Finite Group Databasebeta

← #79779

Subgroups — #79779

Order 384.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5578C211yesmaximal, characteristic, complemented
2C4×D3×D4C211yesmaximal, characteristic
3C3×#440C211yesmaximal, characteristic, complemented
4#5169C211yesmaximal, characteristic
5#5162C211yesmaximal, characteristic, complemented
6#5026C211yesmaximal, characteristic, complemented
7#5018C211yesmaximal, characteristic, complemented
8#4890C211yesmaximal, characteristic, complemented
9#4884C211yesmaximal, characteristic, complemented
10#4818C211yesmaximal, characteristic, complemented
11#4811C211yesmaximal, characteristic, complemented
12#4776C211yesmaximal, characteristic
13#4759C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15#4705C211yesmaximal, characteristic, complemented
16#3110—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C22×C4×D3—21no—
19C2×C3×SD16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic, complemented
21C3×C4×D4C2211yescharacteristic
22#946C2211yescharacteristic
23#944C2211yescharacteristic, complemented
24C2×#215C2211yescharacteristic
25D4×Dic3C2211yescharacteristic
26C2×#214C2211yescharacteristic
27C4×#67—21no—
28C2×#204C2211yescharacteristic
29#897C2211yescharacteristic
30#896—21no—
31#893C2211yescharacteristic
32D3×C4⋊C4C2211yescharacteristic
33#889—21no—
34#888C2211yescharacteristic, complemented
35#67⋊C4—21no—
36D3×C22⋊C4—21no—
37C4×D12C2211yescharacteristic
38#872—21no—
39C42×D3C2211yescharacteristic
40C4×Dic6—21no—
41C4×Dic6C2211yescharacteristic, complemented
42C3×#107C2211yescharacteristic, complemented
43C3×#103C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic
45C3×#97C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#821C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#818C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53Dic3⋊C8C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#807C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#533—31no—
61C2×C4×D4—31no—
62#484—31no—
63#484—31no—
64#474—31no—
65#474—31no—
66#461—31no—
67#461—31no—
68#450—31no—
69#450—31no—
70#440—31no—
71#440—31no—
72#440—31no—
73#440D311yescharacteristic, complemented
74#432—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C22×Dic3—21no—
81D3×D4—22no—
82D3×D4—22no—
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3—44no—
88C2×C4×D3D411yescharacteristic
89C2×C4×D3D411yescharacteristic
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C2×Dic6C2311yescharacteristic
93C3×SD16—22no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C42—21no—
101C3×C42C2311yescharacteristic
102#217—21no—
103#215—22no—
104#214—22no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118C4×Dic3—21no—
119C4×Dic3D411yescharacteristic
120C4×Dic3D411yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124#204—22no—
125C22×D4—31no—
126C23×C4—61no—
127C2×SD16—31no—
128C2×SD16—31no—
129C2×SD16—31no—
130C2×SD16C2×D311yescharacteristic
131C4⋊Q8—31no—
132C4⋊Q8—31no—
133#125—61no—
134C22⋊Q8—31no—
135C22⋊Q8—31no—
136C22⋊Q8—61no—
137C4×Q8—31no—
138C4×Q8—61no—
139C4×Q8C2×D311yescharacteristic, complemented
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145C4×D4C2×D311yescharacteristic
146#117—61no—
147C2×C4⋊C4—31no—
148C2×C22⋊C4—61no—
149C2×C42—31no—
150#107—31no—
151#107—31no—
152#107—31no—
153#107C2×D311yescharacteristic, complemented
154C4⋊C8—31no—
155C4⋊C8—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic, complemented
160#102—31no—
161#102—31no—
162#102—31no—
163#102C2×D311yescharacteristic
164C22⋊C8—31no—
165C22⋊C8—31no—
166#97—31no—
167#97C2×D311yescharacteristic, complemented
168C23×C3—21no—
169C22×D3—24no—
170C22×D3—21no—
171C22×D3—42no—
172C22×D3—42no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×D4—21no—
177C3×D4#4112yes—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—42no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—24no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—42no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193D12—21no—
194D12#4112yes—
195C4×D3—22no—
196C4×D3—21no—
197C4×D3—21no—
198C4×D3—22no—
199C4×D3—42no—
200C4×D3—42no—
201C4×D3—42no—
202Dic6—21no—
203Dic6—21no—
204Dic6—42no—
205C3×C8—21no—
206C3×C8—21no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C24—61no—
210C2×Q8—31no—
211C2×Q8—61no—
212C2×Q8C22×D311yescharacteristic
213C2×D4—31no—
214C2×D4—61no—
215C2×D4—62no—
216C2×D4—62no—
217C2×D4C22×D311yescharacteristic
218C22×C4—21no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—124no—
227SD16—22no—
228SD16—62no—
229SD16—62no—
230SD16—62no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—21no—
254C42—31no—
255C42—31no—
256C42—31no—
257C42—61no—
258C42C22×D311yescharacteristic
259C22×C3—22no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—22no—
266C2×D3—48no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3#13712yes—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4C2×#4111yescharacteristic
276C3×C4C2×#4111yescharacteristic
277Dic3—21no—
278Dic3—21no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3AGL(1,ℤ/8)12yes—
283C23—21no—
284C23—31no—
285C23—64no—
286C23—61no—
287C23—122no—
288C23—122no—
289Q8—21no—
290Q8—21no—
291Q8—61no—
292Q8—61no—
293Q8—122no—
294D4—21no—
295D4—32no—
296D4—64no—
297D4—61no—
298D4#23512yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—42no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4D3×D411yescharacteristic
318C2×C4D3×D411yescharacteristic
319C2×C4C23×D311yescharacteristic
320C8—21no—
321C8—21no—
322C8—61no—
323C8—61no—
324C2×C3—42no—
325C2×C3#52511yescharacteristic
326C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
327C2×C3C2×#13711yescharacteristic
328D3—22no—
329D3—42no—
330C22—22no—
331C22—32no—
332C22—41no—
333C22—41no—
334C22—61no—
335C22—61no—
336C22—62no—
337C22—128no—
338C22—121no—
339C22—121no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—21no—
343C4—32no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—61no—
348C4—61no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23511yescharacteristic
353C4D3×#4111yescharacteristic
354C3#311011yescharacteristic, complemented, Sylow
355C2—42no—
356C2—62no—
357C2—122no—
358C2#554711yescharacteristic
359C2D3×AGL(1,ℤ/8)11yescharacteristic
360C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radicalC2×D3×D4C22
hypercenter#440D3
hypocenterC3#3110
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5578C2semidirect
3. C3×#440C2semidirect
5. #5162C2semidirect
6. #5026C2semidirect
7. #5018C2semidirect
8. #4890C2semidirect
9. #4884C2semidirect
10. #4818C2semidirect
11. #4811C2semidirect
13. #4759C2semidirect
14. #4710C2semidirect
15. #4705C2semidirect
20. C3×C4×Q8C22semidirect
23. #944C22semidirect
34. #888C22semidirect
41. C4×Dic6C22semidirect
42. C3×#107C22semidirect
43. C3×#103C22semidirect
45. C3×#97C22semidirect
46. #835C22semidirect
51. #818C22semidirect
52. #816C22semidirect
53. Dic3⋊C8C22semidirect
54. Dic3⋊C8C22semidirect
55. #810C22semidirect
57. #807C22semidirect
58. #807C22semidirect
59. #803C22semidirect
73. #440D3semidirect
139. C4×Q8C2×D3semidirect
153. #107C2×D3semidirect
159. #103C2×D3semidirect
167. #97C2×D3semidirect
354. C3#3110semidirect

Maximal subgroups

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

subgroupindex
#55782
C4×D3×D42
C3×#4402
#51692
#51622
#50262
#50182
#48902
#48842
#48182
#48112
#47762
#47592
#47102
#47052
#31103