Finite Group Databasebeta

← #81570

Subgroups — #81570

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5733C211yesmaximal, characteristic, complemented
2#5715C211yesmaximal, characteristic, complemented
3C3×#486C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5074C211yesmaximal, characteristic, complemented
6#4787C211yesmaximal, characteristic, complemented
7#4786C211yesmaximal, characteristic, complemented
8#4783C211yesmaximal, characteristic, complemented
9#4781C211yesmaximal, characteristic, complemented
10#4776C211yesmaximal, characteristic, complemented
11#4773C211yesmaximal, characteristic, complemented
12#4767C211yesmaximal, characteristic, complemented
13#4762C211yesmaximal, characteristic, complemented
14#4696C211yesmaximal, characteristic, complemented
15#4690C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×#127C2211yescharacteristic
21#947—21no—
22#946—21no—
23#940C2211yescharacteristic
24#939—21no—
25#939—21no—
26#937—21no—
27#937—21no—
28#936C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#214C2211yescharacteristic, complemented
31C2×#213C2211yescharacteristic
32C2×#213C2211yescharacteristic, complemented
33C2×Dic12C2211yescharacteristic, complemented
34C2×D24C2211yescharacteristic
35C2×#204C2211yescharacteristic
36C2×#204C2211yescharacteristic, complemented
37#898—21no—
38#897C2211yescharacteristic
39#895C2211yescharacteristic
40#894—21no—
41#893C2211yescharacteristic, complemented
42#887C2211yescharacteristic, complemented
43#875C2211yescharacteristic
44#875C2211yescharacteristic, complemented
45#872C2211yescharacteristic
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×#102C2211yescharacteristic, complemented
49C3×#102C2211yescharacteristic, complemented
50C3×C4×C8C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59#803C2211yescharacteristic, complemented
60#542—31no—
61#535—31no—
62#496—31no—
63#495—31no—
64#489—31no—
65#486D311yescharacteristic, complemented
66#464—31no—
67#463—31no—
68#460—31no—
69#459—31no—
70#452—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8—21no—
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#237—41no—
87#237—41no—
88D3×D4—41no—
89D3×D4—41no—
90C2×D12—21no—
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×Dic6C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C42C2311yescharacteristic
103#217—21no—
104#217—21no—
105#214—41no—
106#214—41no—
107#213—41no—
108#213—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4D411yescharacteristic
117(C2×D3)⋊C4D411yescharacteristic
118C4⋊Dic3—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126Dic12—41no—
127D24—41no—
128#204—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×D8—31no—
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8C2×D311yescharacteristic, complemented
141#127—31no—
142#127C2×D311yescharacteristic
143#126—61no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#123—61no—
149C22⋊Q8—31no—
150C22⋊Q8—61no—
151#121—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167#102C2×D311yescharacteristic, complemented
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8C2×D311yescharacteristic, complemented
173C23×C3—21no—
174C23×C3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C3×Q8—81no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—81no—
187C3×D4—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—42no—
195#67—81no—
196#67—81no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—42no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—21no—
206D12—21no—
207D12—81no—
208C4×D3—21no—
209C4×D3—21no—
210C4×D3—81no—
211Dic6—21no—
212Dic6—21no—
213Dic6—21no—
214Dic6—21no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—121no—
221#41—121no—
222C2×Q8—21no—
223C2×Q8—31no—
224C2×Q8—31no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—31no—
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4C22×D311yescharacteristic
237C2×D4C22×D311yescharacteristic
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241Q16—121no—
242SD16—121no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246D8—121no—
247D8—121no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C4⋊C4C22×D311yescharacteristic
260C22⋊C4—31no—
261C22⋊C4—31no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—31no—
272C42—61no—
273C42C22×D311yescharacteristic
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3C22×D411yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—84no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4H(5,𝔽2)11yescharacteristic
296C3×C4H(5,𝔽2)11yescharacteristic
297Dic3—21no—
298Dic3—21no—
299Dic3—21no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—31no—
305C23—61no—
306C23—61no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—61no—
311Q8—61no—
312Q8—61no—
313Q8—61no—
314Q8—81no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—81no—
324D4—81no—
325D4—122no—
326D4—122no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—122no—
340C2×C4—241no—
341C2×C4D3×D411yescharacteristic
342C2×C4D3×D411yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—21no—
345C8—21no—
346C8—61no—
347C8—61no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3#53511yescharacteristic
351C2×C3#57611yescharacteristic
352C2×C3#57711yescharacteristic
353D3—41no—
354D3—81no—
355D3—81no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—244no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—61no—
378C4—61no—
379C4—61no—
380C4—121no—
381C4—121no—
382C4#100911yescharacteristic
383C4#100911yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—81no—
386C2—81no—
387C2—121no—
388C2—241no—
389C2—241no—
390C2#567411yescharacteristic
391C2#575711yescharacteristic
392C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
hypercenter#486D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5733C2semidirect
2. #5715C2semidirect
3. C3×#486C2semidirect
4. #5089C2semidirect
5. #5074C2semidirect
6. #4787C2semidirect
7. #4786C2semidirect
8. #4783C2semidirect
9. #4781C2semidirect
10. #4776C2semidirect
11. #4773C2semidirect
12. #4767C2semidirect
13. #4762C2semidirect
14. #4696C2semidirect
15. #4690C2semidirect
19. C3×C4⋊Q8C22semidirect
30. C2×#214C22semidirect
32. C2×#213C22semidirect
33. C2×Dic12C22semidirect
36. C2×#204C22semidirect
41. #893C22semidirect
42. #887C22semidirect
44. #875C22semidirect
48. C3×#102C22semidirect
49. C3×#102C22semidirect
50. C3×C4×C8C22semidirect
53. Dic3⋊C8C22semidirect
54. Dic3⋊C8C22semidirect
55. #810C22semidirect
56. #810C22semidirect
59. #803C22semidirect
65. #486D3semidirect
140. C4⋊Q8C2×D3semidirect
167. #102C2×D3semidirect
168. #102C2×D3semidirect
172. C4×C8C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#57332
#57152
C3×#4862
#50892
#50742
#47872
#47862
#47832
#47812
#47762
#47732
#47672
#47622
#46962
#46902
#32243