Finite Group Databasebeta

← #80681

Subgroups — #80681

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5630C211yesmaximal, characteristic, complemented
2#5618C211yesmaximal, characteristic, complemented
3C3×#465C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5165C211yesmaximal, characteristic, complemented
6C2×#821C211yesmaximal, characteristic, complemented
7C8×#67C211yesmaximal, characteristic, complemented
8#5044C211yesmaximal, characteristic, complemented
9#5036C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4866C211yesmaximal, characteristic, complemented
12#4804C211yesmaximal, characteristic, complemented
13#4800C211yesmaximal, characteristic, complemented
14#4775C211yesmaximal, characteristic, complemented
15#4756C211yesmaximal, characteristic, complemented
16#3297—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic, complemented
20C22×C3×C8C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22C3×#121C2211yescharacteristic
23#947C2211yescharacteristic
24#939—21no—
25#938C2211yescharacteristic, complemented
26D4×Dic3—21no—
27D4×Dic3C2211yescharacteristic, complemented
28C2×#213C2211yescharacteristic
29C2×#213C2211yescharacteristic
30C4×#67C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic, complemented
33#895—21no—
34#895C2211yescharacteristic
35#894—21no—
36#893C2211yescharacteristic
37#892C2211yescharacteristic, complemented
38#890C2211yescharacteristic, complemented
39#886—21no—
40#882—21no—
41D3×C22⋊C4—21no—
42#879—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#830C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#553—31no—
61#546—31no—
62#488—31no—
63#484—31no—
64#482—31no—
65#478—31no—
66#466—31no—
67#465D311yescharacteristic, complemented
68#463—31no—
69#449—31no—
70#447—31no—
71C4×D8—31no—
72C8×D4—31no—
73#416—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67D411yescharacteristic
82C22×Dic3—21no—
83C22×Dic3—21no—
84#239—22no—
85#239—41no—
86C2×D12—22no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—41no—
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C3×SD16—41no—
97C2×C3×C8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105#217—21no—
106#217—21no—
107#217D411yescharacteristic
108#213—22no—
109#213—41no—
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)⋊C4D411yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×SD16C2×D311yescharacteristic, complemented
130C2×D8—31no—
131C2×D8—31no—
132C22×C8—31no—
133C22×C8C2×D311yescharacteristic
134#126—61no—
135#125—31no—
136#124—31no—
137#123—61no—
138#123—61no—
139C22⋊Q8C2×D311yescharacteristic
140#121—31no—
141#121—31no—
142#121—61no—
143#121—61no—
144#121C2×D311yescharacteristic
145#120—61no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—61no—
153#107—31no—
154#107—31no—
155#106C2×D311yescharacteristic, complemented
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic, complemented
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic, complemented
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—81no—
177C22×D3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—42no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—22no—
191#67—81no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199D12—21no—
200D12—21no—
201D12—21no—
202D12—42no—
203D12—42no—
204D12—81no—
205D12#4112yes—
206C4×D3—21no—
207C4×D3—21no—
208C4×D3—42no—
209C4×D3—81no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C24—61no—
215#41—62no—
216#41—121no—
217C2×Q8C22×D311yescharacteristic
218C2×D4—21no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—62no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—121no—
228C2×D4C22×D311yescharacteristic
229C22×C4—31no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235SD16—41no—
236D8—62no—
237D8—121no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—41no—
241C2×C8—121no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4C22×D311yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—31no—
256C22⋊C4—31no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—31no—
265C42—61no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3#13512yes—
272C22×C3C22×D411yescharacteristic
273C2×D3—21no—
274C2×D3—22no—
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—84no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4C2×#4111yescharacteristic
288C3×C4H(5,𝔽2)11yescharacteristic
289Dic3—21no—
290Dic3—21no—
291Dic3—21no—
292Dic3—41no—
293Dic3—41no—
294C23—21no—
295C23—31no—
296C23—61no—
297C23—61no—
298C23—122no—
299C23—122no—
300C23—241no—
301C23D3×D411yescharacteristic
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—21no—
306D4—32no—
307D4—42no—
308D4—62no—
309D4—61no—
310D4—61no—
311D4—61no—
312D4—122no—
313D4—122no—
314D4—241no—
315D4—241no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—122no—
329C2×C4—241no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4D3×D411yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—21no—
335C8—21no—
336C8—61no—
337C8—61no—
338C2×C3—22no—
339C2×C3—81no—
340C2×C3#54611yescharacteristic
341C2×C3C2×#13511yescharacteristic
342C2×C3#57611yescharacteristic
343D3—41no—
344D3—42no—
345D3—81no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—61no—
352C22—61no—
353C22—61no—
354C22—62no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—244no—
361C22#91612yes—
362C22C2×D3×D411yescharacteristic
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—61no—
368C4—61no—
369C4—61no—
370C4—121no—
371C4—121no—
372C4#100411yescharacteristic
373C4D3×#4111yescharacteristic
374C3#329711yescharacteristic, complemented, Sylow
375C2—22no—
376C2—81no—
377C2—121no—
378C2—122no—
379C2—241no—
380C2#561011yescharacteristic
381C2C2×#91611yescharacteristic
382C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3297
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5630C2semidirect
2. #5618C2semidirect
3. C3×#465C2semidirect
4. #5172C2semidirect
5. #5165C2semidirect
6. C2×#821C2semidirect
7. C8×#67C2semidirect
8. #5044C2semidirect
9. #5036C2semidirect
10. #4871C2semidirect
11. #4866C2semidirect
12. #4804C2semidirect
13. #4800C2semidirect
14. #4775C2semidirect
15. #4756C2semidirect
19. C2×C3×SD16C22semidirect
25. #938C22semidirect
27. D4×Dic3C22semidirect
32. C2×D3×C8C22semidirect
37. #892C22semidirect
38. #890C22semidirect
43. C3×#106C22semidirect
44. C3×#103C22semidirect
45. C3×#102C22semidirect
46. #835C22semidirect
47. #835C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. Dic3⋊C8C22semidirect
55. C8×Dic3C22semidirect
58. #807C22semidirect
59. #807C22semidirect
67. #465D3semidirect
129. C2×SD16C2×D3semidirect
155. #106C2×D3semidirect
159. #103C2×D3semidirect
166. #102C2×D3semidirect
374. C3#3297semidirect

Maximal subgroups

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

subgroupindex
#56302
#56182
C3×#4652
#51722
#51652
C2×#8212
C8×#672
#50442
#50362
#48712
#48662
#48042
#48002
#47752
#47562
#32973