Finite Group Databasebeta

← #86127

Subgroups — #86127

Order 384.

1010 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#5702C214yesmaximal, complemented
3#5688C212yesmaximal, complemented
4#5680C212yesmaximal, complemented
5#5662C214yesmaximal, complemented
6#5641C212yesmaximal, complemented
7#5616C212yesmaximal, complemented
8#5584C211yesmaximal, characteristic, complemented
9Q8×Dic6C211yesmaximal, characteristic, complemented
10#5563C211yesmaximal, characteristic, complemented
11#5552C212yesmaximal, complemented
12#5542C212yesmaximal, complemented
13D4×Dic6C211yesmaximal, characteristic, complemented
14#5534C212yesmaximal, complemented
15#5516C212yesmaximal, complemented
16C4×#235C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3518—31nomaximal, Sylow
19C2×#235C2211yescharacteristic
20C22×Dic6C2212yes—
21C3×#126C2212yes—
22C3×#125C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×#123C2212yes—
25C3×C22⋊Q8C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic
29C3×C4×D4C2212yes—
30C3×#117C2212yes—
31C2×C3×C42C2211yescharacteristic
32#946C2212yes—
33Q8×Dic3C2212yes—
34#939C2214yes—
35#938C2212yes—
36#936C2214yes—
37D4×Dic3C2212yes—
38#930C2212yes—
39C4×#67C2212yes—
40C4×#67C2212yes—
41#926C2212yes—
42#926C2211yescharacteristic
43#926C2212yes—
44#924C2212yes—
45#924C2212yes—
46#924C2212yes—
47#898C2214yes—
48#898C2214yes—
49#896C2214yes—
50#896C2214yes—
51#892C2212yes—
52#892C2212yes—
53#888C2212yes—
54#888C2214yes—
55#888C2212yes—
56#887C2214yes—
57#887C2214yes—
58#887C2214yes—
59#885C2214yes—
60#885C2214yes—
61#885C2214yes—
62#883C2214yes—
63#67⋊C4C2214yes—
64#879C2214yes—
65#879C2214yes—
66#878C2214yes—
67#877C2214yes—
68#877C2214yes—
69#875C2212yes—
70C4×D12C2211yescharacteristic
71C4×D12C2211yescharacteristic
72#872C2212yes—
73C42×D3C2212yes—
74#870C2212yes—
75#869C2211yescharacteristic
76#869C2211yescharacteristic
77#869C2212yes—
78C4×Dic6C2212yes—
79C4×Dic6C2212yes—
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84Q82—31no—
85#556—34no—
86#556—32no—
87#556—31no—
88#556—32no—
89#555—32no—
90#555—31no—
91D4×Q8—31no—
92#548—32no—
93#542—34no—
94#542—32no—
95#533—32no—
96#533—31no—
97#529—32no—
98#529D311yescharacteristic, complemented
99#519—32no—
100C4×#41—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67C2312yes—
109C2×#67C2314yes—
110C22×Dic3C2314yes—
111#235—24no—
112C2×D12C2311yescharacteristic
113C2×C4×D3C2312yes—
114C2×C4×D3C2312yes—
115C2×C4×D3C2312yes—
116C2×Dic6—28no—
117C2×Dic6C2312yes—
118C2×Dic6C2314yes—
119C2×Dic6C2312yes—
120C2×Dic6C2314yes—
121C2×Dic6C2311yescharacteristic
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×C42C2312yes—
135C3×C42C2311yescharacteristic
136C3×C42C2311yescharacteristic
137C3×C42C2311yescharacteristic
138#217C2314yes—
139#217C2314yes—
140#217C2312yes—
141#217C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2314yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2314yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161C4×Dic3C2314yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2314yes—
165C4×Dic3C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C2×#41—31no—
169C22×Q8—32no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—34no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4⋊Q8—32no—
176#126—34no—
177#126—34no—
178#126—34no—
179#126—34no—
180#126C2×D312yes—
181#125—32no—
182#125C2×D311yescharacteristic
183#124—34no—
184#124—34no—
185#124—34no—
186#124—34no—
187#124—32no—
188#124C2×D311yescharacteristic
189#123—34no—
190#123C2×D312yes—
191C22⋊Q8—32no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—34no—
196C22⋊Q8—34no—
197C22⋊Q8—34no—
198C22⋊Q8C2×D311yescharacteristic
199#121—32no—
200#121—34no—
201#121—32no—
202#121C2×D311yescharacteristic
203C4×Q8—32no—
204C4×Q8—34no—
205C4×Q8—34no—
206C4×Q8—34no—
207C4×Q8—31no—
208C4×Q8—31no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×Q8—32no—
213C4×Q8C2×D311yescharacteristic
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—31no—
219C4×D4—31no—
220C4×D4C2×D312yes—
221C4×D4C2×D311yescharacteristic
222#117—31no—
223#117—32no—
224#117—32no—
225#117—32no—
226#117—32no—
227#117—34no—
228#117—34no—
229#117—32no—
230#117C2×D312yes—
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—24no—
253#67—44no—
254C2×Dic3—28no—
255C2×Dic3C2412yes—
256C2×Dic3C2414yes—
257C2×Dic3C2414yes—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260D12—22no—
261C4×D3—24no—
262C4×D3—24no—
263C4×D3—42no—
264Dic6—24no—
265Dic6—28no—
266Dic6—44no—
267Dic6—42no—
268Dic6#4114yes—
269#41—64no—
270C2×Q8—31no—
271C2×Q8—34no—
272C2×Q8—32no—
273C2×Q8—32no—
274C2×Q8—34no—
275C2×Q8—68no—
276C2×Q8C22×D311yescharacteristic
277C2×D4—32no—
278C2×D4—34no—
279C2×D4—31no—
280C2×D4C22×D311yescharacteristic
281C2×D4C22×D312yes—
282C22×C4—34no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—32no—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D312yes—
289C22×C4C22×D311yescharacteristic
290C4⋊C4—32no—
291C4⋊C4—31no—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C4⋊C4—32no—
295C4⋊C4—31no—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4C22×D311yescharacteristic
305C4⋊C4C22×D311yescharacteristic
306C4⋊C4C22×D312yes—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D312yes—
310C22⋊C4—34no—
311C22⋊C4—32no—
312C22⋊C4—34no—
313C22⋊C4—34no—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—34no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4C22×D312yes—
320C22⋊C4C22×D312yes—
321C22⋊C4C22×D312yes—
322C22⋊C4C22×D312yes—
323C22⋊C4C22×D312yes—
324C42—32no—
325C42—32no—
326C42—32no—
327C42—34no—
328C42—34no—
329C42—34no—
330C42—34no—
331C42C22×D311yescharacteristic
332C42C22×D311yescharacteristic
333C42C22×D311yescharacteristic
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C22×C3—22no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3C2511yescharacteristic
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C3×C4—22no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—21no—
353C3×C4C2×#4112yes—
354C3×C42−1+412yes—
355Dic3—24no—
356Dic3—22no—
357Dic3—22no—
358Dic3—24no—
359Dic3C2×#4114yes—
360C23—32no—
361C23C23×D311yescharacteristic
362C23C23×D312yes—
363Q8—22no—
364Q8—34no—
365Q8—68no—
366Q8—64no—
367Q8—122no—
368Q8—124no—
369D4—22no—
370D4—42no—
371D4—64no—
372D4—62no—
373D4—124no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—34no—
380C2×C4—34no—
381C2×C4—32no—
382C2×C4—41no—
383C2×C4—64no—
384C2×C4—68no—
385C2×C4—64no—
386C2×C4—122no—
387C2×C4C23×D311yescharacteristic
388C2×C4C23×D311yescharacteristic
389C2×C4C23×D311yescharacteristic
390C2×C4C23×D312yes—
391C2×C4C23×D312yes—
392C2×C4C23×D312yes—
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—42no—
397C2×C3—41no—
398C2×C3C22×#4111yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400C2×C3#58511yescharacteristic
401D3—42no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—21no—
407C22—22no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22C24×D311yescharacteristic
412C4—21no—
413C4—22no—
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—34no—
420C4—64no—
421C4—62no—
422C4—62no—
423C4—64no—
424C4D3×#4112yes—
425C4#101012yes—
426C3#351811yescharacteristic, complemented, Sylow
427C2—42no—
428C2—41no—
429C2—122no—
430C2C2×D3×#4111yescharacteristic
431C2C2×#101011yescharacteristic
432C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3518
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. #5702C2semidirect
3. #5688C2semidirect
4. #5680C2semidirect
5. #5662C2semidirect
6. #5641C2semidirect
7. #5616C2semidirect
8. #5584C2semidirect
9. Q8×Dic6C2semidirect
10. #5563C2semidirect
11. #5552C2semidirect
12. #5542C2semidirect
13. D4×Dic6C2semidirect
14. #5534C2semidirect
15. #5516C2semidirect
16. C4×#235C2semidirect
17. #5470C2semidirect
98. #529D3semidirect
426. C3#3518semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57022
#56882
#56802
#56622
#56412
#56162
#55842
Q8×Dic62
#55632
#55522
#55422
D4×Dic62
#55342
#55162
C4×#2352
#54702
#35183