Finite Group Databasebeta

← #86268

Subgroups — #86268

Order 384.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5731C212yesmaximal, complemented
3#5729C212yesmaximal, complemented
4#5696C214yesmaximal, complemented
5#5646C214yesmaximal, complemented
6#5641C212yesmaximal, complemented
7#5639C212yesmaximal, complemented
8#5584C212yesmaximal, complemented
9#5584C211yesmaximal, characteristic, complemented
10#5579C211yesmaximal, characteristic, complemented
11#5570C211yesmaximal, characteristic, complemented
12#5567C212yesmaximal, complemented
13Q8×Dic6C211yesmaximal, characteristic, complemented
14#5534C212yesmaximal
15#5518C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic
17#5470C211yesmaximal, characteristic, complemented
18#3555—31nomaximal, Sylow
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4⋊Q8C2211yescharacteristic
21C3×#125C2212yes—
22C3×C22⋊Q8C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24C3×C22⋊Q8C2212yes—
25C3×C4×Q8C2211yescharacteristic
26C3×C4×Q8C2212yes—
27C3×C4×Q8C2211yescharacteristic
28C3×#117C2212yes—
29C2×C3×C42C2211yescharacteristic
30#946C2212yes—
31#946C2212yes—
32#946C2214yes—
33Q8×Dic3C2212yes—
34Q8×Dic3C2212yes—
35#944C2214yes—
36#930C2211yescharacteristic
37#929C2212yes—
38C4×#67C2212yes—
39#926C2212yes—
40#924C2212yes—
41#924C2212yes—
42#924C2211yescharacteristic
43#898C2214yes—
44#898C2214yes—
45#898C2214yes—
46#898C2214yes—
47#896C2214yes—
48#896C2214yes—
49#892C2214yes—
50#892C2212yes—
51D3×C4⋊C4C2212yes—
52#890C2212yes—
53#890C2214yes—
54#889C2214yes—
55#889C2214yes—
56#889C2214yes—
57#889C2214yes—
58#888C2212yes—
59#888C2214yes—
60#887C2214yes—
61#885C2214yes—
62#883C2214yes—
63#879C2214yes—
64#879C2214yes—
65#876C2212yes—
66#875C2211yescharacteristic
67#875C2212yes—
68C4×D12C2211yescharacteristic
69C4×D12C2212yes—
70#872C2212yes—
71#870C2212yes—
72#870C2211yescharacteristic
73#870C2212yes—
74#869C2211yescharacteristic
75#869C2211yescharacteristic
76#869C2212yes—
77#869C2212yes—
78C4×Dic6C2212yes—
79C4×Dic6C2212yes—
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84#563—34no—
85Q82—31no—
86#557—32no—
87#557—31no—
88#556—32no—
89#556—32no—
90#556—31no—
91#554—31no—
92#552—32no—
93#544—32no—
94#542—32no—
95#541—32no—
96#541—34no—
97#541—32no—
98#533—31no—
99#533D311yescharacteristic, complemented
100#529—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2312yes—
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C2×#67C2312yes—
108C2×D12C2311yescharacteristic
109C2×C4×D3C2312yes—
110C2×C4×D3C2314yes—
111C2×Dic6C2312yes—
112C2×Dic6C2311yescharacteristic
113C2×Dic6C2312yes—
114C2×Dic6C2314yes—
115C3×C4⋊C4C2312yes—
116C3×C4⋊C4C2311yescharacteristic
117C3×C4⋊C4C2312yes—
118C3×C4⋊C4C2312yes—
119C3×C4⋊C4C2312yes—
120C3×C4⋊C4C2312yes—
121C3×C4⋊C4C2312yes—
122C3×C4⋊C4C2311yescharacteristic
123C3×C4⋊C4C2312yes—
124C3×C22⋊C4C2312yes—
125C3×C22⋊C4C2312yes—
126C3×C42C2312yes—
127C3×C42C2311yescharacteristic
128C3×C42C2311yescharacteristic
129C3×C42C2311yescharacteristic
130C3×C42C2312yes—
131C3×C42C2311yescharacteristic
132#217C2312yes—
133#217C2314yes—
134(C2×D3)⋊C4C2312yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2311yescharacteristic
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2314yes—
148C4⋊Dic3C2314yes—
149C4⋊Dic3C2314yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2312yes—
161C4×Dic3C2314yes—
162C4×Dic3C2312yes—
163C4×Dic3C2314yes—
164C4×Dic3C2314yes—
165C4×Dic3C2314yes—
166C4⋊Q8—34no—
167C4⋊Q8—34no—
168C4⋊Q8—32no—
169C4⋊Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8C2×D311yescharacteristic
174C4⋊Q8C2×D311yescharacteristic
175#126—34no—
176#126—34no—
177#126—34no—
178#126—34no—
179#126—34no—
180#126—34no—
181#126—32no—
182#125—34no—
183#125—34no—
184#125—34no—
185#125—34no—
186#125—34no—
187#125—32no—
188#125—31no—
189#125—32no—
190#125—32no—
191#125C2×D312yes—
192#124—34no—
193#124—31no—
194#124—32no—
195#123—32no—
196#123—34no—
197C22⋊Q8—32no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—31no—
201C22⋊Q8—32no—
202C22⋊Q8—32no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8C2×D311yescharacteristic
206C22⋊Q8C2×D311yescharacteristic
207C22⋊Q8C2×D312yes—
208#121—31no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—34no—
212C4×Q8—31no—
213C4×Q8—32no—
214C4×Q8—32no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×Q8C2×D311yescharacteristic
219C4×Q8C2×D312yes—
220C4×Q8C2×D311yescharacteristic
221C4×D4—32no—
222C4×D4—31no—
223C4×D4—32no—
224#117—32no—
225#117—34no—
226#117—32no—
227#117—32no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C42C2×D311yescharacteristic
231C23×C3C2411yescharacteristic
232C22×D3C2412yes—
233C3×Q8—22no—
234C3×Q8—22no—
235C3×Q8—42no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2412yes—
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249#67—42no—
250C2×Dic3C2412yes—
251C2×Dic3C2414yes—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254C2×Dic3C2414yes—
255D12—22no—
256C4×D3—24no—
257C4×D3—44no—
258Dic6—24no—
259Dic6—22no—
260Dic6—42no—
261Dic6—44no—
262C2×Q8—31no—
263C2×Q8—32no—
264C2×Q8—32no—
265C2×Q8—34no—
266C2×Q8C22×D311yescharacteristic
267C2×Q8C22×D311yescharacteristic
268C2×Q8C22×D312yes—
269C2×D4—32no—
270C2×D4—31no—
271C22×C4—32no—
272C22×C4—34no—
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C22×C4C22×D311yescharacteristic
276C4⋊C4—32no—
277C4⋊C4—34no—
278C4⋊C4—32no—
279C4⋊C4—32no—
280C4⋊C4—34no—
281C4⋊C4—32no—
282C4⋊C4—31no—
283C4⋊C4—32no—
284C4⋊C4—32no—
285C4⋊C4—31no—
286C4⋊C4—34no—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—32no—
292C4⋊C4—34no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4C22×D311yescharacteristic
297C4⋊C4C22×D312yes—
298C4⋊C4C22×D311yescharacteristic
299C4⋊C4C22×D312yes—
300C4⋊C4C22×D312yes—
301C4⋊C4C22×D312yes—
302C4⋊C4C22×D312yes—
303C4⋊C4C22×D312yes—
304C4⋊C4C22×D312yes—
305C22⋊C4—34no—
306C22⋊C4—32no—
307C22⋊C4—32no—
308C22⋊C4—32no—
309C22⋊C4—32no—
310C22⋊C4—34no—
311C22⋊C4—34no—
312C22⋊C4—32no—
313C22⋊C4—34no—
314C22⋊C4C22×D312yes—
315C22⋊C4C22×D312yes—
316C42—34no—
317C42—32no—
318C42—34no—
319C42—34no—
320C42—34no—
321C42C22×D311yescharacteristic
322C42C22×D311yescharacteristic
323C42C22×D311yescharacteristic
324C42C22×D311yescharacteristic
325C42C22×D312yes—
326C42C22×D312yes—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3C2511yescharacteristic
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—22no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C42−1+412yes—
343C3×C42−1+412yes—
344Dic3—24no—
345Dic3—22no—
346Dic3—22no—
347Dic3—24no—
348Dic3—22no—
349C23—32no—
350C23C23×D311yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—42no—
354Q8—62no—
355Q8—64no—
356Q8—122no—
357Q8—124no—
358D4—62no—
359D4—122no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—34no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—34no—
367C2×C4—41no—
368C2×C4—64no—
369C2×C4—124no—
370C2×C4C23×D311yescharacteristic
371C2×C4C23×D312yes—
372C2×C4C23×D312yes—
373C2×C4C23×D312yes—
374C2×C4C23×D311yescharacteristic
375C2×C4C23×D311yescharacteristic
376C2×C4C23×D311yescharacteristic
377C2×C4C23×D312yes—
378C2×C4C23×D311yescharacteristic
379C2×C4C23×D311yescharacteristic
380C2×C3—41no—
381C2×C3C2×2−1+411yescharacteristic
382C2×C3C2×2−1+411yescharacteristic
383C2×C3#58511yescharacteristic
384D3—42no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—62no—
389C22—62no—
390C22—62no—
391C22C24×D311yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—22no—
400C4—64no—
401C4—62no—
402C4—62no—
403C4—64no—
404C4—62no—
405C4#100712yes—
406C4#101012yes—
407C3#355511yescharacteristic, complemented, Sylow
408C2—41no—
409C2—122no—
410C2C2×#100711yescharacteristic
411C2C2×#101011yescharacteristic
412C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#533C2
prime radical#930C22
hypercenter#533D3
hypocenterC3#3555
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5731C2semidirect
3. #5729C2semidirect
4. #5696C2semidirect
5. #5646C2semidirect
6. #5641C2semidirect
7. #5639C2semidirect
8. #5584C2semidirect
9. #5584C2semidirect
10. #5579C2semidirect
11. #5570C2semidirect
12. #5567C2semidirect
13. Q8×Dic6C2semidirect
15. #5518C2semidirect
17. #5470C2semidirect
99. #533D3semidirect
407. C3#3555semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57312
#57292
#56962
#56462
#56412
#56392
#55842
#55792
#55702
#55672
Q8×Dic62
#55342
#55182
#54792
#54702
#35553