Finite Group Databasebeta

← #80039

Subgroups — #80039

Order 384.

738 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#915C212yesmaximal, complemented
3#5606C211yesmaximal, characteristic
4C3×#448C211yesmaximal, characteristic, complemented
5#5228C211yesmaximal, characteristic, complemented
6#5170C212yesmaximal, complemented
7#5040C211yesmaximal, characteristic, complemented
8#4809C212yesmaximal, complemented
9#4802C212yesmaximal, complemented
10#4739C211yesmaximal, characteristic, complemented
11#4726C211yesmaximal, characteristic, complemented
12#3004—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×D3×Q8—28no—
15C2×D3×Q8—21no—
16C2×D3×Q8—21no—
17C2×D3×Q8C2212yes—
18C2×D3×Q8C2212yes—
19C2×#237C2211yescharacteristic
20C22×C4×D3—21no—
21C22×C4×D3C2211yescharacteristic
22C22×Dic6—21no—
23C22×Dic6C2211yescharacteristic
24C2×C3×SD16C2212yescomplemented
25C3×#121C2211yescharacteristic
26C2×#216C2212yescomplemented
27#938C2211yescharacteristic
28#937—21no—
29D4×Dic3—21no—
30C2×#214C2212yescomplemented
31#924C2211yescharacteristic
32#915—28no—
33C2×Dic12C2212yescomplemented
34C2×ΓL(1,𝔽25)C2212yes—
35#897C2211yescharacteristic
36#892C2211yescharacteristic
37#883—21no—
38#67⋊C4—21no—
39C3×#103C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic
41#835C2212yescomplemented
42#830C2211yescharacteristic
43(C2×D3)⋊C8C2212yes—
44#816C2212yescomplemented
45#810C2212yescomplemented
46C23×Q8—31no—
47C2×#137—32no—
48#525—31no—
49#451—31no—
50#451—32no—
51#451—31no—
52#448—32no—
53#448—31no—
54#448D311yescharacteristic, complemented
55#419—32no—
56#407—31no—
57C23×D3D411yescharacteristic
58C2×C3×Q8—24no—
59C2×C3×Q8—21no—
60C2×C3×Q8C2312yes—
61C2×C3×D4—21no—
62C2×C3×D4C2311yescharacteristic
63C22×C3×C4—21no—
64C22×C3×C4C2311yescharacteristic
65C2×#67—21no—
66C2×#67—21no—
67C22×Dic3—21no—
68C22×Dic3—21no—
69C22×Dic3D411yescharacteristic
70D3×Q8—28no—
71D3×Q8—28no—
72D3×Q8—24no—
73D3×Q8—24no—
74D3×Q8—24no—
75D3×Q8D418yescomplemented
76#237—24no—
77C2×C4×D3—28no—
78C2×C4×D3—22no—
79C2×C4×D3—22no—
80C2×C4×D3—24no—
81C2×C4×D3D411yescharacteristic
82C2×C4×D3D411yescharacteristic
83C2×C4×D3C2311yescharacteristic
84C2×C4×D3C2311yescharacteristic
85C2×Dic6—28no—
86C2×Dic6—21no—
87C2×Dic6—22no—
88C2×Dic6—22no—
89C2×Dic6—24no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2312yes—
92C3×SD16—24no—
93C2×C3×C8C2312yes—
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96#217—21no—
97#217—21no—
98#216—24no—
99#214—24no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104Dic3⋊C4—21no—
105C4×Dic3—21no—
106C2×C3⋊C8C2312yes—
107Dic12—24no—
108ΓL(1,𝔽25)—24no—
109C2×#41—31no—
110C22×Q8—32no—
111C22×Q8—32no—
112C22×Q8—31no—
113C22×Q8—61no—
114C22×Q8—61no—
115C22×Q8—61no—
116C22×Q8—68no—
117C22×Q8C2×D311yescharacteristic
118C23×C4—31no—
119C23×C4—61no—
120#137—68no—
121C2×Q16—32no—
122C2×Q16—32no—
123C2×SD16—32no—
124C2×SD16C2×D312yescomplemented
125C2×M16—32no—
126#123—61no—
127C22⋊Q8—31no—
128C22⋊Q8—31no—
129#121—31no—
130#121C2×D311yescharacteristic
131#120—61no—
132C4×D4—61no—
133C4×D4—61no—
134#117—31no—
135#103—32no—
136#103—32no—
137#103—32no—
138#103C2×D312yescomplemented
139C22⋊C8—31no—
140C22⋊C8—32no—
141C22⋊C8C2×D311yescharacteristic
142C23×C3—21no—
143C23×C3C2×D411yescharacteristic
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3C2×D411yescharacteristic
148C22×D3C2×D411yescharacteristic
149C3×Q8—24no—
150C3×Q8—22no—
151C3×Q8C2×D414yes—
152C3×D4—22no—
153C3×D4—42no—
154C2×C3×C4—24no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4C2×D411yescharacteristic
159C2×C3×C4C2411yescharacteristic
160#67—42no—
161#67—42no—
162C2×Dic3—24no—
163C2×Dic3—22no—
164C2×Dic3—22no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—42no—
168C2×Dic3C2×D411yescharacteristic
169C2×Dic3C2×D411yescharacteristic
170C4×D3—28no—
171C4×D3—28no—
172C4×D3—22no—
173C4×D3—22no—
174C4×D3—22no—
175C4×D3C2×D414yes—
176Dic6—28no—
177Dic6—24no—
178Dic6—22no—
179Dic6—28no—
180Dic6—22no—
181Dic6C2×D414yes—
182C3×C8—22no—
183C3⋊C8—22no—
184C24—31no—
185#41—64no—
186C2×Q8—24no—
187C2×Q8—21no—
188C2×Q8—38no—
189C2×Q8—32no—
190C2×Q8—31no—
191C2×Q8—62no—
192C2×Q8—62no—
193C2×Q8—64no—
194C2×Q8—68no—
195C2×Q8—64no—
196C2×Q8—64no—
197C2×Q8—68no—
198C2×Q8—68no—
199C2×Q8—64no—
200C2×Q8—61no—
201C2×Q8C22×D312yes—
202C2×D4—21no—
203C2×D4—61no—
204C2×D4—61no—
205C2×D4C22×D311yescharacteristic
206C22×C4—21no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—64no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4—68no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4C22×D311yescharacteristic
219Q16—64no—
220Q16—64no—
221SD16—24no—
222SD16—64no—
223M16—64no—
224C2×C8—32no—
225C2×C8C22×D312yes—
226C4⋊C4—31no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4C22×D311yescharacteristic
230C22⋊C4—21no—
231C22⋊C4—61no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C42—61no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3#13712yescomplemented
242C22×C3C22×D411yescharacteristic
243C2×D3—28no—
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—21no—
247C2×D3—21no—
248C2×D3#12012yes—
249C2×D3#13712yescomplemented
250C3×C4—24no—
251C3×C4—21no—
252C3×C4—41no—
253C3×C4C22×D412yes—
254Dic3—21no—
255Dic3—24no—
256Dic3—41no—
257Dic3—41no—
258Dic3#12012yes—
259C23—21no—
260C23—31no—
261C23—31no—
262C23—62no—
263C23—62no—
264C23—62no—
265C23D3×D411yescharacteristic
266Q8—24no—
267Q8—22no—
268Q8—34no—
269Q8—68no—
270Q8—68no—
271Q8—64no—
272Q8—62no—
273Q8—62no—
274Q8D3×D414yes—
275D4—22no—
276D4—42no—
277D4—122no—
278D4—122no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—31no—
284C2×C4—34no—
285C2×C4—31no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—68no—
289C2×C4—68no—
290C2×C4—62no—
291C2×C4—64no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—122no—
297C2×C4D3×D411yescharacteristic
298C2×C4C23×D311yescharacteristic
299C8—22no—
300C8—62no—
301C2×C3—22no—
302C2×C3—81no—
303C2×C3C2×#12011yescharacteristic
304C2×C3C2×#13711yescharacteristic
305C2×C3C2×#13711yescharacteristic
306D3—22no—
307D3—22no—
308C22—21no—
309C22—21no—
310C22—32no—
311C22—32no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—68no—
316C22—61no—
317C22—61no—
318C22—61no—
319C22—61no—
320C22#91512yescomplemented
321C22C2×D3×D411yescharacteristic
322C4—24no—
323C4—21no—
324C4—32no—
325C4—41no—
326C4—61no—
327C4—64no—
328C4—121no—
329C4—121no—
330C4C2×D3×D412yes—
331C3#300411yescharacteristic, complemented, Sylow
332C2—22no—
333C2—62no—
334C2—62no—
335C2—81no—
336C2D3×#12011yescharacteristic
337C2C2×#91511yescharacteristic
338C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5606C2
hypercenter#448D3
hypocenterC3#3004
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q8C2semidirect
2. C2×#915C2semidirect
4. C3×#448C2semidirect
5. #5228C2semidirect
6. #5170C2semidirect
7. #5040C2semidirect
8. #4809C2semidirect
9. #4802C2semidirect
10. #4739C2semidirect
11. #4726C2semidirect
24. C2×C3×SD16C22semidirect
26. C2×#216C22semidirect
30. C2×#214C22semidirect
33. C2×Dic12C22semidirect
39. C3×#103C22semidirect
41. #835C22semidirect
44. #816C22semidirect
45. #810C22semidirect
54. #448D3semidirect
75. D3×Q8D4semidirect
124. C2×SD16C2×D3semidirect
138. #103C2×D3semidirect
241. C22×C3#137semidirect
249. C2×D3#137semidirect
320. C22#915semidirect
331. C3#3004semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×#9152
#56062
C3×#4482
#52282
#51702
#50402
#48092
#48022
#47392
#47262
#30043