Finite Group Databasebeta

← C2×#5706 #86104

Subgroups — C2×#5706 #86104

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic, complemented
2#5706C2116yesmaximal, direct product factor
3C2×#896C211yesmaximal, characteristic, complemented
4C2×#894C211yesmaximal, characteristic, complemented
5C2×#893C211yesmaximal, characteristic, complemented
6C2×#892C211yesmaximal, characteristic, complemented
7C2×#889C211yesmaximal, characteristic, complemented
8C2×#887C211yesmaximal, characteristic, complemented
9C2×#885C211yesmaximal, characteristic, complemented
10C2×#884C211yesmaximal, characteristic
11C2×#883C211yesmaximal, characteristic, complemented
12C2×#67⋊C4C211yesmaximal, characteristic, complemented
13C2×#67⋊C4C211yesmaximal, characteristic, complemented
14C2×#877C211yesmaximal, characteristic, complemented
15C2×#876C211yesmaximal, characteristic, complemented
16C2×C42×D3C211yesmaximal, characteristic, complemented
17C2×#529—31nomaximal, Sylow
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×Dic6C2211yescharacteristic
27C3×#126C2218yescomplemented
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C4⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C42C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×C4⋊Dic3C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50C2×C4×Dic3C2211yescharacteristic
51C2×C4×Dic3C2211yescharacteristic
52C2×C4×Dic3C2211yescharacteristic
53C2×C4×Dic3C2211yescharacteristic
54#896C2218yescomplemented
55#894C2218yescomplemented
56#893C2218yescomplemented
57#892C2218yescomplemented
58#889C2218yescomplemented
59#887C2218yescomplemented
60#885C2218yescomplemented
61#884C2218yes—
62#883C2218yescomplemented
63#67⋊C4C2218yescomplemented
64#67⋊C4C2218yescomplemented
65#877C2218yescomplemented
66#876C2218yescomplemented
67C42×D3C2218yescomplemented
68#529—316no—
69C2×#126—31no—
70C2×#126D311yescharacteristic, complemented
71C2×#125—31no—
72C2×#124—31no—
73C2×#123—31no—
74C2×#123—31no—
75C2×C22⋊Q8—31no—
76C2×#121—31no—
77C2×C4×Q8—31no—
78C2×C4×D4—31no—
79C2×C4×D4—31no—
80C2×C4×D4—31no—
81C2×#117—31no—
82C2×#117—31no—
83C22×C42—31no—
84C24×C3C2311yescharacteristic
85C23×D3C2311yescharacteristic
86C23×D3C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C2×#67—24no—
94C2×#67—24no—
95C2×#67C2314yes—
96C2×#67C2314yes—
97C22×Dic3—24no—
98C22×Dic3C2314yes—
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C2×D12—24no—
106C2×D12C2314yes—
107C2×C4×D3—24no—
108C2×C4×D3—24no—
109C2×C4×D3—24no—
110C2×C4×D3—24no—
111C2×C4×D3C2314yes—
112C2×C4×D3C2314yes—
113C2×C4×D3C2314yes—
114C2×C4×D3C2314yes—
115C2×Dic6—24no—
116C2×Dic6C2314yes—
117C3×C4⋊C4C2314yes—
118C3×C4⋊C4C2314yes—
119C3×C4⋊C4C2314yes—
120C3×C22⋊C4C2314yes—
121C3×C22⋊C4C2314yes—
122C3×C22⋊C4C2314yes—
123C3×C42C2314yes—
124#217C2314yes—
125(C2×D3)⋊C4C2314yes—
126(C2×D3)⋊C4C2314yes—
127(C2×D3)⋊C4C2314yes—
128(C2×D3)⋊C4C2314yes—
129(C2×D3)⋊C4C2314yes—
130(C2×D3)⋊C4C2314yes—
131C4⋊Dic3C2314yes—
132Dic3⋊C4C2314yes—
133Dic3⋊C4C2314yes—
134Dic3⋊C4C2314yes—
135Dic3⋊C4C2314yes—
136Dic3⋊C4C2314yes—
137Dic3⋊C4C2314yes—
138C4×Dic3C2314yes—
139C4×Dic3C2314yes—
140C4×Dic3C2314yes—
141C4×Dic3C2314yes—
142C4×Dic3C2314yes—
143C22×Q8—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4—31no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—31no—
152#126—38no—
153#126C2×D318yescomplemented
154#125—38no—
155#124—38no—
156#123—38no—
157#123—38no—
158C22⋊Q8—38no—
159#121—38no—
160C4×Q8—38no—
161C4×D4—38no—
162C4×D4—38no—
163C4×D4—38no—
164#117—38no—
165#117—38no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4C2×D311yescharacteristic
174C2×C4⋊C4C2×D311yescharacteristic
175C2×C4⋊C4C2×D311yescharacteristic
176C2×C22⋊C4—31no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4C2×D311yescharacteristic
184C2×C22⋊C4C2×D311yescharacteristic
185C2×C22⋊C4C2×D311yescharacteristic
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—31no—
189C2×C42—31no—
190C2×C42—31no—
191C2×C42—38no—
192C2×C42C2×D311yescharacteristic
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3C2412yes—
197C23×C3C2411yescharacteristic
198C22×D3—22no—
199C22×D3—22no—
200C22×D3—22no—
201C22×D3—22no—
202C22×D3—22no—
203C22×D3#4114yes—
204C22×D3C2412yes—
205C22×D3C2412yes—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4C2412yes—
213C2×C3×C4C2412yes—
214C2×C3×C4C2412yes—
215C2×C3×C4C2412yes—
216C2×C3×C4C2412yes—
217C2×C3×C4C2412yes—
218#67—28no—
219#67—28no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—28no—
225C2×Dic3#4114yes—
226C2×Dic3#4114yes—
227C2×Dic3C2412yes—
228C2×Dic3C2412yes—
229C2×Dic3C2412yes—
230C2×Dic3C2412yes—
231C2×Dic3C2412yes—
232C2×Dic3C2412yes—
233D12—28no—
234C4×D3—28no—
235C4×D3—28no—
236C4×D3—28no—
237C4×D3—28no—
238Dic6—28no—
239C24—31no—
240C24—31no—
241C24C22×D311yescharacteristic
242C2×Q8—34no—
243C2×Q8—64no—
244C2×D4—34no—
245C2×D4—34no—
246C2×D4—34no—
247C2×D4—64no—
248C2×D4—64no—
249C2×D4—64no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—34no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—34no—
258C22×C4—34no—
259C22×C4—34no—
260C22×C4—34no—
261C22×C4—64no—
262C22×C4—64no—
263C22×C4—64no—
264C22×C4—64no—
265C22×C4—64no—
266C22×C4C22×D311yescharacteristic
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C22×C4C22×D311yescharacteristic
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C4⋊C4—34no—
273C4⋊C4—34no—
274C4⋊C4—34no—
275C4⋊C4—34no—
276C4⋊C4—34no—
277C4⋊C4—34no—
278C4⋊C4—34no—
279C4⋊C4C22×D314yes—
280C4⋊C4C22×D314yes—
281C4⋊C4C22×D314yes—
282C22⋊C4—34no—
283C22⋊C4—34no—
284C22⋊C4—34no—
285C22⋊C4—34no—
286C22⋊C4—34no—
287C22⋊C4—34no—
288C22⋊C4—34no—
289C22⋊C4C22×D314yes—
290C22⋊C4C22×D314yes—
291C22⋊C4C22×D314yes—
292C42—34no—
293C42—34no—
294C42—34no—
295C42—34no—
296C42—34no—
297C42C22×D314yes—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—44no—
302C22×C3C2×#4112yes—
303C22×C3C2×#4112yes—
304C22×C3C2×#4112yes—
305C22×C3C2511yescharacteristic
306C2×D3—28no—
307C2×D3—22no—
308C2×D3—22no—
309C2×D3—22no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—44no—
313C2×D3C2×#4114yes—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—22no—
320Dic3—22no—
321Dic3—22no—
322Dic3—22no—
323Dic3—22no—
324Dic3C2×#4114yes—
325Dic3C2×#4114yes—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—34no—
330C23—32no—
331C23—32no—
332C23—62no—
333C23—62no—
334C23—62no—
335C23—62no—
336C23—62no—
337C23C23×D312yes—
338C23C23×D311yescharacteristic
339Q8—68no—
340D4—68no—
341D4—68no—
342D4—68no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—34no—
350C2×C4—34no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—32no—
357C2×C4—68no—
358C2×C4—68no—
359C2×C4—68no—
360C2×C4—68no—
361C2×C4—68no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4C23×D312yes—
367C2×C4C23×D312yes—
368C2×C4C23×D312yes—
369C2×C4C23×D312yes—
370C2×C4C23×D312yes—
371C2×C4C23×D312yes—
372C2×C3—42no—
373C2×C3#52914yescomplemented
374C2×C3C22×#4111yescharacteristic
375C2×C3C22×#4111yescharacteristic
376C2×C3C22×#4111yescharacteristic
377D3—24no—
378D3—42no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—34no—
383C22—44no—
384C22—62no—
385C22—62no—
386C22—68no—
387C22—62no—
388C22—62no—
389C22—62no—
390C22—124no—
391C22D3×#4112yes—
392C22D3×#4112yes—
393C22D3×#4112yes—
394C22C24×D311yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—34no—
402C4—34no—
403C4—62no—
404C4—62no—
405C4—62no—
406C4—62no—
407C3C2×#52911yescharacteristic, complemented, Sylow
408C2—42no—
409C2—64no—
410C2—122no—
411C2#570614yesdirect product factor
412C2C2×D3×#4111yescharacteristic
413C2C2×D3×#4111yescharacteristic
414C2C2×D3×#4111yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#126C2
socleC23×C3C24
prime radicalC2×#884C2
hypercenterC2×#126D3
hypocenterC3C2×#529
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. #5706C2direct (N × H)
3. C2×#896C2semidirect
4. C2×#894C2semidirect
5. C2×#893C2semidirect
6. C2×#892C2semidirect
7. C2×#889C2semidirect
8. C2×#887C2semidirect
9. C2×#885C2semidirect
11. C2×#883C2semidirect
12. C2×#67⋊C4C2semidirect
13. C2×#67⋊C4C2semidirect
14. C2×#877C2semidirect
15. C2×#876C2semidirect
16. C2×C42×D3C2semidirect
27. C3×#126C22semidirect
54. #896C22semidirect
55. #894C22semidirect
56. #893C22semidirect
57. #892C22semidirect
58. #889C22semidirect
59. #887C22semidirect
60. #885C22semidirect
62. #883C22semidirect
63. #67⋊C4C22semidirect
64. #67⋊C4C22semidirect
65. #877C22semidirect
66. #876C22semidirect
67. C42×D3C22semidirect
70. C2×#126D3semidirect
153. #126C2×D3semidirect
373. C2×C3#529semidirect
407. C3C2×#529semidirect
411. C2#5706direct (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
#57062
C2×#8962
C2×#8942
C2×#8932
C2×#8922
C2×#8892
C2×#8872
C2×#8852
C2×#8842
C2×#8832
C2×#67⋊C42
C2×#8772
C2×#8762
C2×C42×D32
C2×#5293