Finite Group Databasebeta

← #74906

Subgroups — #74906

Order 384.

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

normal subgroups only

This group has 702 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×#937C211yesmaximal, characteristic
2C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
3C3×#386C211yesmaximal, characteristic, complemented
4#5220C211yesmaximal, characteristic, complemented
5#5210C211yesmaximal, characteristic, complemented
6#4961C212yesmaximal, complemented
7#4959C212yesmaximal, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4950C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4674C212yesmaximal, complemented
12#2622—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yescomplemented
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yescomplemented
24#937—24no—
25C2×(C2×D3)⋊C4—24no—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×C4⋊Dic3C2212yescomplemented
33C2×Dic3⋊C4C2212yescomplemented
34C3×#95C2212yescomplemented
35C3⋊#95C2212yescomplemented
36C3⋊#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C2×#120—31no—
39C22×C22⋊C4—31no—
40#397—32no—
41#397—32no—
42#394—32no—
43#394—32no—
44#386—31no—
45#386D311yescharacteristic, complemented
46#385—31no—
47#385—31no—
48#379—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C23×D3—22no—
52C23×D3—22no—
53C23×D3—42no—
54C23×D3—42no—
55C23×D3D414yes—
56C23×D3C2311yescharacteristic
57C23×D3C2311yescharacteristic
58C2×C3×D4—22no—
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—42no—
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2312yes—
65C22×C3×C4C2312yes—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—42no—
69C2×#67—42no—
70C22×Dic3—22no—
71C22×Dic3—42no—
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C3×C22⋊C4—22no—
77C3×C22⋊C4—42no—
78#217—22no—
79#217—22no—
80#217—42no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—42no—
86(C2×D3)⋊C4—42no—
87C4⋊Dic3—42no—
88Dic3⋊C4—42no—
89C25—31no—
90C22×D4—31no—
91C22×D4—31no—
92C22×D4C2×D311yescharacteristic
93C23×C4—31no—
94C23×C4C2×D311yescharacteristic
95#120—64no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—31no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—64no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D312yescomplemented
110#95—32no—
111#95—32no—
112#95—32no—
113#95C2×D312yescomplemented
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3C2×D412yes—
122C23×C3#4112yes—
123C23×C3#4112yes—
124C23×C3C2411yescharacteristic
125C22×D3—28no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—44no—
135C22×D3—44no—
136C22×D3—48no—
137C22×D3—48no—
138C22×D3C2×D412yes—
139C22×D3C2×D412yes—
140C3×D4—44no—
141C2×C3×C4—22no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—44no—
151#67—44no—
152#67—44no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—22no—
163C2×Dic3—44no—
164C24—31no—
165C24—31no—
166C24—34no—
167C24—62no—
168C24—62no—
169C24—122no—
170C24—122no—
171C24C22×D311yescharacteristic
172C24C22×D311yescharacteristic
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—122no—
179C2×D4—122no—
180C22×C4—22no—
181C22×C4—32no—
182C22×C4—32no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—42no—
186C22×C4—62no—
187C22×C4—122no—
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D312yes—
190C22×C4C22×D312yes—
191C4⋊C4—122no—
192C4⋊C4—122no—
193C22⋊C4—22no—
194C22⋊C4—42no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—62no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22⋊C4—122no—
202C22⋊C4—122no—
203C22⋊C4—122no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3#12314yes—
214C22×C3C22×D411yescharacteristic
215C22×C3C2×#4111yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3H(5,𝔽2)12yes—
218C22×C3H(5,𝔽2)12yes—
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—22no—
224C2×D3—22no—
225C2×D3—416no—
226C2×D3—42no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C3×C4—41no—
231C3×C4—42no—
232C3×C4—42no—
233Dic3—42no—
234Dic3—41no—
235Dic3—41no—
236Dic3—42no—
237C23—22no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—32no—
245C23—32no—
246C23—68no—
247C23—61no—
248C23—61no—
249C23—61no—
250C23—61no—
251C23—62no—
252C23—62no—
253C23—62no—
254C23—62no—
255C23—124no—
256C23—124no—
257C23—128no—
258C23—128no—
259C23#23512yes—
260C23D3×D412yes—
261C23#23712yes—
262C23C23×D311yescharacteristic
263D4—44no—
264D4—124no—
265D4—124no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—44no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—124no—
287C2×C3—24no—
288C2×C3—81no—
289C2×C3C2×#12311yescharacteristic
290C2×C3#53411yescharacteristic
291C2×C3#53411yescharacteristic
292C2×C3#53812yes—
293C2×C3#53812yes—
294D3—42no—
295D3—42no—
296C22—22no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—62no—
306C22—62no—
307C22—62no—
308C22—62no—
309C22—62no—
310C22—62no—
311C22—1216no—
312C22—122no—
313C22—122no—
314C22—122no—
315C22—122no—
316C22#88314yes—
317C22C2×#23511yescharacteristic
318C22C2×D3×D411yescharacteristic
319C22C2×#23711yescharacteristic
320C22#100412yes—
321C22#100912yes—
322C4—41no—
323C4—42no—
324C4—42no—
325C4—122no—
326C4—121no—
327C4—122no—
328C4—121no—
329C3#262211yescharacteristic, complemented, Sylow
330C2—24no—
331C2—81no—
332C2—122no—
333C2—122no—
334C2C2×#88311yescharacteristic
335C2#556012yes—
336C2#558911yescharacteristic
337C2#561712yes—
338C2#565311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC2×#937C2
hypercenter#386D3
hypocenterC3#2622
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×(C2×D3)⋊C4C2semidirect
3. C3×#386C2semidirect
4. #5220C2semidirect
5. #5210C2semidirect
6. #4961C2semidirect
7. #4959C2semidirect
8. #4955C2semidirect
9. #4950C2semidirect
10. #4675C2semidirect
11. #4674C2semidirect
20. C2×C3×C22⋊C4C22semidirect
23. C2×#217C22semidirect
32. C2×C4⋊Dic3C22semidirect
33. C2×Dic3⋊C4C22semidirect
34. C3×#95C22semidirect
35. C3⋊#95C22semidirect
36. C3⋊#95C22semidirect
37. C3⋊#95C22semidirect
45. #386D3semidirect
109. C2×C22⋊C4C2×D3semidirect
113. #95C2×D3semidirect
329. C3#2622semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C22×(C2×D3)⋊C42
C3×#3862
#52202
#52102
#49612
#49592
#49552
#49502
#46752
#46742
#26223