Finite Group Databasebeta

← #79994

Subgroups — #79994

Order 384.

532 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#940C211yesmaximal, characteristic, complemented
2#5596C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5151C212yesmaximal, complemented
6#5150C212yesmaximal, complemented
7#5036C211yesmaximal, characteristic, complemented
8#4786C212yesmaximal, complemented
9#4760C212yesmaximal, complemented
10#4732C211yesmaximal, characteristic, complemented
11#4718C211yesmaximal, characteristic, complemented
12#3205—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C22×#67—21no—
16C2×#237—21no—
17C22×D12C2211yescharacteristic
18C2×C3×D8C2212yescomplemented
19C3×#121C2211yescharacteristic
20#940—21no—
21#940—21no—
22#940C2212yes—
23#940C2212yes—
24#940C2211yescharacteristic, complemented
25#939—21no—
26D4×Dic3C2211yescharacteristic, complemented
27C2×#213C2212yescomplemented
28C2×#213C2212yescomplemented
29#930C2211yescharacteristic
30C2×C4×Dic3C2211yescharacteristic
31C2×D24C2212yescomplemented
32#893C2211yescharacteristic, complemented
33#888C2211yescharacteristic, complemented
34#885—21no—
35#67⋊C4—21no—
36C3×#102C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#832C2212yescomplemented
39#830C2211yescharacteristic
40#821C2212yescomplemented
41Dic3⋊C8C2212yescomplemented
42C8×Dic3C2212yescomplemented
43#809C2212yescomplemented
44#532—31no—
45C2×#127—31no—
46#493—32no—
47#486—32no—
48#459—32no—
49#459—32no—
50#447—31no—
51#447—31no—
52#447—31no—
53#447D311yescharacteristic, complemented
54#431—31no—
55C24×C3—21no—
56C23×D3—21no—
57C2×C3×D4—21no—
58C2×C3×D4—21no—
59C2×C3×D4—42no—
60C2×C3×D4C2312yes—
61C2×C3×D4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×#67—21no—
65C2×#67—21no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—22no—
69C2×#67—22no—
70C2×#67—44no—
71C2×#67—44no—
72C22×Dic3—21no—
73C22×Dic3D411yescharacteristic
74C22×Dic3D411yescharacteristic
75#237—44no—
76C2×D12—21no—
77C2×D12—42no—
78C2×D12C2312yes—
79C2×D12C2311yescharacteristic
80C2×C4×D3—21no—
81C2×Dic6—21no—
82C3×D8—42no—
83C2×C3×C8C2312yes—
84C3×C4⋊C4C2311yescharacteristic
85C3×C22⋊C4—21no—
86#217—21no—
87#213—42no—
88#213—42no—
89(C2×D3)⋊C4—21no—
90(C2×D3)⋊C4—21no—
91C4⋊Dic3C2311yescharacteristic
92Dic3⋊C4—21no—
93C4×Dic3D411yescharacteristic
94C4×Dic3D411yescharacteristic
95C4×Dic3C2311yescharacteristic
96C4×Dic3C2311yescharacteristic
97C2×C3⋊C8C2312yes—
98D24—42no—
99C2×#41—61no—
100C22×D4—31no—
101C22×D4—61no—
102C22×D4—61no—
103C22×D4C2×D311yescharacteristic
104C2×D8—32no—
105C2×D8—32no—
106C2×D8—32no—
107C2×D8C2×D312yescomplemented
108C4⋊Q8—31no—
109#127—32no—
110#127—32no—
111#127—31no—
112#127—61no—
113#127—61no—
114#124—61no—
115#121—31no—
116#121—61no—
117#121C2×D311yescharacteristic
118C4×D4—31no—
119C4×D4—31no—
120C4×D4—61no—
121C2×C42—31no—
122C4⋊C8—32no—
123#102—32no—
124#102—32no—
125#102—32no—
126#102C2×D312yescomplemented
127C22⋊C8—31no—
128C22⋊C8C2×D311yescharacteristic
129C4×C8—32no—
130C23×C3—21no—
131C23×C3—22no—
132C23×C3—42no—
133C23×C3—42no—
134C23×C3—42no—
135C23×C3C2×D411yescharacteristic
136C22×D3—21no—
137C22×D3—22no—
138C22×D3—42no—
139C22×D3—42no—
140C22×D3—42no—
141C3×D4—22no—
142C3×D4—22no—
143C3×D4—22no—
144C3×D4—42no—
145C3×D4—42no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4C2×D411yescharacteristic
149C2×C3×C4C2411yescharacteristic
150#67—44no—
151#67—42no—
152#67—42no—
153#67—44no—
154#67—44no—
155#67—42no—
156#67—44no—
157C2×Dic3—22no—
158C2×Dic3—21no—
159C2×Dic3—42no—
160C2×Dic3D814yescomplemented
161C2×Dic3C2×D411yescharacteristic
162C2×Dic3C2×D411yescharacteristic
163C2×Dic3C2×D411yescharacteristic
164C2×Dic3C2×D411yescharacteristic
165D12—22no—
166D12—22no—
167D12—22no—
168D12—42no—
169C4×D3—42no—
170Dic6—42no—
171C3×C8—22no—
172C3⋊C8—22no—
173C24—21no—
174C24—61no—
175#41—124no—
176C2×Q8—61no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—32no—
180C2×D4—31no—
181C2×D4—42no—
182C2×D4—61no—
183C2×D4—61no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—61no—
189C2×D4—61no—
190C2×D4—122no—
191C2×D4—124no—
192C2×D4—124no—
193C2×D4C22×D312yes—
194C2×D4C22×D311yescharacteristic
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4C22×D311yescharacteristic
200D8—42no—
201D8—122no—
202D8—122no—
203D8—122no—
204C2×C8—32no—
205C2×C8C22×D312yes—
206C4⋊C4—31no—
207C4⋊C4—61no—
208C4⋊C4C22×D311yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—61no—
211C22⋊C4—61no—
212C22⋊C4—61no—
213C42—31no—
214C42—31no—
215C42—31no—
216C42—31no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—42no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—41no—
225C22×C3—84no—
226C22×C3C2×D812yes—
227C22×C3C22×D411yescharacteristic
228C2×D3—42no—
229C2×D3—41no—
230C2×D3—41no—
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—41no—
234C2×D3—84no—
235C3×C4—21no—
236C3×C4—41no—
237C3×C4H(5,𝔽2)12yes—
238Dic3—21no—
239Dic3—41no—
240Dic3C2×D812yes—
241Dic3C2×D812yes—
242Dic3AGL(1,ℤ/8)12yes—
243C23—21no—
244C23—22no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—61no—
249C23—62no—
250C23—122no—
251C23—122no—
252C23—122no—
253C23D3×D411yescharacteristic
254Q8—122no—
255D4—22no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—42no—
260D4—62no—
261D4—62no—
262D4—62no—
263D4—122no—
264D4—124no—
265D4—124no—
266D4—124no—
267D4—122no—
268D4—122no—
269D4—124no—
270D4—122no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—34no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4D3×D411yescharacteristic
283C2×C4C23×D311yescharacteristic
284C8—22no—
285C8—62no—
286C2×C3—22no—
287C2×C3—81no—
288C2×C3—82no—
289C2×C3#53511yescharacteristic
290C2×C3C22×D811yescharacteristic
291C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
292D3—81no—
293D3—82no—
294C22—21no—
295C22—21no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—41no—
302C22—84no—
303C22—121no—
304C22—122no—
305C22—122no—
306C22—121no—
307C22—122no—
308C22—121no—
309C22—244no—
310C22D3×D812yes—
311C22C2×D3×D411yescharacteristic
312C4—21no—
313C4—32no—
314C4—32no—
315C4—32no—
316C4—41no—
317C4—61no—
318C4—121no—
319C4#100412yes—
320C3#320511yescharacteristic, complemented, Sylow
321C2—22no—
322C2—82no—
323C2—81no—
324C2—242no—
325C2—241no—
326C2#559211yescharacteristic
327C2C2×D3×D811yescharacteristic
328C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3205
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#940C2semidirect
2. #5596C2semidirect
3. C3×#447C2semidirect
4. #5217C2semidirect
5. #5151C2semidirect
6. #5150C2semidirect
7. #5036C2semidirect
8. #4786C2semidirect
9. #4760C2semidirect
10. #4732C2semidirect
11. #4718C2semidirect
18. C2×C3×D8C22semidirect
24. #940C22semidirect
26. D4×Dic3C22semidirect
27. C2×#213C22semidirect
28. C2×#213C22semidirect
31. C2×D24C22semidirect
32. #893C22semidirect
33. #888C22semidirect
36. C3×#102C22semidirect
38. #832C22semidirect
40. #821C22semidirect
41. Dic3⋊C8C22semidirect
42. C8×Dic3C22semidirect
43. #809C22semidirect
53. #447D3semidirect
107. C2×D8C2×D3semidirect
126. #102C2×D3semidirect
160. C2×Dic3D8semidirect
320. C3#3205semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
#55962
C3×#4472
#52172
#51512
#51502
#50362
#47862
#47602
#47322
#47182
#32053