Finite Group Databasebeta

← #86022

Subgroups — #86022

Order 384.

1492 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic, complemented
2C23×Dic6C211yesmaximal, characteristic, complemented
3C3×#525C211yesmaximal, characteristic, complemented
4#5825C211yesmaximal, characteristic, complemented
5#5791C211yesmaximal, characteristic, complemented
6#5648C214yesmaximal, complemented
7#5632C214yesmaximal, complemented
8#5598C214yesmaximal, complemented
9#5586C214yesmaximal, complemented
10#5552C214yesmaximal, complemented
11D4×Dic6C214yesmaximal, complemented
12#5526C211yesmaximal, characteristic, complemented
13#5516C211yesmaximal, characteristic, complemented
14#3486—31nomaximal, Sylow
15C2×C3×#41C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C23×Dic3C2212yes—
18#1010—28no—
19C2×D3×Q8C2212yescomplemented
20C2×#237C2214yes—
21C2×#237C2212yescomplemented
22C2×#235C2211yescharacteristic
23C2×#235C2212yescomplemented
24C22×Dic6—28no—
25C22×Dic6C2214yes—
26C22×Dic6C2212yes—
27C22×Dic6C2211yescharacteristic
28C22×Dic6C2214yes—
29C22×Dic6C2211yescharacteristic
30C22×Dic6C2212yescomplemented
31C3×#123C2212yes—
32C3×C22⋊Q8C2212yes—
33C3×#121C2212yes—
34C3×#120C2212yes—
35C3×C4×D4C2214yes—
36C3×#117C2211yescharacteristic
37#953C2212yes—
38#944C2212yescomplemented
39#939C2218yes—
40#936C2212yescomplemented
41#935C2214yes—
42D4×Dic3C2218yes—
43#930C2212yes—
44#929C2212yes—
45C4×#67C2214yes—
46#926C2211yescharacteristic
47#924C2212yes—
48#924C2214yescomplemented
49#897C2214yescomplemented
50#896C2218yescomplemented
51#888C2218yescomplemented
52#887C2214yescomplemented
53#886C2214yes—
54#885C2218yescomplemented
55#67⋊C4C2218yes—
56#878C2218yes—
57#878C2218yescomplemented
58#877C2214yes—
59#875C2212yescomplemented
60#869C2212yescomplemented
61C4×Dic6C2214yescomplemented
62C2×2−1+4—31no—
63C23×Q8—31no—
64D4×Q8—34no—
65D4×Q8—34no—
66#548—34no—
67#548—34no—
68#536—31no—
69#536—34no—
70#536—31no—
71#525—31no—
72#525—34no—
73#525D311yescharacteristic, complemented
74#519—31no—
75C24×C3C2311yescharacteristic
76C3×#41—24no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2314yes—
79C2×C3×D4C2312yes—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—24no—
82C22×C3×C4C2312yes—
83C22×C3×C4C2312yes—
84C22×C3×C4C2312yes—
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C2×#67C2314yes—
88C2×#67C2314yes—
89C2×#67C2312yes—
90C22×Dic3—28no—
91C22×Dic3C2318yes—
92C22×Dic3C2314yes—
93C22×Dic3C2312yes—
94C22×Dic3C2312yes—
95D3×Q8—28no—
96#237—216no—
97#237—28no—
98#235—24no—
99#235—28no—
100C2×D12C2311yescharacteristic
101C2×C4×D3C2314yes—
102C2×C4×D3C2312yes—
103C2×Dic6—24no—
104C2×Dic6—28no—
105C2×Dic6—216no—
106C2×Dic6—216no—
107C2×Dic6—28no—
108C2×Dic6D418yescomplemented
109C2×Dic6C2311yescharacteristic
110C2×Dic6C2312yes—
111C2×Dic6C2314yes—
112C2×Dic6C2318yes—
113C2×Dic6C2312yes—
114C2×Dic6C2312yes—
115C2×Dic6C2312yes—
116C3×C4⋊C4C2312yes—
117C3×C4⋊C4C2314yes—
118C3×C22⋊C4C2312yes—
119C3×C22⋊C4C2314yes—
120C3×C22⋊C4C2314yes—
121C3×C42C2312yes—
122#217C2312yes—
123#217C2314yes—
124#217C2318yes—
125(C2×D3)⋊C4C2318yes—
126(C2×D3)⋊C4C2314yes—
127C4⋊Dic3C2312yes—
128C4⋊Dic3C2318yes—
129Dic3⋊C4C2318yes—
130Dic3⋊C4C2318yes—
131Dic3⋊C4C2314yes—
132C4×Dic3C2318yes—
133C4×Dic3C2312yes—
1342−1+4—68no—
135C2×#41—34no—
136C2×#41—32no—
137C2×#41—31no—
138C2×#41—32no—
139C2×#41C2×D311yescharacteristic
140C22×Q8—32no—
141C22×Q8—31no—
142C22×Q8—32no—
143C22×Q8—31no—
144C22×Q8—34no—
145C22×Q8—34no—
146C22×Q8—32no—
147C22×Q8—68no—
148C23×C4—32no—
149C23×C4C2×D311yescharacteristic
150C4⋊Q8—32no—
151C4⋊Q8—38no—
152C4⋊Q8—32no—
153#124—32no—
154#124—38no—
155#124—32no—
156#123—34no—
157#123—32no—
158#123—34no—
159#123C2×D312yes—
160C22⋊Q8—32no—
161C22⋊Q8—34no—
162C22⋊Q8—38no—
163C22⋊Q8—34no—
164C22⋊Q8—38no—
165C22⋊Q8—38no—
166C22⋊Q8C2×D312yes—
167#121—38no—
168#121—32no—
169#121C2×D312yes—
170#120—32no—
171#120C2×D312yes—
172C4×Q8—34no—
173C4×Q8—34no—
174C4×D4—38no—
175C4×D4—34no—
176C4×D4—38no—
177C4×D4C2×D314yes—
178#117—31no—
179#117—34no—
180#117C2×D311yescharacteristic
181C23×C3—22no—
182C23×C3—24no—
183C23×C3C2412yes—
184C23×C3C2411yescharacteristic
185C23×C3C2412yes—
186C22×D3C2412yes—
187C3×Q8—22no—
188C3×D4—24no—
189C3×D4—28no—
190C3×D4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4C2×D414yes—
196C2×C3×C4C2414yes—
197C2×C3×C4C2412yes—
198C2×C3×C4C2412yes—
199C2×C3×C4C2411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—24no—
202#67—28no—
203#67—28no—
204C2×Dic3—24no—
205C2×Dic3—216no—
206C2×Dic3—28no—
207C2×Dic3C2×D418yes—
208C2×Dic3C2414yes—
209C2×Dic3C2412yes—
210C2×Dic3C2414yes—
211C2×Dic3C2414yes—
212D12—22no—
213C4×D3—24no—
214C4×D3—28no—
215Dic6—24no—
216Dic6—24no—
217Dic6—22no—
218Dic6—216no—
219Dic6—28no—
220Dic6C2×D418yes—
221Dic6C2×D418yes—
222C24C22×D311yescharacteristic
223#41—24no—
224#41—68no—
225#41—616no—
226#41—68no—
227#41—64no—
228C2×Q8—38no—
229C2×Q8—32no—
230C2×Q8—32no—
231C2×Q8—38no—
232C2×Q8—32no—
233C2×Q8—32no—
234C2×Q8—34no—
235C2×Q8—31no—
236C2×Q8—68no—
237C2×Q8—616no—
238C2×Q8—616no—
239C2×Q8—68no—
240C2×Q8—68no—
241C2×Q8—64no—
242C2×Q8C22×D311yescharacteristic
243C2×D4—34no—
244C2×D4—32no—
245C2×D4—34no—
246C2×D4—31no—
247C2×D4C22×D314yes—
248C2×D4C22×D312yes—
249C2×D4C22×D311yescharacteristic
250C22×C4—24no—
251C22×C4—32no—
252C22×C4—38no—
253C22×C4—34no—
254C22×C4—32no—
255C22×C4—34no—
256C22×C4—32no—
257C22×C4—68no—
258C22×C4C22×D312yes—
259C22×C4C22×D312yes—
260C22×C4C22×D312yes—
261C22×C4C22×D311yescharacteristic
262C22×C4C22×D311yescharacteristic
263C4⋊C4—38no—
264C4⋊C4—32no—
265C4⋊C4—38no—
266C4⋊C4—34no—
267C4⋊C4—38no—
268C4⋊C4C22×D312yes—
269C4⋊C4C22×D314yes—
270C22⋊C4—32no—
271C22⋊C4—34no—
272C22⋊C4—38no—
273C22⋊C4—34no—
274C22⋊C4—38no—
275C22⋊C4C22×D312yes—
276C22⋊C4C22×D314yes—
277C22⋊C4C22×D314yes—
278C42—38no—
279C42—32no—
280C42C22×D312yes—
281C22×C3—28no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—24no—
287C22×C3C22×D412yes—
288C22×C32−1+414yescomplemented
289C22×C3C2511yescharacteristic
290C2×D3—22no—
291C2×D3—24no—
292C3×C4—22no—
293C3×C4—22no—
294C3×C4—24no—
295C3×C4C22×D412yes—
296C3×C4C22×D412yes—
297Dic3—24no—
298Dic3—22no—
299Dic3—24no—
300Dic3C22×D418yes—
301C23—22no—
302C23—24no—
303C23—32no—
304C23C23×D312yes—
305C23C23×D311yescharacteristic
306C23C23×D312yes—
307Q8—22no—
308Q8—38no—
309Q8—38no—
310Q8—616no—
311Q8—68no—
312Q8—64no—
313Q8—64no—
314Q8—62no—
315D4—24no—
316D4—28no—
317D4—22no—
318D4—68no—
319D4—64no—
320D4—68no—
321D4—62no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—38no—
327C2×C4—34no—
328C2×C4—34no—
329C2×C4—34no—
330C2×C4—32no—
331C2×C4—68no—
332C2×C4—616no—
333C2×C4—68no—
334C2×C4—64no—
335C2×C4—64no—
336C2×C4D3×D414yes—
337C2×C4C23×D314yes—
338C2×C4C23×D312yes—
339C2×C4C23×D312yes—
340C2×C4C23×D311yescharacteristic
341C2×C4C23×D311yescharacteristic
342C2×C3—22no—
343C2×C3—24no—
344C2×C3—42no—
345C2×C3C23×D411yescharacteristic
346C2×C3C2×2−1+412yes—
347D3—42no—
348C22—28no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—62no—
355C22—64no—
356C22C2×D3×D412yes—
357C22#101014yescomplemented
358C22C24×D311yescharacteristic
359C4—22no—
360C4—22no—
361C4—24no—
362C4—38no—
363C4—62no—
364C4—64no—
365C4—64no—
366C4C2×D3×D412yes—
367C4C2×D3×D412yes—
368C3#348611yescharacteristic, complemented, Sylow
369C2—22no—
370C2—24no—
371C2—42no—
372C2—122no—
373C2C22×D3×D411yescharacteristic
374C2C2×#101012yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#525C2
hypercenter#525D3
hypocenterC3#3486
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C23×Dic6C2semidirect
3. C3×#525C2semidirect
4. #5825C2semidirect
5. #5791C2semidirect
6. #5648C2semidirect
7. #5632C2semidirect
8. #5598C2semidirect
9. #5586C2semidirect
10. #5552C2semidirect
11. D4×Dic6C2semidirect
12. #5526C2semidirect
13. #5516C2semidirect
19. C2×D3×Q8C22semidirect
21. C2×#237C22semidirect
23. C2×#235C22semidirect
30. C22×Dic6C22semidirect
38. #944C22semidirect
40. #936C22semidirect
48. #924C22semidirect
49. #897C22semidirect
50. #896C22semidirect
51. #888C22semidirect
52. #887C22semidirect
54. #885C22semidirect
57. #878C22semidirect
59. #875C22semidirect
60. #869C22semidirect
61. C4×Dic6C22semidirect
73. #525D3semidirect
108. C2×Dic6D4semidirect
288. C22×C32−1+4semidirect
357. C22#1010semidirect
368. C3#3486semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C23×Dic62
C3×#5252
#58252
#57912
#56482
#56322
#55982
#55862
#55522
D4×Dic62
#55262
#55162
#34863