Finite Group Databasebeta

← #73821

Subgroups — #73821

Order 384.

808 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#953C211yesmaximal, characteristic
2C22×#217C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C3×#379C211yesmaximal, characteristic, complemented
5#4958C211yesmaximal, characteristic, complemented
6#4957C212yesmaximal, complemented
7#4956C212yesmaximal, complemented
8#4946C211yesmaximal, characteristic, complemented
9#4945C212yesmaximal, complemented
10#4944C212yesmaximal, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#2321—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2212yes—
16C23×Dic3C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic, complemented
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×C3×C22⋊C4C2212yes—
23#953C418yescomplemented
24C2×#217—24no—
25C2×#217C2212yes—
26C2×#217C2212yes—
27C2×#217C2211yescharacteristic
28C2×#217C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
30C2×(C2×D3)⋊C4C2212yescomplemented
31C2×Dic3⋊C4C2211yescharacteristic, complemented
32C2×Dic3⋊C4C2212yescomplemented
33C2×C4×Dic3C2211yescharacteristic, complemented
34C2×C4×Dic3C2212yescomplemented
35#67⋊C4—24no—
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C2×#120—31no—
40C2×C4×D4—31no—
41C22×C22⋊C4—31no—
42#390—32no—
43#388—32no—
44#388—32no—
45#386—32no—
46#386—31no—
47#385—31no—
48#379D311yescharacteristic, complemented
49C4×C22⋊C4—31no—
50C24×C3—22no—
51C24×C3—22no—
52C24×C3—42no—
53C24×C3—42no—
54C24×C3C2×C414yescomplemented
55C24×C3C2312yes—
56C24×C3C2311yescharacteristic
57C23×D3C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—42no—
64C2×#67—42no—
65C2×#67C2×C414yes—
66C2×#67C2×C418yescomplemented
67C22×Dic3—22no—
68C22×Dic3—42no—
69C22×Dic3—42no—
70C22×Dic3D414yescomplemented
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2311yescharacteristic
76C2×C4×D3—22no—
77C2×C4×D3—42no—
78C3×C22⋊C4—22no—
79C3×C22⋊C4—22no—
80C3×C22⋊C4—42no—
81C3×C22⋊C4—42no—
82#217—24no—
83#217—24no—
84#217C2×C418yescomplemented
85#217C2×C414yescomplemented
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—42no—
88Dic3⋊C4—22no—
89Dic3⋊C4—42no—
90C4×Dic3—22no—
91C4×Dic3—42no—
92C25C2×D311yescharacteristic
93C22×D4—31no—
94C22×D4—32no—
95C23×C4—31no—
96C23×C4—31no—
97C23×C4—31no—
98#120—38no—
99C4×D4—64no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C22⋊C4—31no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—64no—
109C2×C22⋊C4C2×D311yescharacteristic
110C2×C22⋊C4C2×D311yescharacteristic
111C2×C22⋊C4C2×D312yes—
112C2×C22⋊C4C2×D312yes—
113C2×C42—31no—
114C2×C42—32no—
115#95—32no—
116#95—32no—
117#95—32no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—28no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—48no—
128C23×C3—44no—
129C23×C3—44no—
130C23×C3—48no—
131C23×C3C22×C414yes—
132C23×C3C22×C412yes—
133C23×C3C2×D412yes—
134C23×C3#4112yes—
135C23×C3C2411yescharacteristic
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3C22×C412yes—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148#67—28no—
149#67—48no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—44no—
160C2×Dic3—44no—
161C2×Dic3C22×C412yes—
162C2×Dic3C22×C414yes—
163C2×Dic3C2×D412yes—
164C2×Dic3C2×D412yes—
165C4×D3—44no—
166C24—22no—
167C24—22no—
168C24—31no—
169C24—42no—
170C24—42no—
171C24C4×D314yescomplemented
172C24C22×D312yes—
173C24C22×D311yescharacteristic
174C2×D4—34no—
175C2×D4—38no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—122no—
179C2×D4—122no—
180C22×C4—31no—
181C22×C4—32no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—34no—
185C22×C4—32no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4—122no—
189C22×C4—122no—
190C22×C4—122no—
191C22×C4C22×D312yes—
192C22×C4C22×D311yescharacteristic
193C22×C4C22×D311yescharacteristic
194C4⋊C4—62no—
195C4⋊C4—122no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—38no—
199C22⋊C4—34no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—62no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C22⋊C4—122no—
206C42—62no—
207C42—122no—
208C22×C3—24no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—24no—
215C22×C3—416no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3C4×D414yes—
221C22×C3C23×C411yescharacteristic
222C22×C3C22×D411yescharacteristic
223C22×C3C2×#4111yescharacteristic
224C22×C3H(5,𝔽2)12yes—
225C22×C3H(5,𝔽2)12yes—
226C2×D3—22no—
227C2×D3—24no—
228C2×D3—41no—
229C2×D3—41no—
230C2×D3—42no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4—42no—
234Dic3—22no—
235Dic3—24no—
236Dic3—22no—
237Dic3—22no—
238Dic3—42no—
239C23—22no—
240C23—22no—
241C23—28no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—32no—
249C23—48no—
250C23—44no—
251C23—44no—
252C23—48no—
253C23—61no—
254C23—61no—
255C23—62no—
256C23—62no—
257C23C2×C4×D314yes—
258C23C2×C4×D312yes—
259C23D3×D412yes—
260C23#23712yes—
261C23C23×D311yescharacteristic
262D4—68no—
263D4—128no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—32no—
272C2×C4—34no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—124no—
285C2×C4—124no—
286C2×C4—124no—
287C2×C3—24no—
288C2×C3—44no—
289C2×C3C2×C4×D411yescharacteristic
290C2×C3#51812yes—
291C2×C3#53411yescharacteristic
292C2×C3#53511yescharacteristic
293C2×C3#53812yes—
294D3—42no—
295C22—24no—
296C22—22no—
297C22—22no—
298C22—24no—
299C22—24no—
300C22—24no—
301C22—24no—
302C22—416no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—62no—
308C22—64no—
309C22—121no—
310C22—121no—
311C22—122no—
312C22#67⋊C414yes—
313C22C22×C4×D311yescharacteristic
314C22C2×D3×D411yescharacteristic
315C22C2×#23711yescharacteristic
316C22#100412yes—
317C22#100412yes—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—62no—
322C4—64no—
323C4—62no—
324C4—62no—
325C4—122no—
326C3#232111yescharacteristic, complemented, Sylow
327C2—24no—
328C2—44no—
329C2—122no—
330C2C2×#67⋊C411yescharacteristic
331C2#548612yes—
332C2#558911yescharacteristic
333C2#559211yescharacteristic
334C2#559312yes—
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#379C2
socle, cosocleC23×C3C24
prime radicalC2×#953C2
hypercenter#379D3
hypocenterC3#2321
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C22×#217C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#379C2semidirect
5. #4958C2semidirect
6. #4957C2semidirect
7. #4956C2semidirect
8. #4946C2semidirect
9. #4945C2semidirect
10. #4944C2semidirect
11. Dic3×C22⋊C4C2semidirect
18. C22×C4×D3C22semidirect
23. #953C4semidirect
29. C2×(C2×D3)⋊C4C22semidirect
30. C2×(C2×D3)⋊C4C22semidirect
31. C2×Dic3⋊C4C22semidirect
32. C2×Dic3⋊C4C22semidirect
33. C2×C4×Dic3C22semidirect
34. C2×C4×Dic3C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
48. #379D3semidirect
54. C24×C3C2×C4semidirect
66. C2×#67C2×C4semidirect
70. C22×Dic3D4semidirect
84. #217C2×C4semidirect
85. #217C2×C4semidirect
171. C24C4×D3semidirect
326. C3#2321semidirect

Maximal subgroups

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

subgroupindex
C2×#9532
C22×#2172
C2×#67⋊C42
C3×#3792
#49582
#49572
#49562
#49462
#49452
#49442
Dic3×C22⋊C42
#23213