Finite Group Databasebeta

← #80512

Subgroups — #80512

Order 384.

398 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#5680C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5171C211yesmaximal, characteristic, complemented
5#5163C211yesmaximal, characteristic, complemented
6#5157C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5009C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11D3×C4⋊C8C211yesmaximal, characteristic, complemented
12#4807C211yesmaximal, characteristic, complemented
13#4793C211yesmaximal, characteristic, complemented
14#4775C211yesmaximal, characteristic, complemented
15#4761C211yesmaximal, characteristic, complemented
16#3247—31nomaximal, Sylow
17C2×#235—21no—
18C2×C3×SD16C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×#124C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#946C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24C2×#215C2211yescharacteristic
25#940C2211yescharacteristic
26#938C2211yescharacteristic
27#938C2211yescharacteristic
28D4×Dic3C2211yescharacteristic
29C2×#213C2211yescharacteristic
30C4×#67—21no—
31C2×D3×C8C2211yescharacteristic
32C2×D3×C8C2211yescharacteristic
33#895C2211yescharacteristic
34#888C2211yescharacteristic
35#885—21no—
36#884—21no—
37#883—21no—
38#67⋊C4—21no—
39#879—21no—
40#877—21no—
41C4×D12C2211yescharacteristic
42C42×D3C2211yescharacteristic
43#870C2211yescharacteristic
44C3×C4⋊C8C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#818C2211yescharacteristic
54#817C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#532—31no—
61#529—31no—
62#478—31no—
63#478—31no—
64#477—31no—
65#476—31no—
66#466—31no—
67#466—31no—
68#465—31no—
69#465—31no—
70#463—31no—
71#463D311yescharacteristic, complemented
72#460—31no—
73#459—31no—
74C2×C4⋊C8—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84#235—44no—
85C2×D12—21no—
86C2×D12C2311yescharacteristic
87C2×C4×D3—21no—
88C2×C4×D3D411yescharacteristic
89C2×C4×D3D411yescharacteristic
90C2×C4×D3C2311yescharacteristic
91C2×Dic6—21no—
92C3×SD16—41no—
93C3×D8—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4C2311yescharacteristic
97C3×C22⋊C4—21no—
98C3×C22⋊C4—21no—
99C3×C22⋊C4—21no—
100C3×C42C2311yescharacteristic
101#217—21no—
102#217—21no—
103#217—21no—
104#215—41no—
105#213—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3C2311yescharacteristic
111C4⋊Dic3C2311yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3D411yescharacteristic
119C4×Dic3D411yescharacteristic
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123D3×C8—41no—
124D3×C8—41no—
125C2×#41—61no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×D8—31no—
129C2×D8C2×D311yescharacteristic
130C22×C8—31no—
131C22×C8—31no—
132C4⋊Q8—31no—
133#127—31no—
134#126—61no—
135#125—31no—
136#124—61no—
137#124C2×D311yescharacteristic
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144C4×Q8—31no—
145C4×D4—31no—
146C4×D4—31no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150#117—61no—
151C2×C42—31no—
152#107—31no—
153#107—31no—
154#106—31no—
155#106—31no—
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159C4⋊C8C2×D311yescharacteristic
160#103—31no—
161#103C2×D311yescharacteristic
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168C23×C3—21no—
169C23×C3—21no—
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—42no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184#67—42no—
185#67—42no—
186#67—42no—
187#67—42no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—42no—
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195D12—21no—
196D12—21no—
197D12—42no—
198C4×D3—22no—
199C4×D3—21no—
200C4×D3—21no—
201C4×D3—22no—
202C4×D3—42no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—124no—
209C2×Q8—61no—
210C2×Q8C22×D311yescharacteristic
211C2×D4—31no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4C22×D311yescharacteristic
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4—61no—
225SD16—41no—
226SD16—121no—
227D8—41no—
228D8—121no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—121no—
232C2×C8—121no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4C22×D311yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—31no—
254C42—31no—
255C42—31no—
256C42—61no—
257C42C22×D311yescharacteristic
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3C22×D411yescharacteristic
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3AGL(1,ℤ/8)12yes—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4H(5,𝔽2)11yescharacteristic
276C3×C42−1+411yescharacteristic
277Dic3—21no—
278Dic3—21no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282Dic3#13512yes—
283C23—21no—
284C23—21no—
285C23—31no—
286C23—61no—
287Q8—21no—
288Q8—21no—
289Q8—122no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—61no—
295D4—61no—
296D4—122no—
297D4—122no—
298D4—122no—
299D4—122no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—42no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4D3×D411yescharacteristic
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—21no—
321C8—61no—
322C8—61no—
323C2×C3—81no—
324C2×C3—81no—
325C2×C3#53711yescharacteristic
326C2×C3C2×#13511yescharacteristic
327C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
328D3—22no—
329D3—81no—
330C22—32no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—61no—
338C22—61no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—21no—
345C4—32no—
346C4—41no—
347C4—41no—
348C4—61no—
349C4—61no—
350C4—121no—
351C4—121no—
352C4—121no—
353C4#100411yescharacteristic
354C4#101011yescharacteristic
355C3#324711yescharacteristic, complemented, Sylow
356C2—62no—
357C2—81no—
358C2—81no—
359C2—241no—
360C2#560811yescharacteristic
361C2D3×#13511yescharacteristic
362C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3247
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. #5680C2semidirect
2. #5548C2semidirect
3. C3×#463C2semidirect
4. #5171C2semidirect
5. #5163C2semidirect
6. #5157C2semidirect
7. #5150C2semidirect
8. #5057C2semidirect
9. #5009C2semidirect
10. #4847C2semidirect
11. D3×C4⋊C8C2semidirect
12. #4807C2semidirect
13. #4793C2semidirect
14. #4775C2semidirect
15. #4761C2semidirect
71. #463D3semidirect
355. C3#3247semidirect

Maximal subgroups

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

subgroupindex
#56802
#55482
C3×#4632
#51712
#51632
#51572
#51502
#50572
#50092
#48472
D3×C4⋊C82
#48072
#47932
#47752
#47612
#32473