Finite Group Databasebeta

← #86735

Subgroups — #86735

Order 384.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1010C211yesmaximal, characteristic, complemented
2C22×D3×Q8C211yesmaximal, characteristic, complemented
3C3×#548C211yesmaximal, characteristic, complemented
4#5831C211yesmaximal, characteristic, complemented
5Q8×#67C211yesmaximal, characteristic, complemented
6#5677C213yesmaximal, complemented
7#5675C213yesmaximal, complemented
8#5632C213yesmaximal, complemented
9#5628C213yesmaximal, complemented
10#5606C213yesmaximal, complemented
11#5598C213yesmaximal, complemented
12Q8×D12C211yesmaximal, characteristic, complemented
13#5572C211yesmaximal, characteristic, complemented
14#5547C213yesmaximal, complemented
15D4×Dic6C213yesmaximal, complemented
16#3486—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C22×C3×Q8C2211yescharacteristic
19#1010—28no—
20C2×D3×Q8—28no—
21C2×D3×Q8C2213yes—
22C2×D3×Q8C2213yes—
23C2×D3×Q8C2211yescharacteristic
24C2×D3×Q8C2211yescharacteristic
25C2×D3×Q8C2211yescharacteristic, complemented
26C2×#237C2213yes—
27C2×#237C2213yescomplemented
28C2×#235C2213yes—
29C22×C4×D3C2213yes—
30C22×Dic6C2213yes—
31C22×Dic6C2213yescomplemented
32C3×#124C2213yescomplemented
33C3×C22⋊Q8C2213yescomplemented
34C3×#121C2213yes—
35C3×C4×Q8C2211yescharacteristic, complemented
36C3×C4×D4C2213yes—
37#946C2213yescomplemented
38Q8×Dic3C2211yescharacteristic, complemented
39#944C2213yescomplemented
40#939C2213yes—
41#938C2213yes—
42#937C2213yes—
43#936C2213yescomplemented
44D4×Dic3C2216yes—
45C4×#67C2213yes—
46C4×#67C2213yes—
47#924C2213yes—
48#924C2213yescomplemented
49#897C2213yes—
50#897C2213yescomplemented
51#896C2213yes—
52#894C2213yes—
53#892C2213yes—
54#888C2216yescomplemented
55#887C2213yescomplemented
56#885C2216yescomplemented
57#884C2213yes—
58#883C2213yes—
59#883C2216yes—
60#882C2211yescharacteristic
61#67⋊C4C2216yes—
62#878C2213yes—
63#878C2216yescomplemented
64#878C2216yescomplemented
65C4×D12C2213yes—
66#872C2213yes—
67#869C2213yescomplemented
68C4×Dic6C2213yescomplemented
69C2×2−1+4—31no—
70C23×Q8—31no—
71D4×Q8—31no—
72D4×Q8—33no—
73D4×Q8—31no—
74D4×Q8—33no—
75#548—31no—
76#548—33no—
77#548—33no—
78#548D311yescharacteristic, complemented
79#536—33no—
80#536—33no—
81#525—33no—
82#525—33no—
83#519—31no—
84C23×D3C2311yescharacteristic
85C3×#41—24no—
86C2×C3×Q8—24no—
87C2×C3×Q8C2313yes—
88C2×C3×Q8C2311yescharacteristic
89C2×C3×D4C2313yes—
90C2×C3×D4C2313yes—
91C22×C3×C4C2313yes—
92C22×C3×C4C2313yes—
93C2×#67C2313yes—
94C2×#67C2311yescharacteristic
95C2×#67C2313yes—
96C2×#67C2313yes—
97C22×Dic3C2313yes—
98C22×Dic3C2316yes—
99D3×Q8—24no—
100D3×Q8—24no—
101D3×Q8—212no—
102D3×Q8—212no—
103D3×Q8D418yescomplemented
104#237—212no—
105#237—212no—
106#235—212no—
107C2×D12C2311yescharacteristic
108C2×D12C2311yescharacteristic
109C2×C4×D3—212no—
110C2×C4×D3C2313yes—
111C2×C4×D3C2313yes—
112C2×C4×D3C2313yes—
113C2×C4×D3C2313yes—
114C2×C4×D3C2313yes—
115C2×Dic6—212no—
116C2×Dic6—212no—
117C2×Dic6C2313yes—
118C2×Dic6C2313yes—
119C2×Dic6C2313yes—
120C2×Dic6C2316yes—
121C2×Dic6C2313yes—
122C3×C4⋊C4C2313yes—
123C3×C4⋊C4C2313yes—
124C3×C22⋊C4C2311yescharacteristic
125C3×C22⋊C4C2313yes—
126C3×C22⋊C4C2316yes—
127C3×C42C2313yes—
128#217C2313yes—
129#217C2313yes—
130#217C2316yes—
131(C2×D3)⋊C4C2311yescharacteristic
132(C2×D3)⋊C4C2313yes—
133(C2×D3)⋊C4C2313yes—
134(C2×D3)⋊C4C2313yes—
135(C2×D3)⋊C4C2311yescharacteristic
136(C2×D3)⋊C4C2313yes—
137C4⋊Dic3C2313yes—
138C4⋊Dic3C2313yes—
139C4⋊Dic3C2316yes—
140Dic3⋊C4C2313yes—
141Dic3⋊C4C2313yes—
142Dic3⋊C4C2313yes—
143Dic3⋊C4C2313yes—
144Dic3⋊C4C2316yes—
145C4×Dic3C2316yes—
146C4×Dic3C2313yes—
1472−1+4—68no—
148C2×#41—33no—
149C2×#41—33no—
150C2×#41—33no—
151C2×#41C2×D311yescharacteristic, complemented
152C22×Q8—31no—
153C22×Q8—33no—
154C22×Q8—33no—
155C22×Q8—31no—
156C22×Q8—31no—
157C22×Q8—33no—
158C22×Q8—33no—
159C22×Q8—68no—
160C22×Q8C2×D311yescharacteristic
161C23×C4—33no—
162C4⋊Q8—33no—
163C4⋊Q8—36no—
164C4⋊Q8—33no—
165#124—33no—
166#124—36no—
167#124C2×D313yescomplemented
168#123—33no—
169#123—36no—
170#123—33no—
171C22⋊Q8—33no—
172C22⋊Q8—33no—
173C22⋊Q8—33no—
174C22⋊Q8—33no—
175C22⋊Q8—33no—
176C22⋊Q8—33no—
177C22⋊Q8—33no—
178C22⋊Q8—36no—
179C22⋊Q8—36no—
180C22⋊Q8C2×D313yescomplemented
181#121—33no—
182#121—33no—
183#121—33no—
184#121C2×D313yes—
185#120—33no—
186#120—31no—
187C4×Q8—31no—
188C4×Q8—33no—
189C4×Q8—33no—
190C4×Q8C2×D311yescharacteristic, complemented
191C4×D4—36no—
192C4×D4—33no—
193C4×D4—33no—
194C4×D4—36no—
195C4×D4—33no—
196C4×D4C2×D313yes—
197#117—33no—
198#117—33no—
199C23×C3C2413yes—
200C23×C3C2411yescharacteristic
201C22×D3—22no—
202C22×D3—22no—
203C22×D3—22no—
204C22×D3C2411yescharacteristic
205C22×D3C2411yescharacteristic
206C22×D3C2411yescharacteristic
207C3×Q8—26no—
208C3×Q8C2×D414yes—
209C3×D4—26no—
210C3×D4—26no—
211C2×C3×C4—26no—
212C2×C3×C4—26no—
213C2×C3×C4C2413yes—
214C2×C3×C4C2411yescharacteristic
215C2×C3×C4C2413yes—
216C2×C3×C4C2411yescharacteristic
217C2×C3×C4C2413yes—
218#67—22no—
219#67—26no—
220#67—26no—
221#67—26no—
222C2×Dic3—26no—
223C2×Dic3—212no—
224C2×Dic3C2411yescharacteristic
225C2×Dic3C2413yes—
226C2×Dic3C2413yes—
227C2×Dic3C2413yes—
228C2×Dic3C2413yes—
229D12—22no—
230D12—22no—
231C4×D3—26no—
232C4×D3—26no—
233C4×D3—26no—
234C4×D3—26no—
235C4×D3C2×D4112yes—
236Dic6—26no—
237Dic6—26no—
238Dic6—26no—
239Dic6—212no—
240Dic6C2×D4112yes—
241C24—31no—
242#41—24no—
243#41—612no—
244#41—612no—
245#41—612no—
246C2×Q8—24no—
247C2×Q8—38no—
248C2×Q8—33no—
249C2×Q8—36no—
250C2×Q8—33no—
251C2×Q8—33no—
252C2×Q8—33no—
253C2×Q8—612no—
254C2×Q8—612no—
255C2×Q8—612no—
256C2×Q8—612no—
257C2×Q8—64no—
258C2×Q8—64no—
259C2×Q8C22×D313yes—
260C2×Q8C22×D311yescharacteristic
261C2×D4—33no—
262C2×D4—31no—
263C2×D4—33no—
264C2×D4—33no—
265C2×D4—31no—
266C2×D4—31no—
267C2×D4C22×D313yes—
268C2×D4C22×D313yes—
269C22×C4—33no—
270C22×C4—33no—
271C22×C4—36no—
272C22×C4—33no—
273C22×C4—33no—
274C22×C4—33no—
275C22×C4—33no—
276C22×C4—612no—
277C22×C4C22×D313yes—
278C22×C4C22×D313yes—
279C4⋊C4—33no—
280C4⋊C4—33no—
281C4⋊C4—33no—
282C4⋊C4—33no—
283C4⋊C4—33no—
284C4⋊C4—36no—
285C4⋊C4—36no—
286C4⋊C4—33no—
287C4⋊C4C22×D313yes—
288C4⋊C4C22×D313yes—
289C22⋊C4—33no—
290C22⋊C4—36no—
291C22⋊C4—33no—
292C22⋊C4—33no—
293C22⋊C4—31no—
294C22⋊C4—31no—
295C22⋊C4—33no—
296C22⋊C4—33no—
297C22⋊C4—33no—
298C22⋊C4C22×D316yes—
299C22⋊C4C22×D311yescharacteristic
300C22⋊C4C22×D313yes—
301C42—36no—
302C42—33no—
303C42C22×D313yes—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—23no—
307C22×C3—23no—
308C22×C3—23no—
309C22×C32−1+412yescomplemented
310C22×C3C2511yescharacteristic
311C2×D3—28no—
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—21no—
319C2×D3C22×D412yes—
320C2×D32−1+412yescomplemented
321C3×C4—21no—
322C3×C4—23no—
323C3×C4—21no—
324C3×C4—23no—
325C3×C4C22×D416yes—
326Dic3—23no—
327Dic3—23no—
328Dic3—21no—
329Dic3—23no—
330Dic3C22×D416yes—
331C23—31no—
332C23—31no—
333C23—31no—
334C23—62no—
335C23—62no—
336C23—62no—
337C23C23×D313yes—
338C23C23×D311yescharacteristic
339Q8—26no—
340Q8—312no—
341Q8—612no—
342Q8—66no—
343Q8—66no—
344Q8—66no—
345Q8D3×D414yes—
346D4—26no—
347D4—26no—
348D4—62no—
349D4—62no—
350D4—66no—
351D4—66no—
352D4—66no—
353D4—62no—
354C2×C4—26no—
355C2×C4—26no—
356C2×C4—33no—
357C2×C4—312no—
358C2×C4—31no—
359C2×C4—33no—
360C2×C4—33no—
361C2×C4—33no—
362C2×C4—66no—
363C2×C4—66no—
364C2×C4—66no—
365C2×C4—612no—
366C2×C4—66no—
367C2×C4—66no—
368C2×C4C23×D313yes—
369C2×C4C23×D311yescharacteristic
370C2×C4C23×D313yes—
371C2×C4C23×D311yescharacteristic
372C2×C4C23×D313yes—
373C2×C3—22no—
374C2×C3—43no—
375C2×C3C23×D411yescharacteristic
376C2×C3C2×2−1+411yescharacteristic
377C2×C3C2×2−1+411yescharacteristic
378D3—22no—
379D3—22no—
380D3—41no—
381C22—21no—
382C22—21no—
383C22—23no—
384C22—23no—
385C22—23no—
386C22—32no—
387C22—32no—
388C22—68no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22—61no—
395C22—61no—
396C22#101012yescomplemented
397C22C24×D311yescharacteristic
398C4—21no—
399C4—23no—
400C4—23no—
401C4—21no—
402C4—36no—
403C4—63no—
404C4—63no—
405C4—61no—
406C4—63no—
407C4C2×D3×D416yes—
408C3#348611yescharacteristic, complemented, Sylow
409C2—22no—
410C2—43no—
411C2—62no—
412C2—62no—
413C2—121no—
414C2C22×D3×D411yescharacteristic
415C2C2×#101011yescharacteristic
416C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#548C2
hypercenter#548D3
hypocenterC3#3486
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#1010C2semidirect
2. C22×D3×Q8C2semidirect
3. C3×#548C2semidirect
4. #5831C2semidirect
5. Q8×#67C2semidirect
6. #5677C2semidirect
7. #5675C2semidirect
8. #5632C2semidirect
9. #5628C2semidirect
10. #5606C2semidirect
11. #5598C2semidirect
12. Q8×D12C2semidirect
13. #5572C2semidirect
14. #5547C2semidirect
15. D4×Dic6C2semidirect
17. C2×C3×#41C22semidirect
25. C2×D3×Q8C22semidirect
27. C2×#237C22semidirect
31. C22×Dic6C22semidirect
32. C3×#124C22semidirect
33. C3×C22⋊Q8C22semidirect
35. C3×C4×Q8C22semidirect
37. #946C22semidirect
38. Q8×Dic3C22semidirect
39. #944C22semidirect
43. #936C22semidirect
48. #924C22semidirect
50. #897C22semidirect
54. #888C22semidirect
55. #887C22semidirect
56. #885C22semidirect
63. #878C22semidirect
64. #878C22semidirect
67. #869C22semidirect
68. C4×Dic6C22semidirect
78. #548D3semidirect
103. D3×Q8D4semidirect
151. C2×#41C2×D3semidirect
167. #124C2×D3semidirect
180. C22⋊Q8C2×D3semidirect
190. C4×Q8C2×D3semidirect
309. C22×C32−1+4semidirect
320. C2×D32−1+4semidirect
396. C22#1010semidirect
408. C3#3486semidirect

Maximal subgroups

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

subgroupindex
C2×#10102
C22×D3×Q82
C3×#5482
#58312
Q8×#672
#56772
#56752
#56322
#56282
#56062
#55982
Q8×D122
#55722
#55472
D4×Dic62
#34863