Finite Group Databasebeta

← C2×#5709 #86107

Subgroups — C2×#5709 #86107

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×#126C211yesmaximal, characteristic, complemented
2#5709C2116yesmaximal, direct product factor
3C2×#898C211yesmaximal, characteristic, complemented
4C2×#895C211yesmaximal, characteristic, complemented
5C2×#894C211yesmaximal, characteristic, complemented
6C2×#893C211yesmaximal, characteristic, complemented
7C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
8C2×#889C211yesmaximal, characteristic, complemented
9C2×#886C211yesmaximal, characteristic, complemented
10C2×#884C211yesmaximal, characteristic, complemented
11C2×#884C211yesmaximal, characteristic, complemented
12C2×#884C211yesmaximal, characteristic, complemented
13C2×#67⋊C4C211yesmaximal, characteristic, complemented
14C2×#67⋊C4C211yesmaximal, characteristic, complemented
15C2×C4×D12C211yesmaximal, characteristic, complemented
16C2×#872C211yesmaximal, characteristic, complemented
17C2×#553—31nomaximal, Sylow
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C3×#126C2218yescomplemented
31C2×C3×C4⋊C4C2211yescharacteristic
32C2×C3×C4⋊C4C2211yescharacteristic
33C2×C3×C4⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×C3×C42C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×C4⋊Dic3C2211yescharacteristic
46C2×C4⋊Dic3C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×Dic3⋊C4C2211yescharacteristic
50C2×Dic3⋊C4C2211yescharacteristic
51C2×Dic3⋊C4C2211yescharacteristic
52C2×C4×Dic3C2211yescharacteristic
53C2×C4×Dic3C2211yescharacteristic
54#898C2218yescomplemented
55#895C2218yescomplemented
56#894C2218yescomplemented
57#893C2218yescomplemented
58D3×C4⋊C4C2218yescomplemented
59#889C2218yescomplemented
60#886C2218yescomplemented
61#884C2218yescomplemented
62#884C2218yescomplemented
63#884C2218yescomplemented
64#67⋊C4C2218yescomplemented
65#67⋊C4C2218yescomplemented
66C4×D12C2218yescomplemented
67#872C2218yescomplemented
68#553—316no—
69C2×#126—31no—
70C2×#126D311yescharacteristic, complemented
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×D4—31no—
82C2×#117—31no—
83C22×C4⋊C4—31no—
84C24×C3C2311yescharacteristic
85C23×D3C2311yescharacteristic
86C23×D3C2311yescharacteristic
87C23×D3C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C2×#67—24no—
95C2×#67—24no—
96C2×#67—42no—
97C2×#67C2314yes—
98C2×#67C2314yes—
99C2×#67C2314yes—
100C22×Dic3—42no—
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2314yes—
106C22×Dic3C2311yescharacteristic
107C2×D12—24no—
108C2×D12—24no—
109C2×D12—42no—
110C2×D12C2314yes—
111C2×D12C2314yes—
112C2×D12C2314yes—
113C2×C4×D3—24no—
114C2×C4×D3—24no—
115C2×C4×D3—42no—
116C2×C4×D3—42no—
117C2×C4×D3—42no—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×C4×D3C2314yes—
121C2×C4×D3C2314yes—
122C2×C4×D3C2314yes—
123C3×C4⋊C4C2314yes—
124C3×C4⋊C4C2314yes—
125C3×C4⋊C4C2314yes—
126C3×C22⋊C4C2314yes—
127C3×C22⋊C4C2314yes—
128C3×C22⋊C4C2314yes—
129C3×C42C2314yes—
130(C2×D3)⋊C4C2314yes—
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—
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×D4—31no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4—31no—
157C23×C4—31no—
158#126—38no—
159#126C2×D318yescomplemented
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×D4—38no—
171#117—38no—
172C2×C4⋊C4—31no—
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—38no—
180C2×C4⋊C4C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
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×C3C2412yes—
200C23×C3C2411yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3—22no—
205C22×D3—22no—
206C22×D3—22no—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3#4114yes—
210C22×D3C2412yes—
211C22×D3C2412yes—
212C22×D3C2412yes—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4C2412yes—
220C2×C3×C4C2412yes—
221C2×C3×C4C2412yes—
222C2×C3×C4C2412yes—
223C2×C3×C4C2412yes—
224C2×C3×C4C2412yes—
225#67—28no—
226#67—28no—
227#67—44no—
228C2×Dic3—22no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—22no—
232C2×Dic3—44no—
233C2×Dic3#4114yes—
234C2×Dic3C2412yes—
235C2×Dic3C2412yes—
236C2×Dic3C2412yes—
237C2×Dic3C2412yes—
238C2×Dic3C2412yes—
239D12—28no—
240D12—28no—
241D12—44no—
242C4×D3—28no—
243C4×D3—28no—
244C4×D3—44no—
245C4×D3—44no—
246C4×D3—44no—
247C24—31no—
248C24—31no—
249C24—31no—
250C24C22×D311yescharacteristic
251C2×D4—34no—
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—64no—
261C2×D4—122no—
262C2×D4—122no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—34no—
269C22×C4—34no—
270C22×C4—34no—
271C22×C4—34no—
272C22×C4—34no—
273C22×C4—34no—
274C22×C4—64no—
275C22×C4—64no—
276C22×C4—122no—
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×D311yescharacteristic
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4C22×D314yes—
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⋊C4C22×D314yes—
304C22⋊C4C22×D314yes—
305C22⋊C4C22×D314yes—
306C42—34no—
307C42—34no—
308C42C22×D314yes—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—44no—
313C22×C3C2×#4112yes—
314C22×C3C2×#4112yes—
315C22×C3H(5,𝔽2)12yes—
316C22×C3C2511yescharacteristic
317C2×D3—28no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—44no—
327C2×D3—44no—
328C2×D3C2×#4114yes—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—22no—
335Dic3—22no—
336Dic3—22no—
337Dic3—22no—
338Dic3—22no—
339Dic3C2×#4114yes—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—34no—
344C23—32no—
345C23—32no—
346C23—32no—
347C23—62no—
348C23—62no—
349C23—62no—
350C23—62no—
351C23—62no—
352C23—62no—
353C23—62no—
354C23—62no—
355C23C23×D312yes—
356C23C23×D311yescharacteristic
357D4—68no—
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—22no—
369C2×C4—34no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—68no—
376C2×C4—68no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4—124no—
382C2×C4—124no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4C23×D312yes—
386C2×C4C23×D312yes—
387C2×C4C23×D312yes—
388C2×C4C23×D312yes—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C3—42no—
392C2×C3#55314yescomplemented
393C2×C3C22×#4111yescharacteristic
394C2×C3C22×#4111yescharacteristic
395C2×C3C2×H(5,𝔽2)11yescharacteristic
396D3—24no—
397D3—42no—
398D3—42no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—34no—
403C22—44no—
404C22—62no—
405C22—62no—
406C22—68no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22—124no—
414C22—124no—
415C22D3×#4112yes—
416C22D3×#4112yes—
417C22#100912yes—
418C22C24×D311yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—34no—
426C4—62no—
427C4—62no—
428C4—62no—
429C4—62no—
430C3C2×#55311yescharacteristic, complemented, Sylow
431C2—42no—
432C2—64no—
433C2—122no—
434C2—122no—
435C2#570914yesdirect product factor
436C2C2×D3×#4111yescharacteristic
437C2C2×D3×#4111yescharacteristic
438C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
hypercenterC2×#126D3
hypocenterC3C2×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
2. #5709C2direct (N × H)
3. C2×#898C2semidirect
4. C2×#895C2semidirect
5. C2×#894C2semidirect
6. C2×#893C2semidirect
7. C2×D3×C4⋊C4C2semidirect
8. C2×#889C2semidirect
9. C2×#886C2semidirect
10. C2×#884C2semidirect
11. C2×#884C2semidirect
12. C2×#884C2semidirect
13. C2×#67⋊C4C2semidirect
14. C2×#67⋊C4C2semidirect
15. C2×C4×D12C2semidirect
16. C2×#872C2semidirect
30. C3×#126C22semidirect
54. #898C22semidirect
55. #895C22semidirect
56. #894C22semidirect
57. #893C22semidirect
58. D3×C4⋊C4C22semidirect
59. #889C22semidirect
60. #886C22semidirect
61. #884C22semidirect
62. #884C22semidirect
63. #884C22semidirect
64. #67⋊C4C22semidirect
65. #67⋊C4C22semidirect
66. C4×D12C22semidirect
67. #872C22semidirect
70. C2×#126D3semidirect
159. #126C2×D3semidirect
392. C2×C3#553semidirect
430. C3C2×#553semidirect
435. C2#5709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1262
#57092
C2×#8982
C2×#8952
C2×#8942
C2×#8932
C2×D3×C4⋊C42
C2×#8892
C2×#8862
C2×#8842
C2×#67⋊C42
C2×C4×D122
C2×#8722
C2×#5533