Finite Group Databasebeta

← #86930

Subgroups — #86930

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×#554C211yesmaximal, characteristic, complemented
2#5730C211yesmaximal, characteristic, complemented
3#5728C211yesmaximal, characteristic, complemented
4#5698C212yesmaximal, complemented
5#5694C212yesmaximal, complemented
6#5645C214yesmaximal, complemented
7#5642C214yesmaximal, complemented
8#5642C212yesmaximal, complemented
9#5624C212yesmaximal, complemented
10#5580C211yesmaximal, characteristic, complemented
11#5570C211yesmaximal, characteristic, complemented
12#5555C212yesmaximal, complemented
13#5555C211yesmaximal, characteristic, complemented
14#5546C212yesmaximal, complemented
15C4×#237C211yesmaximal, characteristic, complemented
16#5514C212yesmaximal, complemented
17C2×#890C212yesmaximal, complemented
18#3515—31nomaximal, Sylow
19C2×#237C2211yescharacteristic
20C2×#237C2212yes—
21C3×C4⋊Q8C2211yescharacteristic
22C3×#125C2212yes—
23C3×C22⋊Q8C2214yes—
24C3×C22⋊Q8C2212yes—
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2212yes—
28C2×C3×C4⋊C4C2212yes—
29#947C2211yescharacteristic
30#946C2211yescharacteristic
31#946C2214yes—
32#946C2212yes—
33Q8×Dic3C2212yes—
34Q8×Dic3C2212yes—
35D4×Dic3C2211yescharacteristic
36#930C2214yes—
37#930C2212yes—
38C4×#67C2212yes—
39C4×#67C2214yes—
40C2×C4⋊Dic3C2214yes—
41C2×C4⋊Dic3C2212yes—
42C2×Dic3⋊C4C2214yes—
43C2×C4×Dic3C2212yes—
44#898C2212yes—
45#898C2214yes—
46#898C2214yes—
47#898C2214yes—
48#897C2212yes—
49#897C2212yes—
50#897C2212yes—
51#895C2211yescharacteristic
52#894C2212yes—
53#894C2214yes—
54#893C2212yes—
55#892C2214yes—
56#892C2212yes—
57#892C2211yescharacteristic
58D3×C4⋊C4C2212yes—
59#890C2214yes—
60#890C2214yes—
61#890C2214yes—
62#890C2212yes—
63#890C2214yes—
64#890C2212yes—
65#890C2211yescharacteristic
66#890C2212yes—
67#887C2211yescharacteristic
68#887C2212yes—
69#886C2214yes—
70#886C2214yes—
71#886C2214yes—
72#67⋊C4C2212yes—
73#67⋊C4C2214yes—
74#877C2212yes—
75#877C2214yes—
76C4×D12C2212yes—
77C4×D12C2211yescharacteristic
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#869C2212yes—
81#869C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#557—31no—
85#554—31no—
86#554D311yescharacteristic, complemented
87#553—32no—
88#552—32no—
89#552—34no—
90#552—32no—
91#547—32no—
92#547—31no—
93#541—32no—
94#539—34no—
95#533—31no—
96#529—32no—
97#529—31no—
98C2×#125—32no—
99#520—32no—
100C4×#41—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C22×C3×C4C2314yes—
106C2×#67C2312yes—
107C2×#67C2314yes—
108C22×Dic3C2312yes—
109C22×Dic3C2314yes—
110C22×Dic3C2312yes—
111#237—24no—
112#237—28no—
113C2×D12C2311yescharacteristic
114C2×D12C2312yes—
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2311yescharacteristic
117C2×C4×D3C2312yes—
118C2×C4×D3C2311yescharacteristic
119C2×C4×D3C2314yes—
120C2×Dic6C2312yes—
121C2×Dic6C2311yescharacteristic
122C3×C4⋊C4C2312yes—
123C3×C4⋊C4C2311yescharacteristic
124C3×C4⋊C4C2311yescharacteristic
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2314yes—
127C3×C4⋊C4C2312yes—
128C3×C4⋊C4C2314yes—
129C3×C22⋊C4C2314yes—
130C3×C22⋊C4C2312yes—
131C3×C42C2311yescharacteristic
132C3×C42C2312yes—
133#217C2312yes—
134(C2×D3)⋊C4C2312yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143C4⋊Dic3C2311yescharacteristic
144C4⋊Dic3C2314yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2314yes—
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2314yes—
161C4×Dic3C2312yes—
162C4×Dic3C2311yescharacteristic
163C4×Dic3C2311yescharacteristic
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C2×#41—31no—
169C2×#41—32no—
170C4⋊Q8—31no—
171C4⋊Q8—32no—
172C4⋊Q8C2×D311yescharacteristic
173#126—34no—
174#126—32no—
175#126—34no—
176#126—34no—
177#125—31no—
178#125—32no—
179#125—34no—
180#125—34no—
181#125—32no—
182#125—32no—
183#125—34no—
184#125—34no—
185#125C2×D312yes—
186#124—31no—
187#123—32no—
188#123—34no—
189#123—34no—
190#123—34no—
191#123—34no—
192C22⋊Q8—31no—
193C22⋊Q8—32no—
194C22⋊Q8—34no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8C2×D312yes—
199C22⋊Q8C2×D314yes—
200#121—32no—
201#121—34no—
202#121—31no—
203C4×Q8—32no—
204C4×Q8—32no—
205C4×Q8—31no—
206C4×Q8—32no—
207C4×Q8—31no—
208C4×Q8—32no—
209C4×Q8C2×D311yescharacteristic
210C4×D4—31no—
211C4×D4—32no—
212C4×D4—34no—
213C4×D4—32no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—32no—
217C4×D4—31no—
218C4×D4C2×D311yescharacteristic
219C4×D4C2×D312yes—
220#117—31no—
221#117—34no—
222#117—32no—
223#117—32no—
224#117—34no—
225#117—32no—
226C2×C4⋊C4—34no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4C2×D312yes—
231C2×C42—32no—
232C2×C42—31no—
233C23×C3C2412yes—
234C22×D3C2411yescharacteristic
235C22×D3C2412yes—
236C3×Q8—22no—
237C3×Q8—42no—
238C3×D4#4114yes—
239C2×C3×C4—24no—
240C2×C3×C4—44no—
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2414yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248#67—24no—
249#67—28no—
250C2×Dic3—24no—
251C2×Dic3—28no—
252C2×Dic3—42no—
253C2×Dic3C2411yescharacteristic
254C2×Dic3C2411yescharacteristic
255C2×Dic3C2412yes—
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2414yes—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260D12—22no—
261D12—42no—
262C4×D3—22no—
263C4×D3—24no—
264C4×D3—22no—
265C4×D3—44no—
266C4×D3—41no—
267Dic6—24no—
268Dic6—22no—
269#41—64no—
270#41—68no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8C22×D311yescharacteristic
274C2×Q8C22×D312yes—
275C2×D4—32no—
276C2×D4—34no—
277C2×D4—31no—
278C2×D4—32no—
279C2×D4C22×D311yescharacteristic
280C22×C4—32no—
281C22×C4—34no—
282C22×C4—32no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—32no—
286C22×C4—31no—
287C22×C4—34no—
288C22×C4C22×D312yes—
289C22×C4C22×D314yes—
290C4⋊C4—31no—
291C4⋊C4—34no—
292C4⋊C4—32no—
293C4⋊C4—31no—
294C4⋊C4—32no—
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—32no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D311yescharacteristic
309C4⋊C4C22×D312yes—
310C4⋊C4C22×D314yes—
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D314yes—
313C4⋊C4C22×D312yes—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—32no—
317C22⋊C4—32no—
318C22⋊C4—32no—
319C22⋊C4—34no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4C22×D314yes—
325C22⋊C4C22×D312yes—
326C42—31no—
327C42—32no—
328C42—31no—
329C42—31no—
330C42—32no—
331C42—32no—
332C42—34no—
333C42—34no—
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3C2×#4114yes—
339C22×C3C2511yescharacteristic
340C2×D3—21no—
341C2×D3—22no—
342C2×D3—21no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—21no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—24no—
349C3×C4—21no—
350C3×C4—22no—
351C3×C4C2×#4112yes—
352C3×C42−1+412yes—
353Dic3—22no—
354Dic3—24no—
355Dic3—21no—
356Dic3—22no—
357Dic3—21no—
358Dic3—22no—
359Dic3—21no—
360C23—31no—
361C23—32no—
362C23C23×D312yes—
363Q8—22no—
364Q8—42no—
365Q8—62no—
366Q8—64no—
367D4—64no—
368D4—68no—
369D4—62no—
370D4—122no—
371D4#23914yes—
372C2×C4—24no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—34no—
379C2×C4—32no—
380C2×C4—44no—
381C2×C4—64no—
382C2×C4—68no—
383C2×C4—62no—
384C2×C4—64no—
385C2×C4—62no—
386C2×C4—124no—
387C2×C4—121no—
388C2×C4—122no—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C4C23×D311yescharacteristic
392C2×C4C23×D314yes—
393C2×C4C23×D312yes—
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—24no—
397C2×C3C22×#4111yescharacteristic
398C2×C3C2×2−1+411yescharacteristic
399C2×C3#58511yescharacteristic
400D3—41no—
401D3—42no—
402C22—22no—
403C22—22no—
404C22—61no—
405C22—62no—
406C22—61no—
407C22—62no—
408C22—62no—
409C22—61no—
410C22C2×#23914yes—
411C22C24×D311yescharacteristic
412C4—22no—
413C4—22no—
414C4—22no—
415C4—24no—
416C4—21no—
417C4—61no—
418C4—62no—
419C4—62no—
420C4—64no—
421C4—61no—
422C4—62no—
423C4—61no—
424C4C2×#23912yes—
425C4#101012yes—
426C3#351511yescharacteristic, complemented, Sylow
427C2—24no—
428C2—121no—
429C2—122no—
430C2C22×#23911yescharacteristic
431C2C2×#101011yescharacteristic
432C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#554C2
hypercenter#554D3
hypocenterC3#3515
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C3×#554C2semidirect
2. #5730C2semidirect
3. #5728C2semidirect
4. #5698C2semidirect
5. #5694C2semidirect
6. #5645C2semidirect
7. #5642C2semidirect
8. #5642C2semidirect
9. #5624C2semidirect
10. #5580C2semidirect
11. #5570C2semidirect
12. #5555C2semidirect
13. #5555C2semidirect
14. #5546C2semidirect
15. C4×#237C2semidirect
16. #5514C2semidirect
17. C2×#890C2semidirect
86. #554D3semidirect
426. C3#3515semidirect

Maximal subgroups

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

subgroupindex
C3×#5542
#57302
#57282
#56982
#56942
#56452
#56422
#56242
#55802
#55702
#55552
#55462
C4×#2372
#55142
C2×#8902
#35153