Finite Group Databasebeta

← #83375

Subgroups — #83375

Order 384.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#1004C211yesmaximal, characteristic, complemented
2C3×#534C211yesmaximal, characteristic, complemented
3#5804C212yesmaximal, complemented
4C2×#834C211yesmaximal, characteristic, complemented
5#5053C214yesmaximal, complemented
6#5032C214yesmaximal, complemented
7C2×#806C212yesmaximal, complemented
8#3026—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C22×C3×D4C2211yescharacteristic, complemented
11C22×#67—21no—
12C22×#67C2212yes—
13#1004—22no—
14#1004—24no—
15#1004C2214yes—
16C2×#237—22no—
17C2×D3×D4—22no—
18C2×#235—21no—
19C3×#123—21no—
20C3×#123C2212yescomplemented
21C3×#121—22no—
22C3×#120—21no—
23C3×#120C2212yescomplemented
24C2×C3×C22⋊C4C2211yescharacteristic, complemented
25#953C2214yescomplemented
26C2×#217C2212yescomplemented
27#939—24no—
28#937—22no—
29#935—22no—
30#929C2214yescomplemented
31#834C2214yescomplemented
32#806C2218yescomplemented
33C2×H(5,𝔽2)—31no—
34#534—32no—
35#534D311yescharacteristic, complemented
36#458—34no—
37B(4,𝔽2)—34no—
38C2×#99—31no—
39C2×#99—32no—
40C24×C3D411yescharacteristic, complemented
41C24×C3C2311yescharacteristic
42C24×C3C2311yescharacteristic, complemented
43C23×D3—21no—
44C23×D3D412yescomplemented
45C2×C3×D4—22no—
46C2×C3×D4—22no—
47C2×C3×D4—22no—
48C2×C3×D4—22no—
49C2×C3×D4—21no—
50C2×C3×D4—21no—
51C2×C3×D4—22no—
52C2×C3×D4D412yescomplemented
53C2×C3×D4C2312yes—
54C22×C3×C4—21no—
55C22×C3×C4D411yescharacteristic, complemented
56C22×C3×C4C2311yescharacteristic, complemented
57C2×#67—22no—
58C2×#67—24no—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—24no—
64C2×#67—24no—
65C2×#67D414yescomplemented
66C2×#67C2314yes—
67C22×Dic3—21no—
68C22×Dic3—22no—
69C22×Dic3D412yescomplemented
70#237—28no—
71#237—28no—
72D3×D4—28no—
73D3×D4—28no—
74#235—24no—
75#235—24no—
76C2×D12—21no—
77C2×C4×D3—22no—
78C2×Dic6—21no—
79C3×C4⋊C4—22no—
80C3×C22⋊C4—21no—
81C3×C22⋊C4—22no—
82C3×C22⋊C4—22no—
83C3×C22⋊C4C2312yescomplemented
84#217—22no—
85#217—24no—
86#217C2314yescomplemented
87(C2×D3)⋊C4—24no—
88Dic3⋊C4—24no—
89H(5,𝔽2)—34no—
90H(5,𝔽2)—64no—
91H(5,𝔽2)—62no—
92C2×#41—62no—
93C2×#41—61no—
94C22×D4—32no—
95C22×D4—62no—
96C22×D4—61no—
97C22×D4C2×D311yescharacteristic
98C22×D4C2×D311yescharacteristic, complemented
99#123—21no—
100#123—34no—
101#123—62no—
102#123C2×D312yescomplemented
103#121—22no—
104#121—64no—
105#120—21no—
106#120—34no—
107#120—62no—
108#120C2×D312yescomplemented
109C2×C22⋊C4—32no—
110C2×C22⋊C4C2×D311yescharacteristic, complemented
111#99—34no—
112#99—38no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—22no—
116C23×C3—21no—
117C23×C3—81no—
118C23×C3—81no—
119C23×C3—81no—
120C23×C3C2×D412yes—
121C23×C3C2×D412yes—
122C23×C3C2×D412yes—
123C23×C3C2×D411yescharacteristic
124C23×C3C2×D411yescharacteristic
125C23×C3C2411yescharacteristic
126C22×D3—22no—
127C22×D3—24no—
128C22×D3—22no—
129C22×D3—82no—
130C22×D3—82no—
131C22×D3C2×D414yes—
132C3×D4—24no—
133C3×D4—24no—
134C3×D4—41no—
135C3×D4—42no—
136C3×D4—41no—
137C3×D4—82no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—81no—
144C2×C3×C4C2×D412yes—
145#67—28no—
146#67—24no—
147#67—24no—
148#67—24no—
149#67—24no—
150#67—84no—
151C2×Dic3—24no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—24no—
155C2×Dic3—82no—
156C2×Dic3C2×D414yes—
157D12—24no—
158C4×D3—28no—
159Dic6—24no—
160C24—32no—
161C24—61no—
162C24#6711yescharacteristic, complemented
163C24C22×D311yescharacteristic
164C24C22×D311yescharacteristic, complemented
165#41—64no—
166#41—68no—
167#41—68no—
168#41—64no—
169C2×Q8—61no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—22no—
177C2×D4—34no—
178C2×D4—34no—
179C2×D4—61no—
180C2×D4—68no—
181C2×D4—68no—
182C2×D4—62no—
183C2×D4—64no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—64no—
189C2×D4—64no—
190C2×D4#6712yescomplemented
191C2×D4C22×D312yes—
192C22×C4—21no—
193C22×C4—32no—
194C22×C4—62no—
195C22×C4—61no—
196C22×C4—62no—
197C22×C4#6711yescharacteristic, complemented
198C22×C4C22×D311yescharacteristic, complemented
199C4⋊C4—22no—
200C4⋊C4—64no—
201C22⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—34no—
205C22⋊C4—64no—
206C22⋊C4—64no—
207C22⋊C4—62no—
208C22⋊C4C22×D312yescomplemented
209C22×C3—22no—
210C22×C3—42no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—41no—
220C22×C3—81no—
221C22×C3#12011yescharacteristic
222C22×C3#12012yes—
223C22×C3#12011yescharacteristic
224C22×C3C22×D412yes—
225C22×C3C22×D411yescharacteristic
226C2×D3—24no—
227C2×D3—44no—
228C2×D3—44no—
229C2×D3—82no—
230C2×D3—84no—
231C3×C4—22no—
232C3×C4—42no—
233C3×C4—41no—
234Dic3—24no—
235Dic3—44no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—21no—
240C23—34no—
241C23—64no—
242C23—62no—
243C23—62no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—242no—
248C23—242no—
249C23D3×D412yes—
250C23D3×D412yes—
251C23C2×#6712yes—
252C23C2×#6711yescharacteristic
253C23C2×#6711yescharacteristic
254C23C23×D311yescharacteristic
255Q8—64no—
256D4—24no—
257D4—24no—
258D4—41no—
259D4—41no—
260D4—42no—
261D4—64no—
262D4—68no—
263D4—64no—
264D4—64no—
265D4—64no—
266D4—64no—
267D4—82no—
268D4—244no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—34no—
275C2×C4—68no—
276C2×C4—64no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—64no—
280C2×C4—81no—
281C2×C4—242no—
282C2×C4C2×#6712yes—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—22no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—81no—
290C2×C3C2×#12011yescharacteristic
291D3—44no—
292C22—22no—
293C22—41no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—44no—
302C22—41no—
303C22—64no—
304C22—81no—
305C22—124no—
306C22—124no—
307C22—242no—
308C22—244no—
309C22#93711yescharacteristic
310C22#93711yescharacteristic
311C22#93712yes—
312C22C2×D3×D412yes—
313C22C22×#6711yescharacteristic
314C4—22no—
315C4—42no—
316C4—41no—
317C4—64no—
318C4—124no—
319C3#302611yescharacteristic, complemented, Sylow
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—124no—
328C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC3×#534C2
socleC2×C3C2×#120
hypercenter#534D3
hypocenterC3#3026
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C3×#534C2semidirect
3. #5804C2semidirect
4. C2×#834C2semidirect
5. #5053C2semidirect
6. #5032C2semidirect
7. C2×#806C2semidirect
10. C22×C3×D4C22semidirect
20. C3×#123C22semidirect
23. C3×#120C22semidirect
24. C2×C3×C22⋊C4C22semidirect
25. #953C22semidirect
26. C2×#217C22semidirect
30. #929C22semidirect
31. #834C22semidirect
32. #806C22semidirect
35. #534D3semidirect
40. C24×C3D4semidirect
42. C24×C3C23semidirect
44. C23×D3D4semidirect
52. C2×C3×D4D4semidirect
55. C22×C3×C4D4semidirect
56. C22×C3×C4C23semidirect
65. C2×#67D4semidirect
69. C22×Dic3D4semidirect
83. C3×C22⋊C4C23semidirect
86. #217C23semidirect
98. C22×D4C2×D3semidirect
102. #123C2×D3semidirect
108. #120C2×D3semidirect
110. C2×C22⋊C4C2×D3semidirect
162. C24#67semidirect
164. C24C22×D3semidirect
190. C2×D4#67semidirect
197. C22×C4#67semidirect
198. C22×C4C22×D3semidirect
208. C22⋊C4C22×D3semidirect
319. C3#3026semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C3×#5342
#58042
C2×#8342
#50532
#50322
C2×#8062
#30263