Finite Group Databasebeta

← #74690

Subgroups — #74690

Order 384.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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
2C2×#935C211yesmaximal, characteristic, complemented
3C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#386C211yesmaximal, characteristic, complemented
6#5210C211yesmaximal, characteristic, complemented
7#4959C212yesmaximal, complemented
8#4956C212yesmaximal, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4667C212yesmaximal, complemented
12#2412—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×C4×D3C2212yes—
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yescomplemented
22C2×#217C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C2×#217C2212yescomplemented
25#937—24no—
26#935—24no—
27C2×(C2×D3)⋊C4—24no—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2212yes—
32C2×(C2×D3)⋊C4C2212yes—
33C2×Dic3⋊C4C2212yescomplemented
34C2×Dic3⋊C4C2212yescomplemented
35D3×C22⋊C4—24no—
36C3×#95C2212yescomplemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120—31no—
42C22×C22⋊C4—31no—
43C22×C22⋊C4—31no—
44#397—32no—
45#394—32no—
46#386—32no—
47#386D311yescharacteristic, complemented
48#385—31no—
49#381—31no—
50#380—32no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C23×D3—22no—
54C23×D3—22no—
55C23×D3—24no—
56C23×D3—42no—
57C23×D3D414yes—
58C23×D3C2311yescharacteristic
59C23×D3C2311yescharacteristic
60C2×C3×D4—22no—
61C2×C3×D4—22no—
62C2×C3×D4—22no—
63C22×C3×C4—22no—
64C22×C3×C4—42no—
65C22×C3×C4C2312yes—
66C22×C3×C4C2312yes—
67C22×C3×C4C2311yescharacteristic
68C2×#67—22no—
69C2×#67—22no—
70C2×#67—24no—
71C2×#67—42no—
72C22×Dic3—22no—
73C22×Dic3—22no—
74C22×Dic3—22no—
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2312yes—
78C22×Dic3C2312yes—
79C2×C4×D3—24no—
80C2×C4×D3—24no—
81C2×C4×D3—42no—
82C3×C22⋊C4—22no—
83C3×C22⋊C4—42no—
84#217—22no—
85#217—24no—
86#217D414yescomplemented
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—24no—
89(C2×D3)⋊C4—22no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—24no—
92Dic3⋊C4—24no—
93Dic3⋊C4—42no—
94C25—31no—
95C22×D4—31no—
96C22×D4—31no—
97C22×D4C2×D311yescharacteristic
98C23×C4—31no—
99C23×C4—32no—
100C23×C4C2×D311yescharacteristic
101#123—64no—
102#120—64no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—32no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—64no—
114C2×C22⋊C4—64no—
115C2×C22⋊C4C2×D311yescharacteristic
116C2×C22⋊C4C2×D312yescomplemented
117#95—32no—
118#95—32no—
119#95—32no—
120#95C2×D312yescomplemented
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—22no—
126C23×C3—21no—
127C23×C3—22no—
128C23×C3C2×D412yes—
129C23×C3C2×D412yes—
130C23×C3#4112yes—
131C23×C3C2411yescharacteristic
132C22×D3—21no—
133C22×D3—21no—
134C22×D3—22no—
135C22×D3—28no—
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—28no—
141C22×D3—44no—
142C22×D3—48no—
143C22×D3—48no—
144C22×D3C2×D412yes—
145C22×D3C2×D412yes—
146C22×D3#4114yes—
147C3×D4—44no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—21no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—44no—
158#67—44no—
159#67—44no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—21no—
165C2×Dic3—21no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—44no—
170C2×Dic3C2×D414yes—
171C4×D3—48no—
172C24—31no—
173C24—31no—
174C24—34no—
175C24—64no—
176C24—62no—
177C24—62no—
178C24—122no—
179C24C22×D311yescharacteristic
180C24C22×D311yescharacteristic
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—64no—
187C2×D4—122no—
188C22×C4—22no—
189C22×C4—32no—
190C22×C4—31no—
191C22×C4—32no—
192C22×C4—31no—
193C22×C4—42no—
194C22×C4—64no—
195C22×C4—64no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4—122no—
200C22×C4C22×D312yes—
201C22×C4C22×D312yes—
202C22×C4C22×D311yescharacteristic
203C4⋊C4—64no—
204C4⋊C4—122no—
205C22⋊C4—22no—
206C22⋊C4—34no—
207C22⋊C4—42no—
208C22⋊C4—64no—
209C22⋊C4—64no—
210C22⋊C4—64no—
211C22⋊C4—62no—
212C22⋊C4—62no—
213C22⋊C4—64no—
214C22⋊C4—62no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—24no—
218C22×C3—24no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3#12114yes—
225C22×C3C22×D411yescharacteristic
226C22×C3C22×D411yescharacteristic
227C22×C3C2×#4111yescharacteristic
228C22×C3C2×#4112yes—
229C22×C3H(5,𝔽2)12yes—
230C2×D3—22no—
231C2×D3—22no—
232C2×D3—24no—
233C2×D3—24no—
234C2×D3—22no—
235C2×D3—22no—
236C2×D3—22no—
237C2×D3—416no—
238C2×D3—42no—
239C2×D3—42no—
240C2×D3#12314yescomplemented
241C3×C4—41no—
242C3×C4—42no—
243C3×C4—42no—
244Dic3—24no—
245Dic3—42no—
246Dic3—41no—
247Dic3—41no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—22no—
254C23—22no—
255C23—32no—
256C23—34no—
257C23—32no—
258C23—68no—
259C23—61no—
260C23—61no—
261C23—62no—
262C23—68no—
263C23—61no—
264C23—61no—
265C23—62no—
266C23—62no—
267C23—124no—
268C23—128no—
269C23—128no—
270C23#23512yes—
271C23D3×D412yes—
272C23D3×D412yes—
273C23C23×D311yescharacteristic
274D4—44no—
275D4—124no—
276D4—124no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—34no—
287C2×C4—44no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—128no—
298C2×C4—124no—
299C2×C3—24no—
300C2×C3—81no—
301C2×C3C2×#12111yescharacteristic
302C2×C3C2×#12311yescharacteristic
303C2×C3#53411yescharacteristic
304C2×C3#54612yes—
305C2×C3#55112yes—
306D3—24no—
307D3—42no—
308C22—22no—
309C22—24no—
310C22—22no—
311C22—24no—
312C22—34no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—42no—
318C22—62no—
319C22—64no—
320C22—62no—
321C22—64no—
322C22—62no—
323C22—62no—
324C22—62no—
325C22—1216no—
326C22—122no—
327C22—122no—
328C22#88414yes—
329C22C2×#23511yescharacteristic
330C22C2×D3×D411yescharacteristic
331C22C2×D3×D411yescharacteristic
332C22#100412yes—
333C22D3×#4112yes—
334C4—41no—
335C4—42no—
336C4—42no—
337C4—64no—
338C4—121no—
339C4—122no—
340C4—121no—
341C3#241211yescharacteristic, complemented, Sylow
342C2—24no—
343C2—64no—
344C2—81no—
345C2—122no—
346C2C2×#88411yescharacteristic
347C2#555612yes—
348C2#558911yescharacteristic
349C2#561012yes—
350C2D3×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC2×#937C2
hypercenter#386D3
hypocenterC3#2412
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. C22×(C2×D3)⋊C4C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#386C2semidirect
6. #5210C2semidirect
7. #4959C2semidirect
8. #4956C2semidirect
9. #4943C2semidirect
10. #4675C2semidirect
11. #4667C2semidirect
21. C2×C3×C22⋊C4C22semidirect
24. C2×#217C22semidirect
33. C2×Dic3⋊C4C22semidirect
34. C2×Dic3⋊C4C22semidirect
36. C3×#95C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
47. #386D3semidirect
86. #217D4semidirect
116. C2×C22⋊C4C2×D3semidirect
120. #95C2×D3semidirect
240. C2×D3#123semidirect
341. C3#2412semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#9352
C22×(C2×D3)⋊C42
C2×D3×C22⋊C42
C3×#3862
#52102
#49592
#49562
#49432
#46752
#46672
#24123