Finite Group Databasebeta

← #86028

Subgroups — #86028

Order 384.

1022 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#525C211yesmaximal, characteristic
2#5827C211yesmaximal, characteristic, complemented
3#5825C211yesmaximal, characteristic, complemented
4C2×#926C211yesmaximal, characteristic, complemented
5C2×#924C211yesmaximal, characteristic, complemented
6#5650C214yesmaximal, complemented
7#5625C212yesmaximal, complemented
8#5622C212yesmaximal, complemented
9#5603C212yesmaximal, complemented
10#5600C212yesmaximal, complemented
11#5587C214yesmaximal, complemented
12#5543C212yesmaximal, complemented
13#5542C212yesmaximal, complemented
14#5541C212yesmaximal, complemented
15#5541C212yesmaximal, complemented
16#5519C211yesmaximal, characteristic, complemented
17#5516C211yesmaximal, characteristic, complemented
18#3495—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×Dic6C2211yescharacteristic
22C3×#123C2212yes—
23C3×C22⋊Q8C2211yescharacteristic
24C3×C22⋊Q8C2211yescharacteristic
25C3×#121C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×#120C2212yes—
28C3×C4×D4C2212yes—
29C3×C4×D4C2212yes—
30C3×#117C2211yescharacteristic
31C2×#217C2212yes—
32C2×#217C2212yes—
33Q8×Dic3C2212yescomplemented
34#944C2212yescomplemented
35#936C2214yes—
36#936C2212yescomplemented
37#935C2214yes—
38#935C2214yes—
39#935C2214yes—
40D4×Dic3C2214yes—
41D4×Dic3C2214yes—
42D4×Dic3C2212yescomplemented
43#926C2212yes—
44#926C2211yescharacteristic
45#926C2211yescharacteristic
46#926C2211yescharacteristic
47#926C2211yescharacteristic
48#926C2212yes—
49#926C2212yescomplemented
50C2×C4⋊Dic3C2211yescharacteristic
51C2×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊Dic3C2211yescharacteristic
53C2×C4⋊Dic3C2212yescomplemented
54#924C2212yes—
55#924C2212yes—
56#924C2212yes—
57#924C2212yes—
58#924C2212yescomplemented
59#924C2212yescomplemented
60C2×Dic3⋊C4C2212yes—
61C2×C4×Dic3C2212yes—
62#890C2214yescomplemented
63#890C2212yescomplemented
64#889C2214yescomplemented
65#889C2214yescomplemented
66#888C2212yescomplemented
67#887C2214yescomplemented
68#887C2212yescomplemented
69#887C2212yescomplemented
70#879C2214yes—
71#879C2214yes—
72#879C2214yescomplemented
73#879C2214yescomplemented
74#878C2214yes—
75#878C2214yes—
76#878C2214yescomplemented
77#877C2214yes—
78#877C2214yes—
79#877C2214yescomplemented
80#870C2212yescomplemented
81#869C2212yescomplemented
82C4×Dic6C2212yescomplemented
83C4×Dic6C2212yescomplemented
84#556—32no—
85#554—32no—
86#552—34no—
87#552—32no—
88#552—32no—
89#551—34no—
90#544—31no—
91#542—32no—
92#541—32no—
93#536—31no—
94#533—32no—
95#529—32no—
96#525D311yescharacteristic
97C2×C22⋊Q8—31no—
98#520—31no—
99#519—31no—
100C2×#117—31no—
101C24×C3C2311yescharacteristic
102C3×#41—24no—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×D4C2312yes—
105C2×C3×D4C2312yes—
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C22×C3×C4—24no—
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2311yescharacteristic
111C22×C3×C4C2311yescharacteristic
112C22×C3×C4C2311yescharacteristic
113C22×C3×C4C2311yescharacteristic
114C22×C3×C4C2312yes—
115C22×C3×C4C2311yescharacteristic
116C22×Dic3C2312yes—
117C22×Dic3C2312yes—
118C22×Dic3C2314yes—
119C22×Dic3C2312yes—
120C22×Dic3C2312yes—
121C2×Dic6—24no—
122C2×Dic6C2312yes—
123C2×Dic6C2312yes—
124C2×Dic6C2314yes—
125C3×C4⋊C4C2311yescharacteristic
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2312yes—
129C3×C22⋊C4C2312yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C22⋊C4C2312yes—
134C3×C42C2312yes—
135#217C2312yes—
136#217C2314yes—
137#217C2314yes—
138#217C2312yes—
139#217C2314yes—
140#217C2314yes—
141#217C2312yes—
142#217C2312yes—
143#217C2314yes—
144C4⋊Dic3C2311yescharacteristic
145C4⋊Dic3C2311yescharacteristic
146C4⋊Dic3C2311yescharacteristic
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2314yes—
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162C4×Dic3C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C2×#41C2×D311yescharacteristic
170C22×Q8—31no—
171C23×C4C2×D311yescharacteristic
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8—32no—
175#126—34no—
176#126—34no—
177#126—34no—
178#126—34no—
179#125—32no—
180#125—34no—
181#125—34no—
182#125—34no—
183#125—32no—
184#124—32no—
185#124—34no—
186#123—34no—
187#123—34no—
188#123—34no—
189#123C2×D312yes—
190C22⋊Q8—32no—
191C22⋊Q8—32no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—34no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8C2×D311yescharacteristic
200C22⋊Q8C2×D311yescharacteristic
201#121C2×D311yescharacteristic
202#121C2×D311yescharacteristic
203#120C2×D312yes—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8—32no—
207C4×Q8—34no—
208C4×Q8—32no—
209C4×Q8—32no—
210C4×D4—32no—
211C4×D4—34no—
212C4×D4—34no—
213C4×D4C2×D312yes—
214C4×D4C2×D312yes—
215#117—31no—
216#117—31no—
217#117—32no—
218#117—31no—
219#117—32no—
220#117—31no—
221#117—32no—
222#117—34no—
223#117—34no—
224#117—34no—
225#117C2×D311yescharacteristic
226C2×C4⋊C4—31no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—32no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—32no—
233C2×C42—32no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—41no—
237C23×C3C2411yescharacteristic
238C23×C3C2412yes—
239C23×C3C2411yescharacteristic
240C23×C3C2411yescharacteristic
241C3×Q8—22no—
242C3×D4—24no—
243C3×D4—24no—
244C3×D4—22no—
245C3×D4—42no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4#4114yes—
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2412yes—
255C2×C3×C4C2412yes—
256C2×C3×C4C2412yes—
257C2×C3×C4C2411yescharacteristic
258C2×C3×C4C2411yescharacteristic
259C2×C3×C4C2411yescharacteristic
260C2×Dic3—24no—
261C2×Dic3—24no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—44no—
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267C2×Dic3C2412yes—
268C2×Dic3C2414yes—
269C2×Dic3C2412yes—
270C2×Dic3C2414yes—
271Dic6—24no—
272Dic6—24no—
273Dic6—44no—
274C24C22×D311yescharacteristic
275#41—24no—
276C2×Q8—32no—
277C2×Q8—32no—
278C2×Q8—34no—
279C2×Q8—64no—
280C2×Q8C22×D311yescharacteristic
281C2×D4C22×D312yes—
282C2×D4C22×D312yes—
283C2×D4C22×D311yescharacteristic
284C2×D4C22×D312yes—
285C22×C4—24no—
286C22×C4—32no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—32no—
290C22×C4—34no—
291C22×C4C22×D311yescharacteristic
292C22×C4C22×D311yescharacteristic
293C22×C4C22×D311yescharacteristic
294C22×C4C22×D312yes—
295C22×C4C22×D311yescharacteristic
296C22×C4C22×D311yescharacteristic
297C22×C4C22×D311yescharacteristic
298C4⋊C4—31no—
299C4⋊C4—32no—
300C4⋊C4—31no—
301C4⋊C4—32no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—32no—
305C4⋊C4—34no—
306C4⋊C4—32no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—34no—
310C4⋊C4—34no—
311C4⋊C4—34no—
312C4⋊C4—34no—
313C4⋊C4—34no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4C22×D311yescharacteristic
317C4⋊C4C22×D311yescharacteristic
318C4⋊C4C22×D312yes—
319C4⋊C4C22×D312yes—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—32no—
324C22⋊C4—34no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—34no—
328C22⋊C4—34no—
329C22⋊C4C22×D312yes—
330C22⋊C4C22×D312yes—
331C22⋊C4C22×D312yes—
332C22⋊C4C22×D312yes—
333C22⋊C4C22×D312yes—
334C42—32no—
335C42—32no—
336C42—32no—
337C42—32no—
338C42—34no—
339C42—34no—
340C42—34no—
341C42C22×D312yes—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—21no—
352C22×C3—44no—
353C22×C3C2×#4112yes—
354C22×C32−1+412yes—
355C22×C3C2511yescharacteristic
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4C2×#4112yes—
362C3×C4C2×#4112yes—
363Dic3—24no—
364Dic3—22no—
365Dic3—22no—
366Dic3—22no—
367Dic3—24no—
368Dic3—22no—
369C23—22no—
370C23—22no—
371C23—41no—
372C23C23×D311yescharacteristic
373C23C23×D312yes—
374C23C23×D311yescharacteristic
375C23C23×D311yescharacteristic
376Q8—22no—
377Q8—64no—
378Q8—64no—
379Q8—124no—
380D4—24no—
381D4—24no—
382D4—22no—
383D4—42no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—34no—
394C2×C4—32no—
395C2×C4—34no—
396C2×C4—64no—
397C2×C4—64no—
398C2×C4—124no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4#23714yes—
402C2×C4C23×D311yescharacteristic
403C2×C4C23×D312yes—
404C2×C4C23×D312yes—
405C2×C4C23×D312yes—
406C2×C4C23×D311yescharacteristic
407C2×C4C23×D311yescharacteristic
408C2×C4C23×D311yescharacteristic
409C2×C3—22no—
410C2×C3—22no—
411C2×C3—42no—
412C2×C3—41no—
413C2×C3C22×#4111yescharacteristic
414C2×C3C2×2−1+411yescharacteristic
415C2×C3#58511yescharacteristic
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—22no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—44no—
427C22C2×#23712yes—
428C22#101012yes—
429C22C24×D311yescharacteristic
430C4—21no—
431C4—21no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—64no—
436C4—62no—
437C4—62no—
438C4—62no—
439C4—64no—
440C4—62no—
441C4C2×#23712yes—
442C4C2×#23712yes—
443C3#349511yescharacteristic, complemented, Sylow
444C2—22no—
445C2—22no—
446C2—41no—
447C2—42no—
448C2C22×#23711yescharacteristic
449C2C2×#101011yescharacteristic
450C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
Fitting, prime radicalC3×#525C2
hypercenter#525D3
hypocenterC3#3495
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #525

Complements

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

NH ≅ G/Ntype
2. #5827C2semidirect
3. #5825C2semidirect
4. C2×#926C2semidirect
5. C2×#924C2semidirect
6. #5650C2semidirect
7. #5625C2semidirect
8. #5622C2semidirect
9. #5603C2semidirect
10. #5600C2semidirect
11. #5587C2semidirect
12. #5543C2semidirect
13. #5542C2semidirect
14. #5541C2semidirect
15. #5541C2semidirect
16. #5519C2semidirect
17. #5516C2semidirect
33. Q8×Dic3C22semidirect
34. #944C22semidirect
36. #936C22semidirect
42. D4×Dic3C22semidirect
49. #926C22semidirect
53. C2×C4⋊Dic3C22semidirect
58. #924C22semidirect
59. #924C22semidirect
62. #890C22semidirect
63. #890C22semidirect
64. #889C22semidirect
65. #889C22semidirect
66. #888C22semidirect
67. #887C22semidirect
68. #887C22semidirect
69. #887C22semidirect
72. #879C22semidirect
73. #879C22semidirect
76. #878C22semidirect
79. #877C22semidirect
80. #870C22semidirect
81. #869C22semidirect
82. C4×Dic6C22semidirect
83. C4×Dic6C22semidirect
443. C3#3495semidirect

Maximal subgroups

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

subgroupindex
C3×#5252
#58272
#58252
C2×#9262
C2×#9242
#56502
#56252
#56222
#56032
#56002
#55872
#55432
#55422
#55412
#55192
#55162
#34953