Finite Group Databasebeta

← #86253

Subgroups — #86253

Order 384.

1050 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#532C211yesmaximal, characteristic, complemented
2#5833C212yesmaximal, complemented
3#5830C211yesmaximal, characteristic, complemented
4#5825C211yesmaximal, characteristic, complemented
5#5735C212yesmaximal, complemented
6#5720C212yesmaximal, complemented
7#5679C212yesmaximal, complemented
8#5670C212yesmaximal, complemented
9#5615C214yesmaximal, complemented
10#5613C214yesmaximal, complemented
11#5565C212yesmaximal, complemented
12#5552C212yesmaximal, complemented
13#5550C212yesmaximal, complemented
14#5541C212yesmaximal, complemented
15#5479C212yesmaximal, complemented
16#3551—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic, complemented
19C2×#235C2212yes—
20C22×D12C2211yescharacteristic
21C22×Dic6C2211yescharacteristic, complemented
22C3×C4⋊Q8C2211yescharacteristic, complemented
23C3×#127C2211yescharacteristic
24C3×#124C2211yescharacteristic
25C3×#124C2211yescharacteristic, complemented
26C3×#121C2212yes—
27C3×#121C2212yescomplemented
28C3×C4×D4C2212yes—
29C3×C4×D4C2212yescomplemented
30C2×C3×C42C2211yescharacteristic, complemented
31#947C2212yes—
32#946C2214yescomplemented
33#944C2212yescomplemented
34#940C2212yes—
35#939C2214yescomplemented
36#939C2214yescomplemented
37#939C2214yescomplemented
38#939C2214yescomplemented
39#938C2214yescomplemented
40#937C2212yes—
41#937C2214yes—
42#937C2212yes—
43#936C2212yes—
44#935C2212yes—
45#935C2214yes—
46#935C2212yescomplemented
47#930C2212yes—
48#930C2212yes—
49#930C2212yes—
50#929C2214yescomplemented
51C4×#67C2214yescomplemented
52#926C2211yescharacteristic
53#926C2211yescharacteristic, complemented
54C2×C4⋊Dic3C2212yescomplemented
55#924C2212yes—
56#924C2212yescomplemented
57#924C2212yescomplemented
58#898C2214yescomplemented
59#896C2214yescomplemented
60#894C2214yes—
61#889C2214yescomplemented
62#885C2214yescomplemented
63#884C2214yes—
64#883C2214yescomplemented
65#67⋊C4C2214yescomplemented
66#67⋊C4C2214yescomplemented
67D3×C22⋊C4C2214yes—
68#879C2214yes—
69#877C2214yescomplemented
70#876C2214yescomplemented
71#875C2212yes—
72C4×D12C2212yes—
73#872C2214yescomplemented
74#870C2212yescomplemented
75C4×Dic6C2212yescomplemented
76#563—32no—
77#560—32no—
78#555—32no—
79#553—32no—
80#552—32no—
81#551—32no—
82#548—32no—
83#546—32no—
84#540—34no—
85#538—34no—
86#537—32no—
87#536—31no—
88#535—31no—
89#532D311yescharacteristic, complemented
90#529—32no—
91C23×D3C2312yes—
92C3×#41—42no—
93C3×#41—42no—
94C2×C3×Q8C2311yescharacteristic
95C2×C3×Q8C2311yescharacteristic
96C2×C3×D4C2312yes—
97C2×C3×D4C2311yescharacteristic
98C2×C3×D4C2311yescharacteristic
99C2×C3×D4C2311yescharacteristic
100C2×C3×D4C2312yes—
101C2×C3×D4C2312yes—
102C2×C3×D4C2311yescharacteristic
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2312yes—
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2312yes—
108C2×#67C2314yes—
109C2×#67C2314yes—
110C2×#67C2314yes—
111C2×#67C2314yes—
112C22×Dic3C2314yes—
113C22×Dic3C2312yes—
114#235—44no—
115C2×D12—42no—
116C2×D12C2312yes—
117C2×D12C2312yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×Dic6—42no—
121C2×Dic6C2312yes—
122C2×Dic6C2312yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C22⋊C4C2312yes—
126C3×C22⋊C4C2312yes—
127C3×C22⋊C4C2312yes—
128C3×C22⋊C4C2312yes—
129C3×C42C2312yes—
130C3×C42C2311yescharacteristic
131C3×C42C2311yescharacteristic
132#217C2314yes—
133#217C2312yes—
134#217C2314yes—
135#217C2312yes—
136#217C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2314yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2314yes—
146Dic3⋊C4C2314yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152C4×Dic3C2314yes—
153C4×Dic3C2312yes—
154C4×Dic3C2312yes—
155C2×#41—32no—
156C2×#41C2×D311yescharacteristic
157C2×#41C2×D311yescharacteristic, complemented
158C22×Q8—31no—
159C22×D4—31no—
160C4⋊Q8—32no—
161C4⋊Q8C2×D311yescharacteristic, complemented
162#127—32no—
163#127C2×D311yescharacteristic
164#126—34no—
165#126—34no—
166#126—34no—
167#125—34no—
168#125—32no—
169#124—32no—
170#124—32no—
171#124—34no—
172#124—32no—
173#124C2×D311yescharacteristic
174#124C2×D311yescharacteristic, complemented
175#123—32no—
176#123—32no—
177#123—34no—
178#123—34no—
179#123—34no—
180#123—34no—
181C22⋊Q8—34no—
182C22⋊Q8—32no—
183C22⋊Q8—32no—
184C22⋊Q8—32no—
185C22⋊Q8—34no—
186#121—34no—
187#121—34no—
188#121—34no—
189#121—34no—
190#121—34no—
191#121—32no—
192#121—32no—
193#121—32no—
194#121—34no—
195#121C2×D312yes—
196#121C2×D312yescomplemented
197#120—32no—
198#120—32no—
199#120—34no—
200C4×Q8—32no—
201C4×D4—34no—
202C4×D4—34no—
203C4×D4—34no—
204C4×D4—32no—
205C4×D4C2×D312yes—
206C4×D4C2×D312yescomplemented
207#117—31no—
208#117—31no—
209#117—34no—
210#117—34no—
211C2×C4⋊C4—32no—
212C2×C22⋊C4—34no—
213C2×C42C2×D311yescharacteristic, complemented
214C23×C3C2411yescharacteristic
215C23×C3C2411yescharacteristic
216C23×C3C2411yescharacteristic
217C23×C3C2412yes—
218C22×D3—42no—
219C22×D3—44no—
220C22×D3C2414yes—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×D4—42no—
224C3×D4—42no—
225C3×D4—41no—
226C3×D4—42no—
227C3×D4—41no—
228C3×D4—41no—
229C3×D4—41no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—42no—
235C2×C3×C4C2411yescharacteristic
236C2×C3×C4C2412yes—
237C2×C3×C4C2411yescharacteristic
238C2×C3×C4C2412yes—
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2412yes—
241C2×C3×C4C2411yescharacteristic
242#67—44no—
243#67—44no—
244#67—44no—
245#67—44no—
246C2×Dic3—42no—
247C2×Dic3—44no—
248C2×Dic3C2414yes—
249C2×Dic3C2414yes—
250C2×Dic3C2414yes—
251D12—24no—
252D12—42no—
253C4×D3—44no—
254C4×D3—44no—
255Dic6—24no—
256Dic6—42no—
257C24—32no—
258#41—42no—
259#41—42no—
260#41—124no—
261C2×Q8—32no—
262C2×Q8—32no—
263C2×Q8—122no—
264C2×Q8C22×D311yescharacteristic
265C2×Q8C22×D311yescharacteristic
266C2×D4—34no—
267C2×D4—34no—
268C2×D4—34no—
269C2×D4—34no—
270C2×D4—32no—
271C2×D4—32no—
272C2×D4—122no—
273C2×D4C22×D312yes—
274C2×D4C22×D312yes—
275C2×D4C22×D311yescharacteristic
276C2×D4C22×D311yescharacteristic
277C2×D4C22×D312yes—
278C2×D4C22×D311yescharacteristic
279C2×D4C22×D311yescharacteristic
280C22×C4—34no—
281C22×C4—32no—
282C22×C4—34no—
283C22×C4—34no—
284C22×C4C22×D311yescharacteristic
285C22×C4C22×D312yes—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D312yes—
289C4⋊C4—32no—
290C4⋊C4—32no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—34no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4—34no—
298C4⋊C4C22×D312yes—
299C4⋊C4C22×D312yes—
300C22⋊C4—34no—
301C22⋊C4—32no—
302C22⋊C4—34no—
303C22⋊C4—32no—
304C22⋊C4—34no—
305C22⋊C4—34no—
306C22⋊C4—34no—
307C22⋊C4—34no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—34no—
311C22⋊C4C22×D312yes—
312C22⋊C4C22×D312yes—
313C22⋊C4C22×D312yes—
314C22⋊C4C22×D312yes—
315C42—32no—
316C42—34no—
317C42—32no—
318C42C22×D311yescharacteristic
319C42C22×D312yes—
320C42C22×D311yescharacteristic
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3C2511yescharacteristic
331C2×D3—24no—
332C2×D3—24no—
333C2×D3—24no—
334C2×D3—84no—
335C3×C4—21no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4H(5,𝔽2)12yes—
342Dic3—24no—
343Dic3—24no—
344Dic3—24no—
345C23—34no—
346C23—122no—
347C23—124no—
348C23C23×D311yescharacteristic
349C23C23×D311yescharacteristic
350C23C23×D311yescharacteristic
351C23C23×D312yes—
352Q8—41no—
353Q8—41no—
354Q8—64no—
355Q8—122no—
356D4—42no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—42no—
362D4—42no—
363D4—64no—
364D4—122no—
365D4—124no—
366D4—124no—
367D4—124no—
368D4—124no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—34no—
374C2×C4—34no—
375C2×C4—34no—
376C2×C4—42no—
377C2×C4—124no—
378C2×C4—124no—
379C2×C4—122no—
380C2×C4—124no—
381C2×C4C23×D312yes—
382C2×C4C23×D311yescharacteristic
383C2×C4C23×D311yescharacteristic
384C2×C4C23×D312yes—
385C2×C4C23×D311yescharacteristic
386C2×C4C23×D312yes—
387C2×C4C23×D311yescharacteristic
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—42no—
391C2×C3—41no—
392C2×C3C2×H(5,𝔽2)11yescharacteristic
393C2×C3#58512yes—
394D3—44no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—64no—
405C22—64no—
406C22—64no—
407C22—244no—
408C22C24×D311yescharacteristic
409C4—22no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—22no—
414C4—22no—
415C4—64no—
416C4—64no—
417C4—64no—
418C4#100412yes—
419C3#355111yescharacteristic, complemented, Sylow
420C2—41no—
421C2—41no—
422C2—41no—
423C2—42no—
424C2—124no—
425C2C2×#100411yescharacteristic
426C2#596412yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3551
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5833C2semidirect
3. #5830C2semidirect
4. #5825C2semidirect
5. #5735C2semidirect
6. #5720C2semidirect
7. #5679C2semidirect
8. #5670C2semidirect
9. #5615C2semidirect
10. #5613C2semidirect
11. #5565C2semidirect
12. #5552C2semidirect
13. #5550C2semidirect
14. #5541C2semidirect
15. #5479C2semidirect
18. C2×C3×#41C22semidirect
21. C22×Dic6C22semidirect
22. C3×C4⋊Q8C22semidirect
25. C3×#124C22semidirect
27. C3×#121C22semidirect
29. C3×C4×D4C22semidirect
30. C2×C3×C42C22semidirect
32. #946C22semidirect
33. #944C22semidirect
35. #939C22semidirect
36. #939C22semidirect
37. #939C22semidirect
38. #939C22semidirect
39. #938C22semidirect
46. #935C22semidirect
50. #929C22semidirect
51. C4×#67C22semidirect
53. #926C22semidirect
54. C2×C4⋊Dic3C22semidirect
56. #924C22semidirect
57. #924C22semidirect
58. #898C22semidirect
59. #896C22semidirect
61. #889C22semidirect
62. #885C22semidirect
64. #883C22semidirect
65. #67⋊C4C22semidirect
66. #67⋊C4C22semidirect
69. #877C22semidirect
70. #876C22semidirect
73. #872C22semidirect
74. #870C22semidirect
75. C4×Dic6C22semidirect
89. #532D3semidirect
157. C2×#41C2×D3semidirect
161. C4⋊Q8C2×D3semidirect
174. #124C2×D3semidirect
196. #121C2×D3semidirect
206. C4×D4C2×D3semidirect
213. C2×C42C2×D3semidirect
419. C3#3551semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#58332
#58302
#58252
#57352
#57202
#56792
#56702
#56152
#56132
#55652
#55522
#55502
#55412
#54792
#35513