Finite Group Databasebeta

← #80243

Subgroups — #80243

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5829C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5630C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5191C211yesmaximal, characteristic, complemented
7#5187C211yesmaximal, characteristic, complemented
8#5165C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4812C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3273—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic, complemented
20C22×D12C2211yescharacteristic, complemented
21C2×C3×Q16C2211yescharacteristic, complemented
22C2×C3×SD16C2211yescharacteristic, complemented
23C3×C22⋊Q8C2211yescharacteristic, complemented
24#949—24no—
25#947C2211yescharacteristic, complemented
26#947C2211yescharacteristic, complemented
27#946C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29C2×#215C2211yescharacteristic, complemented
30C2×#215C2211yescharacteristic, complemented
31#939—21no—
32#937—21no—
33D4×Dic3C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#930C2211yescharacteristic
37#929—21no—
38C4×#67C2211yescharacteristic
39C2×(C2×D3)⋊C4—21no—
40C2×#208C2211yescharacteristic, complemented
41#914—24no—
42C2×D24C2211yescharacteristic
43C2×D24C2211yescharacteristic
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45#895—21no—
46#895C2211yescharacteristic
47#894—21no—
48#893C2211yescharacteristic, complemented
49#882—21no—
50D3×C22⋊C4—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#821C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×AGL(1,ℤ/8)—31no—
64#546—31no—
65#546—31no—
66#495—31no—
67#468—31no—
68#468—31no—
69#463—31no—
70#463—31no—
71#452—31no—
72#452D311yescharacteristic, complemented
73#447—31no—
74#447—31no—
75C4×D8—31no—
76#436—31no—
77C23×D3—21no—
78C23×D3—21no—
79C3×#41—22no—
80C3×#41—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—21no—
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67D411yescharacteristic
90C22×Dic3—21no—
91#239—22no—
92#239—41no—
93D3×D4—22no—
94D3×D4—41no—
95C2×D12—22no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—41no—
100C2×D12C2311yescharacteristic
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C3×Q16—41no—
107C3×SD16—22no—
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4—21no—
111C3×C4⋊C4C2311yescharacteristic
112C3×C22⋊C4—21no—
113#217—21no—
114#217D411yescharacteristic
115#215—22no—
116#215—22no—
117#215—22no—
118#213—22no—
119#213D414yescomplemented
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4D411yescharacteristic
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3C2311yescharacteristic
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137D24—41no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic
141C22×D4—31no—
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16C2×D311yescharacteristic, complemented
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16C2×D311yescharacteristic, complemented
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C2×M16—31no—
156#124—31no—
157#124—31no—
158#123—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8C2×D311yescharacteristic, complemented
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#120—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4—31no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic, complemented
183C4×C8—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3C2×D411yescharacteristic
196C3×Q8—21no—
197C3×Q8—21no—
198C3×Q8—21no—
199C3×Q8—21no—
200C3×D4—21no—
201C3×D4—42no—
202C3×D4C2×D412yes—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—42no—
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2411yescharacteristic
211#67—41no—
212#67—42no—
213#67—81no—
214C2×Dic3—21no—
215C2×Dic3—81no—
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218C2×Dic3C2×D411yescharacteristic
219D12—21no—
220D12—21no—
221D12—21no—
222D12—21no—
223D12—21no—
224D12—42no—
225D12—42no—
226D12—81no—
227D12C2×D412yes—
228C4×D3—21no—
229C4×D3—21no—
230C4×D3—42no—
231C4×D3—81no—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—61no—
237C24—61no—
238#41—22no—
239#41—41no—
240#41—62no—
241#41—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—21no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—62no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—62no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—121no—
257C2×D4—121no—
258C2×D4C22×D311yescharacteristic
259C22×C4—21no—
260C22×C4—31no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4C22×D311yescharacteristic
265Q16—41no—
266SD16—22no—
267SD16—62no—
268SD16—62no—
269SD16—62no—
270D8—34no—
271D8—62no—
272D8—62no—
273D8—121no—
274M16—62no—
275M16—62no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8C22×D311yescharacteristic
279C2×C8C22×D311yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—31no—
282C4⋊C4—31no—
283C4⋊C4—61no—
284C4⋊C4C22×D311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—31no—
287C22⋊C4—31no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C42—31no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3C22×D411yescharacteristic
305C2×D3—21no—
306C2×D3—22no—
307C2×D3—21no—
308C2×D3—21no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—84no—
318C2×D3—84no—
319C3×C4—21no—
320C3×C4—41no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4C22×D411yescharacteristic
324C3×C4H(5,𝔽2)11yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—21no—
328Dic3—41no—
329C23—21no—
330C23—31no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—122no—
335C23—122no—
336C23—122no—
337C23—122no—
338C23—241no—
339C23—241no—
340C23D3×D411yescharacteristic
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345D4—21no—
346D4—32no—
347D4—42no—
348D4—61no—
349D4—61no—
350D4—61no—
351D4—61no—
352D4—61no—
353D4—122no—
354D4—122no—
355D4—121no—
356D4—122no—
357D4—241no—
358D4—241no—
359D4D3×D412yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—42no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4D3×D411yescharacteristic
376C2×C4C23×D311yescharacteristic
377C8—21no—
378C8—21no—
379C8—32no—
380C8—61no—
381C2×C3—41no—
382C2×C3—42no—
383C2×C3D4211yescharacteristic
384C2×C3#57511yescharacteristic
385C2×C3#57611yescharacteristic
386D3—41no—
387D3—42no—
388D3—81no—
389D3—81no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—61no—
397C22—61no—
398C22—61no—
399C22—62no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—244no—
409C22—244no—
410C22C2×D3×D411yescharacteristic
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—61no—
416C4—61no—
417C4—61no—
418C4—121no—
419C4C2×D3×D411yescharacteristic
420C4#100411yescharacteristic
421C3#327311yescharacteristic, complemented, Sylow
422C2—41no—
423C2—42no—
424C2—121no—
425C2—122no—
426C2—241no—
427C2—241no—
428C2#559011yescharacteristic
429C2#576911yescharacteristic
430C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3273
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #5829C2semidirect
2. C2×#949C2semidirect
3. C2×#914C2semidirect
4. #5630C2semidirect
5. C3×#452C2semidirect
6. #5191C2semidirect
7. #5187C2semidirect
8. #5165C2semidirect
9. #5049C2semidirect
10. #4819C2semidirect
11. #4812C2semidirect
12. #4773C2semidirect
13. #4756C2semidirect
14. #4732C2semidirect
15. #4729C2semidirect
19. C2×D3×D4C22semidirect
20. C22×D12C22semidirect
21. C2×C3×Q16C22semidirect
22. C2×C3×SD16C22semidirect
23. C3×C22⋊Q8C22semidirect
25. #947C22semidirect
26. #947C22semidirect
27. #946C22semidirect
28. C2×#215C22semidirect
29. C2×#215C22semidirect
30. C2×#215C22semidirect
33. D4×Dic3C22semidirect
40. C2×#208C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
48. #893C22semidirect
51. C3×#103C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
54. #835C22semidirect
55. #821C22semidirect
56. #821C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #809C22semidirect
61. #807C22semidirect
72. #452D3semidirect
119. #213D4semidirect
145. C2×Q16C2×D3semidirect
149. C2×SD16C2×D3semidirect
161. C22⋊Q8C2×D3semidirect
176. #103C2×D3semidirect
180. #102C2×D3semidirect
182. C22⋊C8C2×D3semidirect
421. C3#3273semidirect

Maximal subgroups

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

subgroupindex
#58292
C2×#9492
C2×#9142
#56302
C3×#4522
#51912
#51872
#51652
#50492
#48192
#48122
#47732
#47562
#47322
#47292
#32733