Finite Group Databasebeta

← #80546

Subgroups — #80546

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#5674C211yesmaximal, characteristic, complemented
2#5561C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5168C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5154C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5015C211yesmaximal, characteristic, complemented
10#4848C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4808C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4776C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic, complemented
20C2×C3×D8C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24C2×#216C2211yescharacteristic, complemented
25#940C2211yescharacteristic
26#938C2211yescharacteristic
27#936C2211yescharacteristic, complemented
28C2×#214C2211yescharacteristic
29C2×#213C2211yescharacteristic
30C2×#213C2211yescharacteristic, complemented
31#929—21no—
32#924—21no—
33C2×#204C2211yescharacteristic
34C2×#204C2211yescharacteristic, complemented
35#897C2211yescharacteristic
36#888C2211yescharacteristic, complemented
37#885—21no—
38#884—21no—
39#884—21no—
40#882—21no—
41#882—21no—
42#879—21no—
43#874C2211yescharacteristic
44#872C2211yescharacteristic
45C4×Dic6C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic
54(C2×D3)⋊C8C2211yescharacteristic
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#809C2211yescharacteristic, complemented
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#542—31no—
61#535—31no—
62#496—31no—
63#495—31no—
64#489—31no—
65#486—31no—
66#464—31no—
67#463D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84#239—41no—
85#239—41no—
86D3×D4—41no—
87D3×D4—41no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×Dic6—21no—
96C2×Dic6C2311yescharacteristic
97C3×SD16—41no—
98C3×D8—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C42C2311yescharacteristic
106#217—21no—
107#217—21no—
108#217—21no—
109#216—41no—
110#214—41no—
111#213—41no—
112#213—41no—
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—
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123Dic3⋊C4D411yescharacteristic
124C4×Dic3—21no—
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
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×SD16C2×D311yescharacteristic, complemented
137C2×D8—31no—
138C2×D8—31no—
139C2×D8C2×D311yescharacteristic
140C4⋊Q8—31no—
141#127—31no—
142#127—31no—
143#126—61no—
144#124—31no—
145#124—31no—
146#124—61no—
147#124C2×D311yescharacteristic
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×D4C2×D311yescharacteristic, complemented
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103C2×D311yescharacteristic, complemented
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
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—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—81no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—81no—
195#67—81no—
196#67—81no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202D12—21no—
203D12—21no—
204D12—21no—
205D12—21no—
206D12—42no—
207D12—81no—
208D12—81no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212Dic6—21no—
213Dic6—21no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—121no—
221#41—121no—
222C2×Q8—31no—
223C2×Q8—61no—
224C2×Q8C22×D311yescharacteristic
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—61no—
228C2×D4—61no—
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—21no—
239C22×C4—31no—
240C22×C4—61no—
241Q16—121no—
242SD16—41no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246D8—41no—
247D8—121no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—31no—
264C22⋊C4—31no—
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—21no—
311Q8—21no—
312Q8—61no—
313Q8—61no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—122no—
324D4—122no—
325D4—241no—
326D4—241no—
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—81no—
340C2×C4—122no—
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#100411yescharacteristic
383C4#100911yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—81no—
386C2—81no—
387C2—121no—
388C2—241no—
389C2—241no—
390C2#560711yescharacteristic
391C2#577011yescharacteristic
392C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5674C2semidirect
2. #5561C2semidirect
3. C3×#463C2semidirect
4. #5172C2semidirect
5. #5168C2semidirect
6. #5156C2semidirect
7. #5154C2semidirect
8. #5054C2semidirect
9. #5015C2semidirect
10. #4848C2semidirect
11. #4840C2semidirect
12. #4808C2semidirect
13. #4794C2semidirect
14. #4776C2semidirect
15. #4760C2semidirect
19. C2×C3×SD16C22semidirect
22. C3×C4×D4C22semidirect
24. C2×#216C22semidirect
27. #936C22semidirect
30. C2×#213C22semidirect
34. C2×#204C22semidirect
36. #888C22semidirect
45. C4×Dic6C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
49. #835C22semidirect
52. #821C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
58. #809C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
67. #463D3semidirect
136. C2×SD16C2×D3semidirect
157. C4×D4C2×D3semidirect
160. C4⋊C8C2×D3semidirect
162. #103C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#56742
#55612
C3×#4632
#51722
#51682
#51562
#51542
#50542
#50152
#48482
#48402
#48082
#47942
#47762
#47602
#32243