Finite Group Databasebeta

← #86220

Subgroups — #86220

Order 384.

1112 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5832C212yesmaximal, complemented
3Dic3×#41C212yesmaximal, complemented
4#5732C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5718C211yesmaximal, characteristic, complemented
7#5713C211yesmaximal, characteristic, complemented
8#5681C212yesmaximal, complemented
9#5671C212yesmaximal, complemented
10#5618C212yesmaximal, complemented
11#5601C214yesmaximal, complemented
12#5598C212yesmaximal, complemented
13#5557C212yesmaximal, complemented
14#5538C214yesmaximal, complemented
15C4×#237C212yesmaximal, complemented
16C4×#235C211yesmaximal, characteristic, complemented
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18#3480—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#237C2212yes—
21C2×#235C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C3×C4⋊Q8C2211yescharacteristic
24C3×#127C2211yescharacteristic
25C3×#124C2212yes—
26C3×#121C2212yes—
27C3×#121C2212yes—
28C3×C4×D4C2212yes—
29C3×C4×D4C2212yes—
30C2×C3×C42C2211yescharacteristic
31#947C2212yes—
32Q8×Dic3C2212yes—
33Q8×Dic3C2212yes—
34#944C2212yes—
35#940C2212yes—
36#939C2214yes—
37#939C2214yes—
38#939C2214yes—
39#938C2212yes—
40#936C2212yes—
41#936C2212yes—
42#935C2214yes—
43D4×Dic3C2212yes—
44D4×Dic3C2212yes—
45D4×Dic3C2212yes—
46D4×Dic3C2214yes—
47D4×Dic3C2214yes—
48C4×#67C2212yes—
49C4×#67C2212yes—
50C4×#67C2214yes—
51C4×#67C2214yes—
52#926C2211yescharacteristic
53#926C2214yes—
54C2×C4⋊Dic3C2211yescharacteristic
55C2×Dic3⋊C4C2212yes—
56C2×C4×Dic3C2212yes—
57C2×C4×Dic3C2214yes—
58#898C2214yes—
59#897C2212yes—
60#892C2212yes—
61#890C2212yes—
62#889C2214yes—
63#887C2212yes—
64#886C2214yes—
65#885C2214yes—
66#883C2214yes—
67#67⋊C4C2214yes—
68#879C2214yes—
69#879C2214yes—
70#878C2214yes—
71#877C2214yes—
72C4×D12C2211yescharacteristic
73C4×D12C2211yescharacteristic
74#872C2212yes—
75C42×D3C2212yes—
76C4×Dic6C2211yescharacteristic
77C4×Dic6C2212yes—
78C4×Dic6C2211yescharacteristic
79C4×Dic6C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#559—31no—
85#559—32no—
86#557—31no—
87#556—31no—
88#556—32no—
89#553—34no—
90#553—32no—
91#550—31no—
92#548—32no—
93#532—32no—
94#532D311yescharacteristic, complemented
95#529—34no—
96#529—32no—
97C4×#41—32no—
98C4×#41—32no—
99C4×#41—31no—
100C2×C4×Q8—31no—
101C3×#41—28no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2312yes—
104C2×C3×D4C2314yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2312yes—
107C22×C3×C4C2314yes—
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2311yescharacteristic
111C2×#67C2314yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C22×Dic3C2314yes—
115C22×Dic3C2312yes—
116C22×Dic3C2314yes—
117C22×Dic3C2312yes—
118#237—28no—
119#235—24no—
120C2×D12C2311yescharacteristic
121C2×C4×D3C2312yes—
122C2×C4×D3C2312yes—
123C2×C4×D3C2312yes—
124C2×Dic6—24no—
125C2×Dic6C2312yes—
126C2×Dic6C2311yescharacteristic
127C2×Dic6C2311yescharacteristic
128C2×Dic6C2311yescharacteristic
129C3×C4⋊C4C2312yes—
130C3×C4⋊C4C2312yes—
131C3×C22⋊C4C2314yes—
132C3×C22⋊C4C2314yes—
133C3×C42C2311yescharacteristic
134C3×C42C2311yescharacteristic
135C3×C42C2311yescharacteristic
136C3×C42C2311yescharacteristic
137#217C2314yes—
138#217C2314yes—
139#217C2314yes—
140#217C2314yes—
141#217C2312yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2314yes—
145(C2×D3)⋊C4C2314yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2311yescharacteristic
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161C4×Dic3C2312yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2314yes—
169C4×Dic3C2314yes—
170C2×#41—32no—
171C2×#41—31no—
172C2×#41C2×D312yes—
173C22×Q8—31no—
174C4⋊Q8—32no—
175C4⋊Q8C2×D311yescharacteristic
176#127—32no—
177#127C2×D311yescharacteristic
178#126—34no—
179#126—34no—
180#126—34no—
181#125—34no—
182#125—32no—
183#124—32no—
184#124—32no—
185#124—32no—
186#124—34no—
187#124C2×D312yes—
188#123—34no—
189#123—34no—
190#123—34no—
191C22⋊Q8—32no—
192C22⋊Q8—34no—
193#121—32no—
194#121—34no—
195#121—34no—
196#121—34no—
197#121C2×D312yes—
198#121C2×D312yes—
199C4×Q8—32no—
200C4×Q8—32no—
201C4×Q8—32no—
202C4×Q8—31no—
203C4×Q8—32no—
204C4×Q8—31no—
205C4×Q8—31no—
206C4×Q8—31no—
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8—31no—
210C4×D4—34no—
211C4×D4—32no—
212C4×D4—34no—
213C4×D4—32no—
214C4×D4—32no—
215C4×D4—34no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—31no—
221C4×D4—31no—
222C4×D4C2×D312yes—
223C4×D4C2×D312yes—
224#117—31no—
225#117—34no—
226#117—32no—
227#117—34no—
228#117—32no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C42—32no—
232C2×C42—34no—
233C2×C42—32no—
234C2×C42C2×D311yescharacteristic
235C23×C3C2414yes—
236C23×C3C2411yescharacteristic
237C22×D3C2412yes—
238C3×Q8—24no—
239C3×D4—28no—
240C3×D4—24no—
241C3×D4—24no—
242C3×D4—42no—
243C2×C3×C4—22no—
244C2×C3×C4—28no—
245C2×C3×C4—41no—
246C2×C3×C4#4114yescomplemented
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2414yes—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254#67—24no—
255#67—28no—
256#67—44no—
257C2×Dic3—24no—
258C2×Dic3—28no—
259C2×Dic3—44no—
260C2×Dic3—42no—
261C2×Dic3C2412yes—
262C2×Dic3C2412yes—
263C2×Dic3C2412yes—
264C2×Dic3C2412yes—
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267C2×Dic3C2412yes—
268D12—22no—
269C4×D3—24no—
270C4×D3—24no—
271C4×D3—42no—
272Dic6—22no—
273Dic6—22no—
274Dic6—24no—
275Dic6#4114yescomplemented
276#41—28no—
277#41—68no—
278#41—64no—
279C2×Q8—31no—
280C2×Q8—32no—
281C2×Q8—31no—
282C2×Q8—31no—
283C2×Q8—64no—
284C2×Q8C22×D312yes—
285C2×D4—32no—
286C2×D4—34no—
287C2×D4—34no—
288C2×D4—31no—
289C2×D4C22×D314yes—
290C2×D4C22×D312yes—
291C2×D4C22×D312yes—
292C2×D4C22×D312yes—
293C22×C4—32no—
294C22×C4—34no—
295C22×C4—32no—
296C22×C4—34no—
297C22×C4—32no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D314yes—
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—32no—
307C4⋊C4—31no—
308C4⋊C4—32no—
309C4⋊C4—32no—
310C4⋊C4—32no—
311C4⋊C4—31no—
312C4⋊C4—31no—
313C4⋊C4—34no—
314C4⋊C4—32no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—34no—
318C4⋊C4—34no—
319C4⋊C4C22×D312yes—
320C4⋊C4C22×D312yes—
321C22⋊C4—32no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—32no—
327C22⋊C4—34no—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4C22×D314yes—
331C22⋊C4C22×D314yes—
332C42—34no—
333C42—32no—
334C42—34no—
335C42—32no—
336C42—32no—
337C42—32no—
338C42—32no—
339C42—32no—
340C42—32no—
341C42C22×D311yescharacteristic
342C42C22×D311yescharacteristic
343C42C22×D311yescharacteristic
344C42C22×D311yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—24no—
350C22×C3C2×#4112yes—
351C22×C3C2511yescharacteristic
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—22no—
355C3×C4—21no—
356C3×C4—24no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4C2×#4112yes—
360C3×C4C2×#4112yes—
361C3×C4C2×#4112yes—
362Dic3—22no—
363Dic3—22no—
364Dic3—22no—
365Dic3—22no—
366Dic3—22no—
367Dic3—22no—
368Dic3C2×#4114yes—
369C23—32no—
370C23C23×D314yes—
371C23C23×D311yescharacteristic
372Q8—24no—
373Q8—34no—
374Q8—64no—
375Q8—62no—
376Q8—62no—
377D4—28no—
378D4—24no—
379D4—24no—
380D4—42no—
381D4—68no—
382D4—64no—
383D4—62no—
384D4—124no—
385C2×C4—28no—
386C2×C4—22no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—32no—
394C2×C4—41no—
395C2×C4—64no—
396C2×C4—68no—
397C2×C4—64no—
398C2×C4—64no—
399C2×C4—122no—
400C2×C4—124no—
401C2×C4—122no—
402C2×C4#23714yescomplemented
403C2×C4C23×D311yescharacteristic
404C2×C4C23×D314yes—
405C2×C4C23×D311yescharacteristic
406C2×C4C23×D311yescharacteristic
407C2×C4C23×D311yescharacteristic
408C2×C4C23×D311yescharacteristic
409C2×C4C23×D311yescharacteristic
410C2×C3—22no—
411C2×C3—44no—
412C2×C3C22×#4111yescharacteristic
413C2×C3C22×#4111yescharacteristic
414C2×C3#58511yescharacteristic
415D3—42no—
416C22—21no—
417C22—21no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—62no—
422C22—62no—
423C22—62no—
424C22C2×#23712yes—
425C22C24×D311yescharacteristic
426C4—24no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—34no—
431C4—62no—
432C4—62no—
433C4—62no—
434C4—62no—
435C4—62no—
436C4—62no—
437C4C2×#23712yes—
438C4C2×#23712yes—
439C4D3×#4112yes—
440C3#348011yescharacteristic, complemented, Sylow
441C2—22no—
442C2—44no—
443C2—122no—
444C2C22×#23711yescharacteristic
445C2C2×D3×#4111yescharacteristic
446C2#596411yescharacteristic
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3480
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5832C2semidirect
3. Dic3×#41C2semidirect
4. #5732C2semidirect
5. #5721C2semidirect
6. #5718C2semidirect
7. #5713C2semidirect
8. #5681C2semidirect
9. #5671C2semidirect
10. #5618C2semidirect
11. #5601C2semidirect
12. #5598C2semidirect
13. #5557C2semidirect
14. #5538C2semidirect
15. C4×#237C2semidirect
16. C4×#235C2semidirect
17. C2×C4×Dic6C2semidirect
94. #532D3semidirect
246. C2×C3×C4#41semidirect
275. Dic6#41semidirect
402. C2×C4#237semidirect
440. C3#3480semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58322
Dic3×#412
#57322
#57212
#57182
#57132
#56812
#56712
#56182
#56012
#55982
#55572
#55382
C4×#2372
C4×#2352
C2×C4×Dic62
#34803