Finite Group Databasebeta

← #80541

Subgroups — #80541

Order 384.

410 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
1D3×#124C211yesmaximal, characteristic, complemented
2#5557C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5170C211yesmaximal, characteristic, complemented
5#5168C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic
7#5155C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4833C211yesmaximal, characteristic, complemented
12#4809C211yesmaximal, characteristic, complemented
13#4800C211yesmaximal, characteristic, complemented
14#4776C211yesmaximal, characteristic, complemented
15#4768C211yesmaximal, characteristic, complemented
16#3211—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic, complemented
20C2×C3×D8C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24C2×#216C2211yescharacteristic, complemented
25#938C2211yescharacteristic
26#936C2211yescharacteristic
27#936C2211yescharacteristic, complemented
28C2×#214C2211yescharacteristic
29C2×#214C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C4×#67—21no—
32#926—21no—
33C2×Dic12C2211yescharacteristic, complemented
34C2×#204C2211yescharacteristic
35#897C2211yescharacteristic
36#890C2211yescharacteristic, complemented
37#885—21no—
38#885—21no—
39#883—21no—
40D3×C22⋊C4—21no—
41D3×C22⋊C4—21no—
42#879—21no—
43#875C2211yescharacteristic
44C42×D3C2211yescharacteristic
45C4×Dic6C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#529—31no—
61C2×#124—31no—
62#495—31no—
63#495—31no—
64#488—31no—
65#488—31no—
66#464—31no—
67#464—31no—
68#463—31no—
69#463D311yescharacteristic, complemented
70#451—31no—
71#450—31no—
72#448—31no—
73#447—31no—
74#431—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C2×#67—21no—
82D3×Q8—41no—
83D3×Q8—41no—
84D3×D4—41no—
85D3×D4—41no—
86C2×D12C2311yescharacteristic
87C2×C4×D3—21no—
88C2×C4×D3D411yescharacteristic
89C2×C4×D3D411yescharacteristic
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×SD16—41no—
95C3×D8—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C42C2311yescharacteristic
103#217—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217—21no—
108#216—41no—
109#214—41no—
110#214—41no—
111#213—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C4×Dic3D411yescharacteristic
124C4×Dic3D411yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C22×Q8—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16C2×D311yescharacteristic, complemented
138C2×D8—31no—
139C2×D8C2×D311yescharacteristic
140#126—61no—
141#125—31no—
142#124—31no—
143#124—31no—
144#124—31no—
145#124—31no—
146#124—61no—
147#124—61no—
148#124C2×D311yescharacteristic
149#123—61no—
150C22⋊Q8—31no—
151#121—31no—
152C4×Q8—31no—
153C4×D4—61no—
154C4×D4C2×D311yescharacteristic, complemented
155#117—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C42—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic, complemented
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171C4×C8—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3C2×D411yescharacteristic
180C3×Q8—21no—
181C3×Q8—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—42no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193#67—42no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202C4×D3—22no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—41no—
206C4×D3—42no—
207Dic6—21no—
208Dic6—21no—
209Dic6—21no—
210Dic6—21no—
211Dic6—42no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216C24—61no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×Q8—61no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—31no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4C22×D311yescharacteristic
229C2×D4C22×D311yescharacteristic
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—61no—
235Q16—121no—
236Q16—121no—
237SD16—41no—
238SD16—121no—
239SD16—121no—
240SD16—121no—
241D8—41no—
242D8—121no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8C22×D311yescharacteristic
246C2×C8C22×D311yescharacteristic
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C42—31no—
267C42—31no—
268C42—31no—
269C42—61no—
270C42C22×D311yescharacteristic
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C22×D411yescharacteristic
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—84no—
284C2×D3#13512yes—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4H(5,𝔽2)11yescharacteristic
290C3×C4H(5,𝔽2)11yescharacteristic
291Dic3—21no—
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—21no—
298C23—21no—
299C23—31no—
300C23—61no—
301C23—122no—
302C23—122no—
303C23—122no—
304Q8—21no—
305Q8—21no—
306Q8—61no—
307Q8—61no—
308Q8—61no—
309Q8—61no—
310Q8—122no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—61no—
316D4—61no—
317D4—122no—
318D4—122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—42no—
325C2×C4—62no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—121no—
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—81no—
342C2×C3#53511yescharacteristic
343C2×C3C2×#13511yescharacteristic
344C2×C3#57511yescharacteristic
345D3—22no—
346D3—81no—
347C22—32no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—61no—
355C22—61no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—244no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—61no—
366C4—61no—
367C4—61no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100911yescharacteristic
373C3#321111yescharacteristic, complemented, Sylow
374C2—62no—
375C2—81no—
376C2—81no—
377C2—241no—
378C2#560711yescharacteristic
379C2D3×#13511yescharacteristic
380C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
prime radical#5156C2
hypercenter#463D3
hypocenterC3#3211
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. D3×#124C2semidirect
2. #5557C2semidirect
3. C3×#463C2semidirect
4. #5170C2semidirect
5. #5168C2semidirect
7. #5155C2semidirect
8. #5054C2semidirect
9. #5016C2semidirect
10. #4845C2semidirect
11. #4833C2semidirect
12. #4809C2semidirect
13. #4800C2semidirect
14. #4776C2semidirect
15. #4768C2semidirect
19. C2×C3×SD16C22semidirect
22. C3×C4×D4C22semidirect
24. C2×#216C22semidirect
27. #936C22semidirect
29. C2×#214C22semidirect
33. C2×Dic12C22semidirect
36. #890C22semidirect
45. C4×Dic6C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
49. #835C22semidirect
54. #816C22semidirect
55. C8×Dic3C22semidirect
56. C8×Dic3C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
69. #463D3semidirect
137. C2×SD16C2×D3semidirect
154. C4×D4C2×D3semidirect
160. C4⋊C8C2×D3semidirect
164. #103C2×D3semidirect
373. C3#3211semidirect

Maximal subgroups

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

subgroupindex
D3×#1242
#55572
C3×#4632
#51702
#51682
#51562
#51552
#50542
#50162
#48452
#48332
#48092
#48002
#47762
#47682
#32113