Finite Group Databasebeta

← #86165

Subgroups — #86165

Order 384.

1014 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#529C211yesmaximal, characteristic, complemented
2#5708C214yesmaximal, complemented
3D3×#125C212yesmaximal, complemented
4#5680C212yesmaximal, complemented
5#5660C214yesmaximal, complemented
6#5639C212yesmaximal, complemented
7#5614C212yesmaximal, complemented
8#5579C211yesmaximal, characteristic, complemented
9#5570C211yesmaximal, characteristic, complemented
10#5551C211yesmaximal, characteristic, complemented
11#5551C212yesmaximal, complemented
12#5543C211yesmaximal, characteristic, complemented
13#5541C212yesmaximal, complemented
14#5523C212yesmaximal, complemented
15#5518C212yesmaximal, complemented
16C4×#235C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3515—31nomaximal, Sylow
19C2×#235C2211yescharacteristic
20C2×#235C2212yes—
21C3×#126C2212yes—
22C3×#125C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×#123C2212yes—
25C3×C22⋊Q8C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2212yes—
29C3×C4×D4C2211yescharacteristic
30C3×#117C2212yes—
31C2×C3×C42C2211yescharacteristic
32#946C2212yes—
33Q8×Dic3C2212yes—
34#938C2214yes—
35#938C2212yes—
36#935C2214yes—
37D4×Dic3C2212yes—
38C4×#67C2212yes—
39C4×#67C2212yes—
40#926C2212yes—
41#926C2211yescharacteristic
42C2×C4⋊Dic3C2212yes—
43#924C2212yes—
44#924C2212yes—
45#924C2212yes—
46#924C2212yes—
47#896C2214yes—
48#892C2214yes—
49D3×C4⋊C4C2212yes—
50D3×C4⋊C4C2214yes—
51D3×C4⋊C4C2214yes—
52D3×C4⋊C4C2212yes—
53#890C2212yes—
54#890C2214yes—
55#890C2212yes—
56#889C2214yes—
57#889C2214yes—
58#889C2214yes—
59#883C2214yes—
60#883C2214yes—
61#883C2214yes—
62#67⋊C4C2214yes—
63#67⋊C4C2214yes—
64#879C2214yes—
65#879C2214yes—
66#879C2214yes—
67#878C2214yes—
68#877C2214yes—
69C4×D12C2211yescharacteristic
70C4×D12C2211yescharacteristic
71C4×D12C2212yes—
72#872C2212yes—
73C42×D3C2212yes—
74#870C2212yes—
75#870C2212yes—
76#869C2211yescharacteristic
77#869C2211yescharacteristic
78#869C2212yes—
79C4×Dic6C2212yes—
80C4×Dic6C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#557—31no—
85#554—31no—
86#554—31no—
87#553—32no—
88#552—34no—
89#552—32no—
90#552—32no—
91#547—31no—
92#547—32no—
93#541—32no—
94#539—34no—
95#533—31no—
96#529—32no—
97#529D311yescharacteristic, complemented
98C2×#125—32no—
99#520—32no—
100C4×#41—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2312yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67C2314yes—
109C2×#67C2312yes—
110C22×Dic3C2314yes—
111#235—24no—
112#235—28no—
113C2×D12C2311yescharacteristic
114C2×C4×D3C2312yes—
115C2×C4×D3C2312yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2312yes—
118C2×C4×D3C2314yes—
119C2×Dic6C2312yes—
120C2×Dic6C2311yescharacteristic
121C2×Dic6C2312yes—
122C3×C4⋊C4C2312yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2311yescharacteristic
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2312yes—
128C3×C22⋊C4C2312yes—
129C3×C22⋊C4C2312yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C42C2311yescharacteristic
134C3×C42C2311yescharacteristic
135C3×C42C2311yescharacteristic
136C3×C42C2312yes—
137C3×C42C2311yescharacteristic
138#217C2314yes—
139#217C2314yes—
140#217C2312yes—
141#217C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145C4⋊Dic3C2311yescharacteristic
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2314yes—
153C4⋊Dic3C2314yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162Dic3⋊C4C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2314yes—
167C4×Dic3C2312yes—
168C2×#41—31no—
169C2×#41—32no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8—32no—
173#126—34no—
174#126—34no—
175#126—34no—
176#126C2×D312yes—
177#125—34no—
178#125—34no—
179#125—32no—
180#125—32no—
181#125—34no—
182#125—34no—
183#125—32no—
184#125—32no—
185#125C2×D311yescharacteristic
186#124C2×D311yescharacteristic
187#123—34no—
188#123—34no—
189#123—34no—
190#123—34no—
191#123C2×D312yes—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8C2×D311yescharacteristic
200#121—32no—
201#121—34no—
202#121C2×D311yescharacteristic
203C4×Q8—32no—
204C4×Q8—31no—
205C4×Q8—31no—
206C4×Q8—32no—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8C2×D311yescharacteristic
210C4×D4—32no—
211C4×D4—32no—
212C4×D4—32no—
213C4×D4—34no—
214C4×D4—34no—
215C4×D4—31no—
216C4×D4—31no—
217C4×D4—32no—
218C4×D4C2×D312yes—
219C4×D4C2×D311yescharacteristic
220#117—31no—
221#117—32no—
222#117—34no—
223#117—34no—
224#117—32no—
225#117C2×D312yes—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—34no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—32no—
231C2×C42—32no—
232C2×C42C2×D311yescharacteristic
233C23×C3C2411yescharacteristic
234C23×C3C2412yes—
235C22×D3C2412yes—
236C3×Q8—22no—
237C3×D4—22no—
238C3×D4—42no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—41no—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2412yes—
247C2×C3×C4C2412yes—
248C2×C3×C4C2412yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252#67—28no—
253#67—24no—
254C2×Dic3—44no—
255C2×Dic3C2412yes—
256C2×Dic3C2412yes—
257C2×Dic3C2414yes—
258C2×Dic3C2412yes—
259C2×Dic3C2414yes—
260D12#4114yes—
261C4×D3—24no—
262C4×D3—28no—
263C4×D3—24no—
264C4×D3—44no—
265C4×D3—42no—
266Dic6—24no—
267Dic6—22no—
268Dic6—42no—
269#41—64no—
270#41—68no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8—32no—
274C2×Q8C22×D311yescharacteristic
275C2×D4—32no—
276C2×D4—34no—
277C2×D4—31no—
278C2×D4C22×D311yescharacteristic
279C2×D4C22×D312yes—
280C22×C4—34no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—34no—
284C22×C4—32no—
285C22×C4—34no—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D312yes—
289C22×C4C22×D311yescharacteristic
290C4⋊C4—31no—
291C4⋊C4—34no—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—34no—
298C4⋊C4—34no—
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—31no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—32no—
307C4⋊C4—34no—
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D312yes—
314C22⋊C4—32no—
315C22⋊C4—34no—
316C22⋊C4—34no—
317C22⋊C4—34no—
318C22⋊C4—32no—
319C22⋊C4—32no—
320C22⋊C4—34no—
321C22⋊C4C22×D312yes—
322C22⋊C4C22×D312yes—
323C22⋊C4C22×D312yes—
324C22⋊C4C22×D312yes—
325C22⋊C4C22×D312yes—
326C42—32no—
327C42—32no—
328C42—32no—
329C42—34no—
330C42—34no—
331C42C22×D311yescharacteristic
332C42C22×D311yescharacteristic
333C42C22×D311yescharacteristic
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3C2511yescharacteristic
343C2×D3—22no—
344C2×D3—22no—
345C2×D3C2×#4114yes—
346C3×C4—21no—
347C3×C4—22no—
348C3×C4—21no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—22no—
353C3×C4C2×#4112yes—
354C3×C42−1+412yes—
355Dic3—22no—
356Dic3—24no—
357Dic3—22no—
358Dic3—24no—
359Dic3—22no—
360C23—32no—
361C23C23×D311yescharacteristic
362C23C23×D312yes—
363Q8—22no—
364Q8—62no—
365Q8—64no—
366Q8—122no—
367D4—22no—
368D4—34no—
369D4—42no—
370D4—64no—
371D4—68no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—34no—
379C2×C4—34no—
380C2×C4—41no—
381C2×C4—64no—
382C2×C4—68no—
383C2×C4—64no—
384C2×C4—122no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4C23×D311yescharacteristic
388C2×C4C23×D311yescharacteristic
389C2×C4C23×D311yescharacteristic
390C2×C4C23×D312yes—
391C2×C4C23×D311yescharacteristic
392C2×C4C23×D312yes—
393C2×C4C23×D312yes—
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—41no—
397C2×C3—42no—
398C2×C3C22×#4111yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400C2×C3#58511yescharacteristic
401D3—24no—
402C22—21no—
403C22—22no—
404C22—21no—
405C22—22no—
406C22—21no—
407C22—22no—
408C22—34no—
409C22—62no—
410C22—62no—
411C22C24×D311yescharacteristic
412C4—21no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—62no—
420C4—64no—
421C4—62no—
422C4—62no—
423C4—64no—
424C4D3×#4112yes—
425C4#101012yes—
426C3#351511yescharacteristic, complemented, Sylow
427C2—41no—
428C2—42no—
429C2—64no—
430C2C2×D3×#4111yescharacteristic
431C2C2×#101011yescharacteristic
432C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3515
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5708C2semidirect
3. D3×#125C2semidirect
4. #5680C2semidirect
5. #5660C2semidirect
6. #5639C2semidirect
7. #5614C2semidirect
8. #5579C2semidirect
9. #5570C2semidirect
10. #5551C2semidirect
11. #5551C2semidirect
12. #5543C2semidirect
13. #5541C2semidirect
14. #5523C2semidirect
15. #5518C2semidirect
16. C4×#235C2semidirect
17. #5470C2semidirect
97. #529D3semidirect
426. C3#3515semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57082
D3×#1252
#56802
#56602
#56392
#56142
#55792
#55702
#55512
#55432
#55412
#55232
#55182
C4×#2352
#54702
#35153