Finite Group Databasebeta

← #82590

Subgroups — #82590

Order 384.

800 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#120C211yesmaximal, characteristic
2C22×#217C211yesmaximal, characteristic, complemented
3C2×#935C211yesmaximal, characteristic, complemented
4C2×D4×Dic3C211yesmaximal, characteristic, complemented
5#5247C211yesmaximal, characteristic, complemented
6#5222C212yesmaximal, complemented
7#5220C212yesmaximal, complemented
8#4948C212yesmaximal, complemented
9#4945C212yesmaximal, complemented
10#4943C211yesmaximal, characteristic, complemented
11#4942C211yesmaximal, characteristic, complemented
12#2534—31nomaximal, Sylow
13C25×C3C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×C3×D4C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic, complemented
19C3×#120—24no—
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217—24no—
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2212yes—
26C2×#217C2212yes—
27C2×#217C2212yes—
28C2×#217C2212yes—
29C2×#217C2212yescomplemented
30C2×#217C2211yescharacteristic, complemented
31#935—24no—
32D4×Dic3—24no—
33C2×C4⋊Dic3C2211yescharacteristic, complemented
34C2×Dic3⋊C4C2212yescomplemented
35C2×Dic3⋊C4C2212yescomplemented
36C2×C4×Dic3C2211yescharacteristic, complemented
37C3⋊#95C2212yescomplemented
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120D311yescharacteristic
42C2×C4×D4—31no—
43C22×C22⋊C4—31no—
44#396—32no—
45#394—32no—
46#388—32no—
47#386—32no—
48#381—31no—
49#380—31no—
50#379—31no—
51C24×C3—22no—
52C24×C3—24no—
53C24×C3—22no—
54C24×C3—22no—
55C24×C3—42no—
56C24×C3C2311yescharacteristic
57C24×C3C2312yes—
58C24×C3C2311yescharacteristic
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C2×C3×D4—24no—
62C2×C3×D4—24no—
63C2×C3×D4—42no—
64C2×C3×D4D414yes—
65C22×C3×C4C2312yes—
66C22×C3×C4C2311yescharacteristic
67C22×Dic3—22no—
68C22×Dic3—22no—
69C22×Dic3—22no—
70C22×Dic3—22no—
71C22×Dic3—42no—
72C22×Dic3—42no—
73C22×Dic3D414yescomplemented
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2312yes—
76C22×Dic3C2311yescharacteristic
77C22×Dic3C2312yes—
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C3×C22⋊C4—22no—
81C3×C22⋊C4—24no—
82#217—22no—
83#217—22no—
84#217—24no—
85#217—24no—
86#217—22no—
87#217—24no—
88#217—24no—
89#217—42no—
90C4⋊Dic3—22no—
91Dic3⋊C4—24no—
92Dic3⋊C4—42no—
93C4×Dic3—22no—
94C25C2×D311yescharacteristic
95C22×D4C2×D312yes—
96C22×D4C2×D311yescharacteristic
97C23×C4—31no—
98C23×C4—31no—
99C23×C4—31no—
100#123—64no—
101#120—24no—
102C4×D4—64no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—64no—
115C2×C22⋊C4C2×D311yescharacteristic
116C2×C22⋊C4C2×D312yes—
117C2×C42—31no—
118#95—32no—
119#95—32no—
120#95—32no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—28no—
124C23×C3—22no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3—21no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—22no—
133C23×C3—22no—
134C23×C3—48no—
135C23×C3—48no—
136C23×C3—44no—
137C23×C3—44no—
138C23×C3C2×D412yes—
139C23×C3C2×D412yes—
140C23×C3C2×D412yes—
141C23×C3#4114yes—
142C23×C3#4112yes—
143C23×C3C2411yescharacteristic
144C3×D4—28no—
145C3×D4—48no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4C2×D412yes—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—44no—
165C2×Dic3—44no—
166C2×Dic3—44no—
167C2×Dic3C2×D412yes—
168C2×Dic3C2×D412yes—
169C24—22no—
170C24—24no—
171C24—22no—
172C24—22no—
173C24—42no—
174C24C22×D312yes—
175C24C22×D311yescharacteristic
176C24C22×D311yescharacteristic
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—42no—
182C2×D4#6714yes—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—34no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—31no—
189C22×C4—32no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4—122no—
195C22×C4—122no—
196C22×C4C22×D312yes—
197C22×C4C22×D311yescharacteristic
198C4⋊C4—62no—
199C4⋊C4—64no—
200C4⋊C4—122no—
201C22⋊C4—22no—
202C22⋊C4—24no—
203C22⋊C4—64no—
204C22⋊C4—64no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C22⋊C4—64no—
208C22⋊C4—62no—
209C22⋊C4—64no—
210C22⋊C4—122no—
211C42—62no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3—24no—
216C22×C3—22no—
217C22×C3—24no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—24no—
221C22×C3—416no—
222C22×C3—42no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3#12114yes—
230C22×C3C22×D411yescharacteristic
231C22×C3C22×D411yescharacteristic
232C22×C3C2×#4111yescharacteristic
233C22×C3C2×#4112yes—
234C22×C3H(5,𝔽2)12yes—
235C3×C4—22no—
236C3×C4—42no—
237Dic3—22no—
238Dic3—22no—
239Dic3—42no—
240Dic3—41no—
241Dic3—41no—
242Dic3—42no—
243C23—22no—
244C23—22no—
245C23—28no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—48no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23D3×D412yes—
261C23#23714yes—
262C23#23712yes—
263C23C2×#6712yes—
264C23C2×#6712yes—
265C23C23×D311yescharacteristic
266D4—28no—
267D4—48no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—32no—
274C2×C4—32no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—61no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—124no—
288C2×C4—124no—
289C2×C4—124no—
290C2×C4C2×#6712yes—
291C2×C3—24no—
292C2×C3—42no—
293C2×C3—44no—
294C2×C3C2×#12111yescharacteristic
295C2×C3#52511yescharacteristic
296C2×C3#53411yescharacteristic
297C2×C3#54612yes—
298C2×C3#55112yes—
299C22—24no—
300C22—22no—
301C22—24no—
302C22—22no—
303C22—22no—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22—416no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22#93914yes—
317C22C2×D3×D411yescharacteristic
318C22C2×#23711yescharacteristic
319C22C2×#23712yes—
320C22#100412yes—
321C22C22×#6711yescharacteristic
322C4—22no—
323C4—42no—
324C4—62no—
325C4—62no—
326C4—122no—
327C4—122no—
328C4—121no—
329C4—121no—
330C3#253411yescharacteristic, complemented, Sylow
331C2—24no—
332C2—44no—
333C2—42no—
334C2#558611yescharacteristic
335C2#558712yes—
336C2#558911yescharacteristic
337C2C2×#93911yescharacteristic
338C2#580612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#120C2
hypercenterC2×#120D3
hypocenterC3#2534
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C22×#217C2semidirect
3. C2×#935C2semidirect
4. C2×D4×Dic3C2semidirect
5. #5247C2semidirect
6. #5222C2semidirect
7. #5220C2semidirect
8. #4948C2semidirect
9. #4945C2semidirect
10. #4943C2semidirect
11. #4942C2semidirect
18. C23×Dic3C22semidirect
29. C2×#217C22semidirect
30. C2×#217C22semidirect
33. C2×C4⋊Dic3C22semidirect
34. C2×Dic3⋊C4C22semidirect
35. C2×Dic3⋊C4C22semidirect
36. C2×C4×Dic3C22semidirect
37. C3⋊#95C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
73. C22×Dic3D4semidirect
330. C3#2534semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C22×#2172
C2×#9352
C2×D4×Dic32
#52472
#52222
#52202
#49482
#49452
#49432
#49422
#25343