Finite Group Databasebeta

← #80608

Subgroups — #80608

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5674C211yesmaximal, characteristic
2#5583C211yesmaximal, characteristic, complemented
3C3×#464C211yesmaximal, characteristic, complemented
4#5190C211yesmaximal, characteristic, complemented
5#5187C211yesmaximal, characteristic, complemented
6#5175C211yesmaximal, characteristic, complemented
7#5169C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5027C211yesmaximal, characteristic, complemented
10#4848C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4808C211yesmaximal, characteristic, complemented
13#4800C211yesmaximal, characteristic, complemented
14#4776C211yesmaximal, characteristic, complemented
15#4764C211yesmaximal, characteristic, complemented
16#3222—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic, complemented
20C2×C3×SD16C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24#946C2211yescharacteristic
25#944C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic, complemented
29#940C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×#204C2211yescharacteristic
32C2×#204C2211yescharacteristic, complemented
33#898—21no—
34#897C2211yescharacteristic
35#890C2211yescharacteristic, complemented
36#889—21no—
37#884—21no—
38#884—21no—
39#882—21no—
40#882—21no—
41#876—21no—
42#874C2211yescharacteristic
43#872C2211yescharacteristic
44#870—21no—
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#541—31no—
61#535—31no—
62#497—31no—
63#492—31no—
64#490—31no—
65#488—31no—
66#464—31no—
67#464D311yescharacteristic, complemented
68#460—31no—
69#460—31no—
70#452—31no—
71#452—31no—
72#450—31no—
73#450—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82#239—41no—
83#239—41no—
84D3×D4—41no—
85D3×D4—41no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×Q16—41no—
95C3×SD16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C42—21no—
103C3×C42C2311yescharacteristic
104#216—41no—
105#215—41no—
106#215—41no—
107#214—41no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
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×Q16C2×D311yescharacteristic, complemented
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16—31no—
138C2×SD16—31no—
139C2×SD16C2×D311yescharacteristic
140C4⋊Q8—31no—
141#127—31no—
142#127—31no—
143#126—61no—
144#126—61no—
145#125—31no—
146#125—61no—
147#125—61no—
148#124—31no—
149#124C2×D311yescharacteristic
150C22⋊Q8—31no—
151C22⋊Q8—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×Q8C2×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—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3—81no—
179C22×D3C2×D411yescharacteristic
180C3×Q8—21no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193#67—81no—
194#67—81no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—42no—
205D12—81no—
206D12—81no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—42no—
210Dic6—21no—
211Dic6—21no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216C24—61no—
217#41—121no—
218#41—121no—
219C2×Q8—31no—
220C2×Q8C22×D311yescharacteristic
221C2×Q8C22×D311yescharacteristic
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4C22×D311yescharacteristic
233C22×C4—31no—
234C22×C4—61no—
235Q16—41no—
236Q16—121no—
237SD16—41no—
238SD16—121no—
239SD16—121no—
240SD16—121no—
241SD16—121no—
242SD16—121no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8C22×D311yescharacteristic
246C2×C8C22×D311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—21no—
268C42—31no—
269C42—61no—
270C42C22×D311yescharacteristic
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3C22×D411yescharacteristic
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—84no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4H(5,𝔽2)11yescharacteristic
291C3×C4H(5,𝔽2)11yescharacteristic
292Dic3—21no—
293Dic3—21no—
294Dic3—21no—
295Dic3—41no—
296Dic3—41no—
297C23—21no—
298C23—31no—
299C23—61no—
300C23—61no—
301C23—122no—
302C23—241no—
303C23—241no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—61no—
309Q8—61no—
310D4—21no—
311D4—21no—
312D4—61no—
313D4—61no—
314D4—61no—
315D4—61no—
316D4—122no—
317D4—122no—
318D4—241no—
319D4—241no—
320D4—241no—
321D4—241no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—122no—
333C2×C4D3×D411yescharacteristic
334C2×C4D3×D411yescharacteristic
335C2×C4C23×D311yescharacteristic
336C8—21no—
337C8—21no—
338C8—61no—
339C8—61no—
340C2×C3—81no—
341C2×C3#53511yescharacteristic
342C2×C3#57711yescharacteristic
343C2×C3#57711yescharacteristic
344D3—41no—
345D3—81no—
346D3—81no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—61no—
351C22—61no—
352C22—61no—
353C22—121no—
354C22—121no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—244no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—61no—
367C4—61no—
368C4—61no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100911yescharacteristic
373C3#322211yescharacteristic, complemented, Sylow
374C2—81no—
375C2—121no—
376C2—241no—
377C2—241no—
378C2#560711yescharacteristic
379C2#577011yescharacteristic
380C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
prime radical#5674C2
hypercenter#464D3
hypocenterC3#3222
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #5583C2semidirect
3. C3×#464C2semidirect
4. #5190C2semidirect
5. #5187C2semidirect
6. #5175C2semidirect
7. #5169C2semidirect
8. #5054C2semidirect
9. #5027C2semidirect
10. #4848C2semidirect
11. #4840C2semidirect
12. #4808C2semidirect
13. #4800C2semidirect
14. #4776C2semidirect
15. #4764C2semidirect
19. C2×C3×Q16C22semidirect
22. C3×C4×Q8C22semidirect
25. #944C22semidirect
26. C2×#216C22semidirect
28. C2×#215C22semidirect
32. C2×#204C22semidirect
35. #890C22semidirect
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. #464D3semidirect
133. C2×Q16C2×D3semidirect
157. C4×Q8C2×D3semidirect
160. C4⋊C8C2×D3semidirect
162. #103C2×D3semidirect
373. C3#3222semidirect

Maximal subgroups

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

subgroupindex
#56742
#55832
C3×#4642
#51902
#51872
#51752
#51692
#50542
#50272
#48482
#48402
#48082
#48002
#47762
#47642
#32223