Finite Group Databasebeta

← #80876

Subgroups — #80876

Order 384.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5614C211yesmaximal, characteristic, complemented
2#5609C211yesmaximal, characteristic, complemented
3C3×#469C211yesmaximal, characteristic, complemented
4#5156C211yesmaximal, characteristic, complemented
5#5150C211yesmaximal, characteristic, complemented
6#5131C211yesmaximal, characteristic, complemented
7#5127C211yesmaximal, characteristic, complemented
8#5036C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4863C211yesmaximal, characteristic, complemented
12#4785C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14D3×#102C211yesmaximal, characteristic, complemented
15#4764C211yesmaximal, characteristic, complemented
16#3318—31nomaximal, Sylow
17C2×D3×D4—21no—
18C2×D3×D4C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic, complemented
22C3×#121C2211yescharacteristic
23C3×#121C2211yescharacteristic, complemented
24#940C2211yescharacteristic
25#938—21no—
26#938C2211yescharacteristic, complemented
27#937—21no—
28#935—21no—
29D4×Dic3C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic, complemented
32C4×#67C2211yescharacteristic
33#926C2211yescharacteristic, complemented
34C2×D3×C8C2211yescharacteristic, complemented
35#895C2211yescharacteristic
36#893C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic, complemented
38#890C2211yescharacteristic, complemented
39#884—21no—
40#882—21no—
41#67⋊C4—21no—
42#879—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#832C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#830C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#553—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#481—31no—
65#478—31no—
66#469D311yescharacteristic, complemented
67#466—31no—
68#459—31no—
69#447—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67D411yescharacteristic
87C22×Dic3—21no—
88D3×D4—22no—
89D3×D4—41no—
90D3×D4—44no—
91C2×D12—22no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—41no—
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3C2311yescharacteristic
100C3×D8—41no—
101C3×M16—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#217D411yescharacteristic
112#213—22no—
113#213—41no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4D411yescharacteristic
117C4⋊Dic3—21no—
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×D8—31no—
132C2×D8—31no—
133C2×D8C2×D311yescharacteristic, complemented
134C2×M16C2×D311yescharacteristic, complemented
135C22×C8—31no—
136#127—31no—
137#126—61no—
138#125—31no—
139#123—61no—
140#121—31no—
141#121—31no—
142#121—61no—
143#121—61no—
144#121C2×D311yescharacteristic
145#121C2×D311yescharacteristic, complemented
146#120—61no—
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152#117—31no—
153C2×C4⋊C4—31no—
154#107—31no—
155#106—31no—
156#106C2×D311yescharacteristic, complemented
157C4⋊C8—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic, complemented
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C22×D3C2×D411yescharacteristic
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—42no—
188C3×D4—81no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—22no—
196#67—42no—
197#67—42no—
198#67—42no—
199#67—81no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—81no—
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206D12—21no—
207D12—21no—
208D12—21no—
209D12—42no—
210D12—42no—
211D12#4112yes—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—42no—
215C4×D3—81no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—61no—
221C24—61no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—62no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—62no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—124no—
238C2×D4C22×D311yescharacteristic
239C2×D4C22×D311yescharacteristic
240C22×C4—31no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4C22×D311yescharacteristic
245D8—41no—
246D8—62no—
247D8—121no—
248M16—41no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—121no—
252C2×C8C22×D311yescharacteristic
253C2×C8C22×D311yescharacteristic
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4C22×D311yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—31no—
265C22⋊C4—31no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—31no—
272C42—61no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3C22×D411yescharacteristic
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—22no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—84no—
292C2×D3—84no—
293C2×D3AGL(1,ℤ/8)12yes—
294C3×C4—21no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4C2×#4111yescharacteristic
298C3×C4H(5,𝔽2)11yescharacteristic
299Dic3—21no—
300Dic3—21no—
301Dic3—21no—
302Dic3—41no—
303Dic3—41no—
304C23—21no—
305C23—21no—
306C23—31no—
307C23—61no—
308C23—61no—
309C23—122no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23—241no—
314C23—241no—
315C23D3×D411yescharacteristic
316D4—21no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—32no—
321D4—42no—
322D4—62no—
323D4—61no—
324D4—61no—
325D4—61no—
326D4—81no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—122no—
332D4—241no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—122no—
345C2×C4—241no—
346C2×C4—241no—
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—61no—
352C8—61no—
353C2×C3—41no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3#54611yescharacteristic
357C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
358C2×C3#57611yescharacteristic
359D3—22no—
360D3—42no—
361D3—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—32no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—61no—
373C22—61no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—244no—
381C22—244no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—41no—
385C4—41no—
386C4—61no—
387C4—61no—
388C4—61no—
389C4—121no—
390C4—121no—
391C4#100411yescharacteristic
392C4D3×#4111yescharacteristic
393C3#331811yescharacteristic, complemented, Sylow
394C2—41no—
395C2—62no—
396C2—81no—
397C2—81no—
398C2—122no—
399C2—241no—
400C2#561011yescharacteristic
401C2D3×AGL(1,ℤ/8)11yescharacteristic
402C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3318
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5614C2semidirect
2. #5609C2semidirect
3. C3×#469C2semidirect
4. #5156C2semidirect
5. #5150C2semidirect
6. #5131C2semidirect
7. #5127C2semidirect
8. #5036C2semidirect
9. #5035C2semidirect
10. #4871C2semidirect
11. #4863C2semidirect
12. #4785C2semidirect
13. #4775C2semidirect
14. D3×#102C2semidirect
15. #4764C2semidirect
20. C2×C3×D8C22semidirect
21. C2×C3×M16C22semidirect
23. C3×#121C22semidirect
26. #938C22semidirect
29. D4×Dic3C22semidirect
31. C2×#213C22semidirect
33. #926C22semidirect
34. C2×D3×C8C22semidirect
37. D3×C4⋊C4C22semidirect
38. #890C22semidirect
43. C3×#106C22semidirect
44. C3×#102C22semidirect
45. C3×#102C22semidirect
46. #832C22semidirect
47. #832C22semidirect
48. #830C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #807C22semidirect
66. #469D3semidirect
133. C2×D8C2×D3semidirect
134. C2×M16C2×D3semidirect
145. #121C2×D3semidirect
156. #106C2×D3semidirect
166. #102C2×D3semidirect
167. #102C2×D3semidirect
393. C3#3318semidirect

Maximal subgroups

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

subgroupindex
#56142
#56092
C3×#4692
#51562
#51502
#51312
#51272
#50362
#50352
#48712
#48632
#47852
#47752
D3×#1022
#47642
#33183