Finite Group Databasebeta

← C2×#5540 #85759

Subgroups — C2×#5540 #85759

Order 384.

1252 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×C4×D4C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#939C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C2×#929C211yesmaximal, characteristic, complemented
6C2×C4×#67C211yesmaximal, characteristic, complemented
7C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
8#5540C2116yesmaximal, direct product factor
9C2×#894C211yesmaximal, characteristic, complemented
10C2×#889C211yesmaximal, characteristic, complemented
11C2×#884C211yesmaximal, characteristic, complemented
12C2×#67⋊C4C211yesmaximal, characteristic, complemented
13C2×#67⋊C4C211yesmaximal, characteristic, complemented
14C2×#879C211yesmaximal, characteristic, complemented
15C2×#876C211yesmaximal, characteristic, complemented
16C2×#872C211yesmaximal, characteristic, complemented
17C2×#553—31nomaximal, Sylow
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C23×Dic3C2211yescharacteristic
26C23×Dic3C2211yescharacteristic
27C22×D12C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C3×C4×D4C2218yescomplemented
31C2×C3×C4⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C42C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37#939C2218yescomplemented
38#939C2218yescomplemented
39#930C2218yescomplemented
40#929C2218yescomplemented
41C4×#67C2218yescomplemented
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×Dic3⋊C4C2211yescharacteristic
50C2×Dic3⋊C4C2211yescharacteristic
51C2×Dic3⋊C4C2211yescharacteristic
52C2×Dic3⋊C4C2211yescharacteristic
53C2×Dic3⋊C4C2211yescharacteristic
54C2×Dic3⋊C4C2211yescharacteristic
55C2×Dic3⋊C4C2211yescharacteristic
56C2×Dic3⋊C4C2211yescharacteristic
57C2×Dic3⋊C4C2218yescomplemented
58C2×C4×Dic3C2211yescharacteristic
59C2×C4×Dic3C2211yescharacteristic
60#894C2218yescomplemented
61#889C2218yescomplemented
62#884C2218yescomplemented
63#67⋊C4C2218yescomplemented
64#67⋊C4C2218yescomplemented
65#879C2218yescomplemented
66#876C2218yescomplemented
67#872C2218yescomplemented
68#553—316no—
69C2×#126—31no—
70C2×#126—31no—
71C2×#125—31no—
72C2×#123—31no—
73C2×#123—31no—
74C2×#121—31no—
75C2×#121—31no—
76C2×#121—31no—
77C2×#121—31no—
78C2×C4×D4—31no—
79C2×C4×D4—31no—
80C2×C4×D4—31no—
81C2×C4×D4D311yescharacteristic, complemented
82C2×#117—31no—
83C22×C4⋊C4—31no—
84C24×C3C2311yescharacteristic
85C24×C3C2311yescharacteristic
86C23×D3C2311yescharacteristic
87C23×D3C2311yescharacteristic
88C2×C3×D4—24no—
89C2×C3×D4C2314yes—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2314yes—
98C22×C3×C4C2314yes—
99C2×#67—24no—
100C2×#67—24no—
101C2×#67—24no—
102C2×#67—42no—
103C2×#67C2314yes—
104C2×#67C2314yes—
105C2×#67C2314yes—
106C2×#67C2314yes—
107C22×Dic3—24no—
108C22×Dic3—42no—
109C22×Dic3C2314yes—
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2314yes—
116C22×Dic3C2311yescharacteristic
117C2×D12—42no—
118C2×D12C2314yes—
119C2×C4×D3—24no—
120C2×C4×D3—42no—
121C2×C4×D3C2314yes—
122C2×C4×D3C2314yes—
123C3×C4⋊C4C2314yes—
124C3×C22⋊C4C2314yes—
125C3×C22⋊C4C2314yes—
126C3×C42C2314yes—
127#217C2314yes—
128#217C2314yes—
129(C2×D3)⋊C4C2314yes—
130(C2×D3)⋊C4C2314yes—
131(C2×D3)⋊C4C2314yes—
132(C2×D3)⋊C4C2314yes—
133(C2×D3)⋊C4C2314yes—
134(C2×D3)⋊C4C2314yes—
135C4⋊Dic3C2314yes—
136Dic3⋊C4C2314yes—
137Dic3⋊C4C2314yes—
138Dic3⋊C4C2314yes—
139Dic3⋊C4C2314yes—
140Dic3⋊C4C2314yes—
141Dic3⋊C4C2314yes—
142Dic3⋊C4C2314yes—
143Dic3⋊C4C2314yes—
144C4×Dic3C2314yes—
145C4×Dic3C2314yes—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—31no—
151C22×D4C2×D311yescharacteristic
152C23×C4—31no—
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4C2×D311yescharacteristic
157C23×C4C2×D311yescharacteristic
158#126—38no—
159#126—38no—
160#125—38no—
161#123—38no—
162#123—38no—
163#121—38no—
164#121—38no—
165#121—38no—
166#121—38no—
167C4×D4—38no—
168C4×D4—38no—
169C4×D4—38no—
170C4×D4C2×D318yescomplemented
171#117—38no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—38no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
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⋊C4C2×D311yescharacteristic
192C2×C22⋊C4C2×D311yescharacteristic
193C2×C42—31no—
194C2×C42—31no—
195C2×C42C2×D311yescharacteristic
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—22no—
201C23×C3#4114yes—
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—28no—
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×C4—44no—
221C2×C3×C4C2412yes—
222C2×C3×C4C2412yes—
223C2×C3×C4C2412yes—
224C2×C3×C4C2412yes—
225C2×C3×C4C2412yes—
226#67—28no—
227#67—28no—
228#67—28no—
229#67—44no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—28no—
236C2×Dic3—44no—
237C2×Dic3#4114yes—
238C2×Dic3C2412yes—
239C2×Dic3C2412yes—
240C2×Dic3C2412yes—
241C2×Dic3C2412yes—
242C2×Dic3C2412yes—
243C2×Dic3C2412yes—
244D12—44no—
245C4×D3—28no—
246C4×D3—44no—
247C24—31no—
248C24—31no—
249C24C22×D311yescharacteristic
250C24C22×D311yescharacteristic
251C2×D4—24no—
252C2×D4—34no—
253C2×D4—34no—
254C2×D4—34no—
255C2×D4—34no—
256C2×D4—34no—
257C2×D4—64no—
258C2×D4—64no—
259C2×D4—64no—
260C2×D4—122no—
261C2×D4—122no—
262C2×D4C22×D314yes—
263C22×C4—31no—
264C22×C4—34no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—34no—
271C22×C4—34no—
272C22×C4—34no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—64no—
276C22×C4—64no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D314yes—
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⋊C4—34no—
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⋊C4C22×D314yes—
305C22⋊C4C22×D314yes—
306C42—34no—
307C42—34no—
308C42C22×D314yes—
309C22×C3—28no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—44no—
316C22×C3C2×#4114yes—
317C22×C3C2×#4112yes—
318C22×C3C2×#4112yes—
319C22×C3H(5,𝔽2)12yes—
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—
339Dic3C2×#4114yes—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—32no—
346C23—32no—
347C23—62no—
348C23—62no—
349C23—62no—
350C23—62no—
351C23—62no—
352C23—62no—
353C23#23514yes—
354C23C23×D312yes—
355C23C23×D312yes—
356C23C23×D311yescharacteristic
357D4—28no—
358D4—68no—
359D4—68no—
360D4—68no—
361D4—124no—
362D4—124no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—34no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—68no—
378C2×C4—68no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—124no—
385C2×C4—124no—
386C2×C4C23×D312yes—
387C2×C4C23×D312yes—
388C2×C4C23×D312yes—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C3—24no—
392C2×C3—42no—
393C2×C3#55314yescomplemented
394C2×C3C22×#4111yescharacteristic
395C2×C3C22×#4111yescharacteristic
396C2×C3C2×H(5,𝔽2)11yescharacteristic
397D3—42no—
398D3—42no—
399C22—22no—
400C22—22no—
401C22—28no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—44no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—124no—
413C22—124no—
414C22C2×#23514yes—
415C22C2×#23512yes—
416C22#100412yes—
417C22D3×#4112yes—
418C22C24×D311yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—34no—
425C4—62no—
426C4—62no—
427C4—62no—
428C4—62no—
429C4—62no—
430C3C2×#55311yescharacteristic, complemented, Sylow
431C2—24no—
432C2—42no—
433C2—122no—
434C2—122no—
435C2#554014yesdirect product factor
436C2C22×#23511yescharacteristic
437C2C2×#100411yescharacteristic
438C2C2×D3×#4111yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
socleC23×C3C24
hypercenterC2×C4×D4D3
hypocenterC3C2×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×#939C2semidirect
3. C2×#939C2semidirect
4. C2×#930C2semidirect
5. C2×#929C2semidirect
6. C2×C4×#67C2semidirect
7. C22×Dic3⋊C4C2semidirect
8. #5540C2direct (N × H)
9. C2×#894C2semidirect
10. C2×#889C2semidirect
11. C2×#884C2semidirect
12. C2×#67⋊C4C2semidirect
13. C2×#67⋊C4C2semidirect
14. C2×#879C2semidirect
15. C2×#876C2semidirect
16. C2×#872C2semidirect
30. C3×C4×D4C22semidirect
37. #939C22semidirect
38. #939C22semidirect
39. #930C22semidirect
40. #929C22semidirect
41. C4×#67C22semidirect
57. C2×Dic3⋊C4C22semidirect
60. #894C22semidirect
61. #889C22semidirect
62. #884C22semidirect
63. #67⋊C4C22semidirect
64. #67⋊C4C22semidirect
65. #879C22semidirect
66. #876C22semidirect
67. #872C22semidirect
81. C2×C4×D4D3semidirect
170. C4×D4C2×D3semidirect
393. C2×C3#553semidirect
430. C3C2×#553semidirect
435. C2#5540direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9392
C2×#9302
C2×#9292
C2×C4×#672
C22×Dic3⋊C42
#55402
C2×#8942
C2×#8892
C2×#8842
C2×#67⋊C42
C2×#8792
C2×#8762
C2×#8722
C2×#5533