Finite Group Databasebeta

← #80404

Subgroups — #80404

Order 384.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#914C211yesmaximal, characteristic, complemented
2C2×D3×SD16C211yesmaximal, characteristic, complemented
3#5717C211yesmaximal, characteristic, complemented
4#5576C211yesmaximal, characteristic, complemented
5C3×#460C211yesmaximal, characteristic, complemented
6#5172C211yesmaximal, characteristic, complemented
7#5170C211yesmaximal, characteristic, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5026C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4775C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3265—31nomaximal, Sylow
17C2×#239C2211yescharacteristic, complemented
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×D3×D4—21no—
20C2×D3×D4C2211yescharacteristic
21C2×D3×D4C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic, complemented
23C2×C3×SD16C2211yescharacteristic, complemented
24C3×#127C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27#938—21no—
28#938C2211yescharacteristic
29#938C2211yescharacteristic
30#937—21no—
31#937—21no—
32C2×#214C2211yescharacteristic, complemented
33C2×#214C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic, complemented
35C2×#213C2211yescharacteristic, complemented
36#914—24no—
37D3×SD16—24no—
38C2×D24C2211yescharacteristic
39C2×#204C2211yescharacteristic
40C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
41C2×D3×C8C2211yescharacteristic, complemented
42#897—21no—
43#897C2211yescharacteristic, complemented
44#895—21no—
45#895C2211yescharacteristic, complemented
46#875—21no—
47#875C2211yescharacteristic, complemented
48C4×D12—21no—
49C4×D12C2211yescharacteristic
50C4×D12C2211yescharacteristic
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#807C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×SD16—31no—
64#548—31no—
65D42—31no—
66#496—31no—
67#468—31no—
68#466—31no—
69#463—31no—
70#460D311yescharacteristic, complemented
71#450—31no—
72#449—31no—
73#448—31no—
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88#239—22no—
89#239—41no—
90D3×Q8—22no—
91D3×Q8—41no—
92D3×D4—22no—
93D3×D4—22no—
94D3×D4—41no—
95D3×D4—41no—
96D3×D4—44no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12D411yescharacteristic
100C2×D12C2311yescharacteristic
101C2×D12C2311yescharacteristic
102C2×C4×D3—21no—
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C2×C4×D3C2311yescharacteristic
107C2×Dic6—21no—
108C2×Dic6C2311yescharacteristic
109C3×SD16—22no—
110C3×SD16—22no—
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4—21no—
114C3×C4⋊C4C2311yescharacteristic
115C3×C42—21no—
116C3×C42C2311yescharacteristic
117#217—21no—
118#217—21no—
119#215D414yescomplemented
120#214—22no—
121#214—41no—
122#213—22no—
123#213—41no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4D411yescharacteristic
130(C2×D3)⋊C4D411yescharacteristic
131C4⋊Dic3—21no—
132C4⋊Dic3D411yescharacteristic
133C4⋊Dic3C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137#204—22no—
138ΓL(1,𝔽25)—22no—
139D3×C8—22no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—64no—
151C2×SD16C2×D311yescharacteristic, complemented
152C2×SD16C2×D311yescharacteristic, complemented
153C2×D8—31no—
154C2×D8—31no—
155C2×D8—31no—
156C2×M16—31no—
157C22×C8—31no—
158#127C2×D311yescharacteristic
159#124—31no—
160#124—61no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—31no—
166#121—61no—
167#121—61no—
168#120—61no—
169#120—61no—
170C4×Q8C2×D311yescharacteristic, complemented
171C4×D4—31no—
172C4×D4—31no—
173C4×D4—61no—
174#107—31no—
175C4⋊C8C2×D311yescharacteristic, complemented
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97—31no—
185C23×C3—21no—
186C23×C3—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
196C22×D3C2×D411yescharacteristic
197C3×Q8—21no—
198C3×Q8C2×D412yescomplemented
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—21no—
202C3×D4—21no—
203C3×D4—42no—
204C3×D4—81no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4C2×D411yescharacteristic
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2411yescharacteristic
210#67—42no—
211#67—42no—
212#67—42no—
213#67—42no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218D12—22no—
219D12—21no—
220D12—21no—
221D12—21no—
222D12—42no—
223D12—42no—
224D12C2×D412yes—
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—21no—
228C4×D3—21no—
229C4×D3—42no—
230C4×D3—42no—
231C4×D3—42no—
232Dic6—21no—
233Dic6—21no—
234Dic6—42no—
235C3×C8—21no—
236C3×C8—21no—
237C3⋊C8—21no—
238C3⋊C8C2×D412yescomplemented
239C24—61no—
240C24—61no—
241#41—62no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—121no—
247C2×Q8C22×D311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—31no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—124no—
264C2×D4C22×D311yescharacteristic
265C2×D4C22×D311yescharacteristic
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271SD16—22no—
272SD16—22no—
273SD16—34no—
274SD16—62no—
275SD16—62no—
276SD16—121no—
277D8—62no—
278D8—62no—
279D8—121no—
280M16—62no—
281C2×C8—31no—
282C2×C8—31no—
283C2×C8—62no—
284C2×C8C22×D311yescharacteristic
285C2×C8C22×D311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—61no—
290C4⋊C4C22×D311yescharacteristic
291C22⋊C4—31no—
292C22⋊C4—31no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C42—21no—
301C42C22×D311yescharacteristic
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3C22×D411yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—21no—
312C2×D3—22no—
313C2×D3—21no—
314C2×D3—21no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—84no—
321C2×D3—84no—
322C2×D3AGL(1,ℤ/8)12yescomplemented
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4C22×D411yescharacteristic
328C3×C4H(5,𝔽2)11yescharacteristic
329Dic3—21no—
330Dic3—21no—
331Dic3—41no—
332Dic3—41no—
333C23—21no—
334C23—21no—
335C23—31no—
336C23—31no—
337C23—61no—
338C23—61no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—241no—
344C23—241no—
345Q8—21no—
346Q8—61no—
347Q8—61no—
348Q8—122no—
349Q8D3×D412yescomplemented
350D4—21no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—32no—
355D4—42no—
356D4—62no—
357D4—61no—
358D4—61no—
359D4—61no—
360D4—81no—
361D4—122no—
362D4—122no—
363D4—122no—
364D4—122no—
365D4—122no—
366D4—122no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4D3×D411yescharacteristic
381C2×C4D3×D411yescharacteristic
382C2×C4C23×D311yescharacteristic
383C8—21no—
384C8—21no—
385C8—32no—
386C8—61no—
387C2×C3—81no—
388C2×C3—81no—
389C2×C3D4211yescharacteristic
390C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
391C2×C3#57711yescharacteristic
392D3—22no—
393D3—41no—
394D3—42no—
395D3—81no—
396C22—32no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—61no—
404C22—61no—
405C22—61no—
406C22—61no—
407C22—61no—
408C22—62no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—244no—
415C22—244no—
416C22C2×D3×D411yescharacteristic
417C4—21no—
418C4—21no—
419C4—22no—
420C4—41no—
421C4—61no—
422C4—61no—
423C4—121no—
424C4—121no—
425C4C2×D3×D411yescharacteristic
426C4#100911yescharacteristic
427C3#326511yescharacteristic, complemented, Sylow
428C2—62no—
429C2—81no—
430C2—81no—
431C2—121no—
432C2—122no—
433C2—241no—
434C2#560911yescharacteristic
435C2#576211yescharacteristic
436C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
hypercenter#460D3
hypocenterC3#3265
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. C2×D3×SD16C2semidirect
3. #5717C2semidirect
4. #5576C2semidirect
5. C3×#460C2semidirect
6. #5172C2semidirect
7. #5170C2semidirect
8. #5074C2semidirect
9. #5026C2semidirect
10. #4844C2semidirect
11. #4836C2semidirect
12. #4780C2semidirect
13. #4776C2semidirect
14. #4775C2semidirect
15. #4773C2semidirect
17. C2×#239C22semidirect
18. C2×D3×Q8C22semidirect
22. C2×C3×SD16C22semidirect
23. C2×C3×SD16C22semidirect
25. C3×C4×Q8C22semidirect
32. C2×#214C22semidirect
33. C2×#214C22semidirect
34. C2×#213C22semidirect
35. C2×#213C22semidirect
40. C2×ΓL(1,𝔽25)C22semidirect
41. C2×D3×C8C22semidirect
43. #897C22semidirect
45. #895C22semidirect
47. #875C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. #835C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #816C22semidirect
60. #807C22semidirect
61. #803C22semidirect
70. #460D3semidirect
119. #215D4semidirect
151. C2×SD16C2×D3semidirect
152. C2×SD16C2×D3semidirect
170. C4×Q8C2×D3semidirect
175. C4⋊C8C2×D3semidirect
180. #102C2×D3semidirect
181. #102C2×D3semidirect
198. C3×Q8C2×D4semidirect
238. C3⋊C8C2×D4semidirect
322. C2×D3AGL(1,ℤ/8)semidirect
349. Q8D3×D4semidirect
427. C3#3265semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
C2×D3×SD162
#57172
#55762
C3×#4602
#51722
#51702
#50742
#50262
#48442
#48362
#47802
#47762
#47752
#47732
#32653