Finite Group Databasebeta

← #81352

Subgroups — #81352

Order 384.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5603C211yesmaximal, characteristic, complemented
2#5521C211yesmaximal, characteristic, complemented
3C3×#482C211yesmaximal, characteristic, complemented
4#5036C211yesmaximal, characteristic
5#4996C211yesmaximal, characteristic, complemented
6#4890C211yesmaximal, characteristic, complemented
7#4889C211yesmaximal, characteristic, complemented
8#4870C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4787C211yesmaximal, characteristic, complemented
11#4787C211yesmaximal, characteristic, complemented
12#4785C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3118—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C3×#121C2211yescharacteristic
19C3×#117C2211yescharacteristic
20#936—21no—
21#936C2211yescharacteristic, complemented
22#935—21no—
23D4×Dic3C2211yescharacteristic, complemented
24C2×#213C2211yescharacteristic
25C2×#213C2211yescharacteristic
26#926C2211yescharacteristic
27#924C2211yescharacteristic
28C2×#204C2211yescharacteristic
29C2×#204C2211yescharacteristic
30#893C2211yescharacteristic
31#893C2211yescharacteristic
32#893C2211yescharacteristic
33#893C2211yescharacteristic
34#888C2211yescharacteristic, complemented
35#887C2211yescharacteristic, complemented
36D3×C22⋊C4—21no—
37D3×C22⋊C4—21no—
38#879—21no—
39#878—21no—
40C4×D12—21no—
41C4×D12—21no—
42C3×#107C2211yescharacteristic, complemented
43C3×#106C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46C3×C22⋊C8C2211yescharacteristic
47#830C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#815C2211yescharacteristic, complemented
51Dic3⋊C8C2211yescharacteristic, complemented
52Dic3⋊C8C2211yescharacteristic, complemented
53C8×Dic3C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#542—31no—
61#518—31no—
62#485—31no—
63#482D311yescharacteristic, complemented
64#476—31no—
65#474—31no—
66#463—31no—
67#463—31no—
68#450—31no—
69#447—31no—
70#442—31no—
71#440—31no—
72C4×SD16—31no—
73C4×D8—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×Dic3—21no—
80C2×D12—21no—
81C2×D12—22no—
82C2×D12—22no—
83C2×D12C2311yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3—21no—
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4C2311yescharacteristic
94C3×C4⋊C4C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97C3×C22⋊C4—21no—
98C3×C42—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217D411yescharacteristic
104#217D411yescharacteristic
105#213—22no—
106#213—22no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111C4⋊Dic3D411yescharacteristic
112C4⋊Dic3D411yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3C2311yescharacteristic
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122#204—22no—
123#204—22no—
124C22×D4—31no—
125C2×SD16—31no—
126C2×SD16—31no—
127C2×D8—31no—
128C2×D8—31no—
129C4⋊Q8—31no—
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121C2×D311yescharacteristic
137C4×Q8—31no—
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145#117—31no—
146#117C2×D311yescharacteristic
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149#107—31no—
150#107C2×D311yescharacteristic, complemented
151#106—31no—
152#106C2×D311yescharacteristic, complemented
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic, complemented
162#102C2×D311yescharacteristic, complemented
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—24no—
172C22×D3—81no—
173C22×D3—81no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—81no—
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191D12—24no—
192D12—21no—
193D12—21no—
194D12#4112yes—
195D12#4112yes—
196C4×D3—81no—
197C4×D3—81no—
198C4×D3—81no—
199C4×D3—81no—
200Dic6—21no—
201Dic6—21no—
202Dic6—81no—
203C3×C8—21no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C24—61no—
208C2×Q8—31no—
209C2×Q8—61no—
210C2×D4—21no—
211C2×D4—31no—
212C2×D4—31no—
213C2×D4—62no—
214C2×D4—62no—
215C2×D4—61no—
216C2×D4C22×D311yescharacteristic
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C22×D311yescharacteristic
223SD16—62no—
224SD16—62no—
225D8—62no—
226D8—62no—
227C2×C8—31no—
228C2×C8—31no—
229C2×C8C22×D311yescharacteristic
230C2×C8C22×D311yescharacteristic
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4C22×D311yescharacteristic
238C4⋊C4C22×D311yescharacteristic
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—21no—
242C22⋊C4—31no—
243C22⋊C4—31no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C42—21no—
253C42—31no—
254C42—31no—
255C42—61no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C2×D3—22no—
264C2×D3—22no—
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—84no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4C2×#4111yescharacteristic
275C3×C4C2×#4111yescharacteristic
276Dic3—21no—
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282C23—21no—
283C23—64no—
284C23—61no—
285C23—61no—
286C23—241no—
287C23—241no—
288C23D3×D411yescharacteristic
289Q8—61no—
290Q8—61no—
291Q8—241no—
292D4—21no—
293D4—21no—
294D4—32no—
295D4—32no—
296D4—64no—
297D4—61no—
298D4—61no—
299D4—81no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—241no—
312C2×C4—241no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4D3×D411yescharacteristic
317C2×C4C23×D311yescharacteristic
318C8—21no—
319C8—21no—
320C8—61no—
321C8—61no—
322C2×C3—41no—
323C2×C3—81no—
324C2×C3#52511yescharacteristic
325C2×C3#57611yescharacteristic
326C2×C3#57711yescharacteristic
327D3—42no—
328D3—42no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—62no—
336C22—62no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22—244no—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—61no—
348C4—61no—
349C4—61no—
350C4—61no—
351C4—121no—
352C4—121no—
353C4D3×#4111yescharacteristic
354C4D3×#4111yescharacteristic
355C3#311811yescharacteristic, complemented, Sylow
356C2—41no—
357C2—81no—
358C2—122no—
359C2—122no—
360C2#565611yescharacteristic
361C2#577011yescharacteristic
362C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#482C2
socleC22×C3C22×D4
prime radical#5036C2
hypercenter#482D3
hypocenterC3#3118
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #5603C2semidirect
2. #5521C2semidirect
3. C3×#482C2semidirect
5. #4996C2semidirect
6. #4890C2semidirect
7. #4889C2semidirect
8. #4870C2semidirect
9. #4852C2semidirect
10. #4787C2semidirect
11. #4787C2semidirect
12. #4785C2semidirect
13. #4756C2semidirect
14. #4731C2semidirect
15. #4720C2semidirect
21. #936C22semidirect
23. D4×Dic3C22semidirect
34. #888C22semidirect
35. #887C22semidirect
42. C3×#107C22semidirect
43. C3×#106C22semidirect
44. C3×#102C22semidirect
45. C3×#102C22semidirect
50. #815C22semidirect
51. Dic3⋊C8C22semidirect
52. Dic3⋊C8C22semidirect
53. C8×Dic3C22semidirect
54. #810C22semidirect
55. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
63. #482D3semidirect
150. #107C2×D3semidirect
152. #106C2×D3semidirect
161. #102C2×D3semidirect
162. #102C2×D3semidirect
355. C3#3118semidirect

Maximal subgroups

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

subgroupindex
#56032
#55212
C3×#4822
#50362
#49962
#48902
#48892
#48702
#48522
#47872
#47852
#47562
#47312
#47202
#31183