Finite Group Databasebeta

← #76758

Subgroups — #76758

Order 384.

900 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 159 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×#938C211yesmaximal, characteristic, complemented
2C2×#937C212yesmaximal, complemented
3C2×#883C212yesmaximal, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#392C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5221C212yesmaximal, complemented
8#4957C212yesmaximal, complemented
9#4949C211yesmaximal, characteristic, complemented
10#4677C211yesmaximal, characteristic, complemented
11#4666C211yesmaximal, characteristic, complemented
12#2539—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×C3×D4C2211yescharacteristic, complemented
16C22×#67C2212yes—
17C22×#67C2212yes—
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yescomplemented
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2212yescomplemented
25C2×#217C2212yescomplemented
26#938—24no—
27#937—28no—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2212yes—
32C2×C4⋊Dic3C2211yescharacteristic, complemented
33C2×C4⋊Dic3C2212yescomplemented
34C2×Dic3⋊C4C2212yescomplemented
35C2×C4×Dic3C2212yescomplemented
36#883—28no—
37D3×C22⋊C4—24no—
38C3×#95C2211yescharacteristic, complemented
39C3⋊#95C2211yescharacteristic, complemented
40C2×#123—32no—
41C2×#121—31no—
42C2×#120—32no—
43C22×C22⋊C4—31no—
44#399—31no—
45#399—31no—
46#392—31no—
47#392D311yescharacteristic, complemented
48#390—32no—
49#390—32no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C23×D3—22no—
54C23×D3—22no—
55C23×D3—22no—
56C23×D3—24no—
57C23×D3—42no—
58C23×D3C2311yescharacteristic
59C23×D3C2311yescharacteristic
60C2×C3×D4—24no—
61C2×C3×D4—24no—
62C2×C3×D4—22no—
63C2×C3×D4—42no—
64C2×C3×D4—42no—
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2312yes—
68C2×#67—24no—
69C2×#67—24no—
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—24no—
73C2×#67—42no—
74C22×Dic3C2312yes—
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2312yes—
77C22×Dic3C2311yescharacteristic
78C2×C4×D3—22no—
79C2×C4×D3—24no—
80C2×C4×D3—42no—
81C2×C4×D3D414yescomplemented
82C3×C22⋊C4—22no—
83C3×C22⋊C4—24no—
84#217—22no—
85#217—24no—
86#217—24no—
87#217—24no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—24no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4D418yescomplemented
92C4⋊Dic3—24no—
93C4⋊Dic3—22no—
94Dic3⋊C4—24no—
95C4×Dic3—42no—
96C25—31no—
97C22×D4—32no—
98C22×D4—32no—
99C22×D4C2×D312yes—
100C22×D4C2×D311yescharacteristic, complemented
101C23×C4—31no—
102C23×C4—31no—
103#123—68no—
104#121—64no—
105#120—68no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—31no—
116C2×C22⋊C4—32no—
117C2×C22⋊C4—64no—
118C2×C22⋊C4C2×D311yescharacteristic
119C2×C22⋊C4C2×D312yescomplemented
120C2×C42—32no—
121#95—31no—
122#95C2×D311yescharacteristic, complemented
123C23×C3—21no—
124C23×C3—22no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—22no—
134C23×C3#4112yes—
135C23×C3C2411yescharacteristic
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—28no—
140C22×D3—21no—
141C22×D3—22no—
142C22×D3—21no—
143C22×D3—22no—
144C22×D3—48no—
145C22×D3—48no—
146C22×D3—44no—
147C22×D3—44no—
148C22×D3C2×D414yes—
149C22×D3C2×D412yes—
150C22×D3#4112yes—
151C3×D4—48no—
152C3×D4—44no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4C2×D414yes—
159C2×C3×C4C2×D412yes—
160#67—48no—
161#67—48no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—21no—
169C2×Dic3—22no—
170C2×Dic3C2×D414yes—
171C2×Dic3C2×D412yes—
172C4×D3—44no—
173C4×D3—44no—
174C24—31no—
175C24—31no—
176C24—62no—
177C24—62no—
178C24—64no—
179C24—62no—
180C24—122no—
181C24C22×D312yes—
182C24C22×D311yescharacteristic
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—64no—
189C2×D4—64no—
190C2×D4—64no—
191C2×D4—64no—
192C2×D4—64no—
193C2×D4—122no—
194C22×C4—32no—
195C22×C4—31no—
196C22×C4—34no—
197C22×C4—31no—
198C22×C4—32no—
199C22×C4—62no—
200C22×C4—64no—
201C22×C4—122no—
202C22×C4C22×D311yescharacteristic
203C22×C4C22×D311yescharacteristic
204C22×C4C22×D312yes—
205C4⋊C4—64no—
206C4⋊C4—62no—
207C4⋊C4—64no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—38no—
211C22⋊C4—62no—
212C22⋊C4—62no—
213C22⋊C4—64no—
214C22⋊C4—64no—
215C22⋊C4—64no—
216C22⋊C4—62no—
217C22⋊C4—64no—
218C42—122no—
219C22×C3—22no—
220C22×C3—24no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3C22×D411yescharacteristic
232C22×C3C22×D412yes—
233C22×C3C2×#4111yescharacteristic
234C22×C3C2×#4111yescharacteristic
235C22×C3H(5,𝔽2)12yes—
236C2×D3—22no—
237C2×D3—22no—
238C2×D3—24no—
239C2×D3—24no—
240C2×D3—22no—
241C2×D3—24no—
242C2×D3—416no—
243C2×D3—41no—
244C2×D3—41no—
245C2×D3—42no—
246C2×D3#12714yes—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—41no—
250Dic3—22no—
251Dic3—24no—
252Dic3—41no—
253Dic3—42no—
254C23—21no—
255C23—22no—
256C23—22no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—34no—
266C23—32no—
267C23—32no—
268C23—68no—
269C23—61no—
270C23—62no—
271C23—61no—
272C23—61no—
273C23—62no—
274C23—61no—
275C23—62no—
276C23—128no—
277C23—124no—
278C23—128no—
279C23—124no—
280C23#23712yes—
281C23C23×D311yescharacteristic
282D4—48no—
283D4—44no—
284D4—128no—
285D4—128no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—34no—
292C2×C4—32no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—61no—
297C2×C4—62no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—124no—
302C2×C4—124no—
303C2×C4D3×D414yes—
304C2×C4D3×D412yes—
305C2×C3—42no—
306C2×C3—82no—
307C2×C3#52511yescharacteristic
308C2×C3C2×#12711yescharacteristic
309C2×C3#53411yescharacteristic
310C2×C3#54612yes—
311C2×C3#54612yes—
312D3—24no—
313D3—42no—
314C22—24no—
315C22—22no—
316C22—34no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—62no—
328C22—64no—
329C22—62no—
330C22—64no—
331C22—62no—
332C22—64no—
333C22—1216no—
334C22—121no—
335C22—121no—
336C22—122no—
337C22C2×D3×D411yescharacteristic
338C22C2×D3×D412yes—
339C22C2×#23711yescharacteristic
340C22#100412yes—
341C22D3×#4111yescharacteristic
342C4—22no—
343C4—24no—
344C4—41no—
345C4—62no—
346C4—64no—
347C4—121no—
348C4—122no—
349C3#253911yescharacteristic, complemented, Sylow
350C2—42no—
351C2—64no—
352C2—82no—
353C2—122no—
354C2#558911yescharacteristic
355C2#559112yes—
356C2#560611yescharacteristic
357C2#561212yes—
358C2D3×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2539
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#938C2semidirect
2. C2×#937C2semidirect
3. C2×#883C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#392C2semidirect
6. #5223C2semidirect
7. #5221C2semidirect
8. #4957C2semidirect
9. #4949C2semidirect
10. #4677C2semidirect
11. #4666C2semidirect
15. C22×C3×D4C22semidirect
21. C2×C3×C22⋊C4C22semidirect
24. C2×#217C22semidirect
25. C2×#217C22semidirect
32. C2×C4⋊Dic3C22semidirect
33. C2×C4⋊Dic3C22semidirect
34. C2×Dic3⋊C4C22semidirect
35. C2×C4×Dic3C22semidirect
38. C3×#95C22semidirect
39. C3⋊#95C22semidirect
47. #392D3semidirect
81. C2×C4×D3D4semidirect
91. (C2×D3)⋊C4D4semidirect
100. C22×D4C2×D3semidirect
119. C2×C22⋊C4C2×D3semidirect
122. #95C2×D3semidirect
349. C3#2539semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#9372
C2×#8832
C2×D3×C22⋊C42
C3×#3922
#52232
#52212
#49572
#49492
#46772
#46662
#25393