Finite Group Databasebeta

← #82598

Subgroups — #82598

Order 384.

688 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 141 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×C3×#120C211yesmaximal, characteristic
2C2×#936C211yesmaximal, characteristic, complemented
3#5247C212yesmaximal, complemented
4#5222C212yesmaximal, complemented
5#5220C211yesmaximal, characteristic, complemented
6#5220C212yesmaximal, complemented
7#4952C212yesmaximal, complemented
8#4947C212yesmaximal, complemented
9#4944C212yesmaximal, complemented
10#2684—31nomaximal, Sylow
11C25×C3C2211yescharacteristic
12C22×C3×D4C2212yes—
13C22×C3×D4C2211yescharacteristic
14C23×Dic3C2211yescharacteristic, complemented
15C22×Dic6C2211yescharacteristic, complemented
16C3×#120—24no—
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yescomplemented
26C2×#217C2211yescharacteristic, complemented
27#936—24no—
28C2×Dic3⋊C4C2212yescomplemented
29C2×C4×Dic3C2211yescharacteristic, complemented
30C3⋊#95C2212yescomplemented
31C3⋊#95C2212yescomplemented
32C3⋊#95C2212yescomplemented
33C3⋊#95C2212yescomplemented
34C2×#124—31no—
35C2×#120D311yescharacteristic
36#397—32no—
37#394—32no—
38#393—32no—
39#388—32no—
40#386—31no—
41#386—32no—
42#379—32no—
43C24×C3—22no—
44C24×C3—24no—
45C24×C3—42no—
46C24×C3—41no—
47C24×C3—41no—
48C24×C3C2311yescharacteristic
49C24×C3C2311yescharacteristic
50C24×C3C2312yes—
51C2×C3×D4—24no—
52C2×C3×D4—24no—
53C2×C3×D4—41no—
54C2×C3×D4—41no—
55C2×C3×D4—42no—
56C2×C3×D4D414yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×Dic3—41no—
60C22×Dic3—42no—
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C2×Dic6—22no—
66C2×Dic6—41no—
67C2×Dic6—41no—
68C3×C22⋊C4—22no—
69C3×C22⋊C4—24no—
70#217—24no—
71#217—24no—
72#217—24no—
73#217—42no—
74#217—42no—
75#217—42no—
76#217—42no—
77#217—41no—
78Dic3⋊C4—42no—
79C4×Dic3—22no—
80C25C2×D311yescharacteristic
81C22×Q8—31no—
82C22×D4C2×D312yes—
83C22×D4C2×D311yescharacteristic
84C23×C4—31no—
85#124—64no—
86#120—24no—
87C2×C4⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4C2×D312yes—
97C2×C22⋊C4C2×D311yescharacteristic
98C2×C42—31no—
99#95—32no—
100#95—32no—
101#95—32no—
102#95—32no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—21no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—21no—
113C23×C3—22no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—22no—
117C23×C3—48no—
118C23×C3—44no—
119C23×C3—44no—
120C23×C3—48no—
121C23×C3—44no—
122C23×C3C2×D412yes—
123C23×C3C2×D412yes—
124C23×C3#4114yes—
125C23×C3C2411yescharacteristic
126C3×D4—44no—
127C3×D4—48no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—44no—
147Dic6—44no—
148C24—24no—
149C24—22no—
150C24—42no—
151C24—41no—
152C24—41no—
153C24C22×D312yes—
154C24C22×D311yescharacteristic
155C24C22×D311yescharacteristic
156C2×Q8—62no—
157C2×Q8—121no—
158C2×Q8—121no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—42no—
164C2×D4#6714yes—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4—121no—
170C22×C4—122no—
171C22×C4C22×D312yes—
172C22×C4C22×D311yescharacteristic
173C4⋊C4—122no—
174C22⋊C4—22no—
175C22⋊C4—24no—
176C22⋊C4—64no—
177C22⋊C4—64no—
178C22⋊C4—64no—
179C22⋊C4—121no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22⋊C4—122no—
183C22⋊C4—122no—
184C42—62no—
185C22×C3—22no—
186C22×C3—22no—
187C22×C3—24no—
188C22×C3—24no—
189C22×C3—24no—
190C22×C3—24no—
191C22×C3—24no—
192C22×C3—416no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—42no—
203C22×C3C22×D411yescharacteristic
204C22×C3C2×#4112yes—
205C22×C3H(5,𝔽2)11yescharacteristic
206C22×C3H(5,𝔽2)11yescharacteristic
207C22×C3H(5,𝔽2)12yes—
208C3×C4—22no—
209C3×C4—42no—
210Dic3—42no—
211Dic3—42no—
212Dic3—42no—
213Dic3—42no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—48no—
229C23—44no—
230C23—44no—
231C23—48no—
232C23—44no—
233C23#23714yes—
234C23C2×#6712yes—
235C23C2×#6712yes—
236C23C23×D311yescharacteristic
237Q8—124no—
238D4—44no—
239D4—48no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—124no—
258C2×C4C2×#6712yes—
259C2×C3—42no—
260C2×C3—42no—
261C2×C3—44no—
262C2×C3#53411yescharacteristic
263C2×C3#53812yes—
264C2×C3#54612yes—
265C2×C3#55111yescharacteristic
266C2×C3H(3,𝔽4)11yescharacteristic
267C22—22no—
268C22—22no—
269C22—24no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—24no—
274C22—416no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22C2×#23712yes—
286C22#100411yescharacteristic
287C22#100411yescharacteristic
288C22#100412yes—
289C22C22×#6711yescharacteristic
290C4—22no—
291C4—42no—
292C4—122no—
293C4—122no—
294C4—122no—
295C4—122no—
296C3#268411yescharacteristic, complemented, Sylow
297C2—42no—
298C2—44no—
299C2—42no—
300C2#558711yescharacteristic
301C2#559312yes—
302C2#559411yescharacteristic
303C2#580411yescharacteristic
304C2#580612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2684
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C2×#936C2semidirect
3. #5247C2semidirect
4. #5222C2semidirect
5. #5220C2semidirect
6. #5220C2semidirect
7. #4952C2semidirect
8. #4947C2semidirect
9. #4944C2semidirect
14. C23×Dic3C22semidirect
15. C22×Dic6C22semidirect
25. C2×#217C22semidirect
26. C2×#217C22semidirect
28. C2×Dic3⋊C4C22semidirect
29. C2×C4×Dic3C22semidirect
30. C3⋊#95C22semidirect
31. C3⋊#95C22semidirect
32. C3⋊#95C22semidirect
33. C3⋊#95C22semidirect
296. C3#2684semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C2×#9362
#52472
#52222
#52202
#49522
#49472
#49442
#26843