×C4C23×D312yes—
415C2×C3—24no—
416C2×C3—42no—
417C2×C3#54714yescomplemented
418C2×C3C23×D411yescharacteristic
419C2×C3C22×#4111yescharacteristic
420C2×C3C2×2−1+411yescharacteristic
421D3—24no—
422D3—42no—
423C22—28no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—34no—
430C22—44no—
431C22—68no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—124no—
438C22C2×D3×D414yes—
439C22C2×D3×D412yes—
440C22D3×#4112yes—
441C22#101012yes—
442C22C24×D311yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—34no—
449C4—34no—
450C4—62no—
451C4—62no—
452C4—62no—
453C4—62no—
454C3C2×#54711yescharacteristic, complemented, Sylow
455C2—24no—
456C2—42no—
457C2—64no—
458C2—122no—
459C2#565514yesdirect product factor
460C2C22×D3×D411yescharacteristic
461C2C2×D3×#4111yescharacteristic
462C2C2×#101011yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#123C2
socleC23×C3C24
hypercenterC2×#123D3
hypocenterC3C2×#547
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C22×#235C2semidirect
3. C2×C3×#123C2semidirect
4. C2×#939C2semidirect
5. C22×Dic3⋊C4C2semidirect
6. #5655C2direct (N × H)
7. C2×#896C2semidirect
8. C2×#894C2semidirect
9. C2×D3×C4⋊C4C2semidirect
10. C2×#888C2semidirect
11. C2×#886C2semidirect
12. C2×#884C2semidirect
13. C2×#883C2semidirect
14. C2×#67⋊C4C2semidirect
15. C2×#67⋊C4C2semidirect
16. C2×#878C2semidirect
28. C2×#237C22semidirect
30. C2×#235C22semidirect
38. C3×#123C22semidirect
45. #939C22semidirect
58. C2×Dic3⋊C4C22semidirect
60. #896C22semidirect
61. #894C22semidirect
62. D3×C4⋊C4C22semidirect
63. #888C22semidirect
64. #886C22semidirect
65. #884C22semidirect
66. #883C22semidirect
67. #67⋊C4C22semidirect
68. #67⋊C4C22semidirect
69. #878C22semidirect
77. C2×#123D3semidirect
102. C2×#67D4semidirect
178. #123C2×D3semidirect
238. #67C2×D4semidirect
247. C2×Dic3#41semidirect
359. Dic3C2×#41semidirect
417. C2×C3#547semidirect
454. C3C2×#547semidirect
459. C2#5655direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2372
C22×#2352
C2×C3×#1232
C2×#9392
C22×Dic3⋊C42
#56552
C2×#8962
C2×#8942
C2×D3×C4⋊C42
C2×#8882
C2×#8862
C2×#8842
C2×#8832
C2×#67⋊C42
C2×#8782
C2×#5473