Finite Group Databasebeta

← #84091

Subgroups — #84091

Order 384.

914 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#41C211yesmaximal, characteristic, complemented
2C2×#939C212yesmaximal, complemented
3C2×#929C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic, complemented
5#5230C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5222C211yesmaximal, characteristic, complemented
8#5220C212yesmaximal, complemented
9#5213C212yesmaximal, complemented
10#5210C212yesmaximal, complemented
11C2×C3⋊#95C211yesmaximal, characteristic, complemented
12#2430—31nomaximal, Sylow
13C2×C3×#41—28no—
14C2×C3×#41—24no—
15C22×C3×Q8C2211yescharacteristic
16C22×C3×D4C2211yescharacteristic
17C22×C3×D4C2212yes—
18C23×C3×C4C2212yes—
19C23×C3×C4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2212yes—
22C23×Dic3C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2211yescharacteristic
26C2×#217C2212yes—
27#939—28no—
28#929—24no—
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C22×#41D311yescharacteristic, complemented
41C2×#123—31no—
42C2×#121—32no—
43#395—31no—
44#394—31no—
45#393—31no—
46#392—31no—
47#386—32no—
48#386—32no—
49#385—32no—
50C2×#95—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C3×#41—232no—
55C2×C3×Q8—24no—
56C2×C3×Q8—22no—
57C2×C3×D4—24no—
58C2×C3×D4—24no—
59C2×C3×D4—24no—
60C2×C3×D4—22no—
61C2×C3×D4D418yescomplemented
62C22×C3×C4—24no—
63C22×C3×C4—24no—
64C22×C3×C4—24no—
65C22×C3×C4—24no—
66C22×C3×C4D414yescomplemented
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2312yes—
69C22×C3×C4C2311yescharacteristic
70C2×#67—22no—
71C2×#67—24no—
72C2×#67—24no—
73C2×#67—24no—
74C2×#67—42no—
75C22×Dic3—24no—
76C22×Dic3—24no—
77C22×Dic3—42no—
78C22×Dic3C2312yes—
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2312yes—
83#217—22no—
84#217—24no—
85#217—24no—
86#217—42no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—24no—
90Dic3⋊C4—22no—
91Dic3⋊C4—22no—
92Dic3⋊C4—24no—
93C2×#41—28no—
94C2×#41—24no—
95C22×Q8C2×D311yescharacteristic
96C22×D4—31no—
97C22×D4—32no—
98C22×D4C2×D311yescharacteristic
99C22×D4C2×D312yes—
100C23×C4—32no—
101C23×C4C2×D312yes—
102C23×C4C2×D311yescharacteristic
103#123—64no—
104#121—68no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—32no—
111C2×C22⋊C4—32no—
112C2×C22⋊C4—31no—
113C2×C22⋊C4—31no—
114C2×C22⋊C4—32no—
115#95—32no—
116#95—31no—
117#95—31no—
118#95—32no—
119#95—32no—
120C23×C3—22no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3C2×D414yes—
130C23×C3C2×D412yes—
131C23×C3C2×D414yes—
132C23×C3C2411yescharacteristic
133C22×D3—21no—
134C22×D3—22no—
135C22×D3—22no—
136C22×D3—21no—
137C22×D3—21no—
138C3×Q8—28no—
139C3×D4—28no—
140C3×D4—216no—
141C2×C3×C4—216no—
142C2×C3×C4—28no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D414yes—
149C2×C3×C4C2×D412yes—
150#67—48no—
151#67—44no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—22no—
161C2×Dic3—21no—
162C2×Dic3—22no—
163C2×Dic3—21no—
164C2×Dic3—48no—
165C2×Dic3#4114yescomplemented
166C24—31no—
167C24C22×D312yes—
168C24C22×D311yescharacteristic
169#41—232no—
170C2×Q8—24no—
171C2×Q8—22no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—64no—
177C2×D4—64no—
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—122no—
181C2×D4#6718yescomplemented
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—32no—
187C22×C4—32no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—64no—
192C22×C4—64no—
193C22×C4—122no—
194C22×C4#6714yescomplemented
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D312yes—
197C22×C4C22×D311yescharacteristic
198C4⋊C4—64no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C22⋊C4—64no—
202C22⋊C4—64no—
203C22⋊C4—62no—
204C22⋊C4—64no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C22⋊C4—122no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—24no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3#12014yes—
220C22×C3C22×D411yescharacteristic
221C22×C3C22×D412yes—
222C22×C3C2×#4112yes—
223C22×C3H(5,𝔽2)11yescharacteristic
224C22×C32−1+411yescharacteristic
225C2×D3—41no—
226C2×D3—41no—
227C2×D3—41no—
228C2×D3—42no—
229C2×D3—42no—
230C3×C4—22no—
231C3×C4—24no—
232C3×C4—22no—
233Dic3—24no—
234Dic3—41no—
235Dic3—41no—
236Dic3—41no—
237Dic3—42no—
238C23—22no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—62no—
248C23—61no—
249C23—61no—
250C23—61no—
251C23—62no—
252C23C2×#6714yes—
253C23C2×#6712yes—
254C23C2×#6714yes—
255C23C23×D311yescharacteristic
256Q8—28no—
257D4—28no—
258D4—216no—
259D4—128no—
260D4—124no—
261C2×C4—216no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—34no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—61no—
280C2×C4—128no—
281C2×C4C2×#6712yes—
282C2×C4C2×#6714yes—
283C2×C4C2×#6712yes—
284C2×C3—24no—
285C2×C3—44no—
286C2×C3C2×#12011yescharacteristic
287C2×C3#53211yescharacteristic
288C2×C3#53711yescharacteristic
289C2×C3#54612yes—
290C2×C3#54712yes—
291D3—81no—
292C22—24no—
293C22—24no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—122no—
304C22—122no—
305C22—121no—
306C22—121no—
307C22—121no—
308C22#95314yes—
309C22D3×#4112yes—
310C22#100911yescharacteristic
311C22#101011yescharacteristic
312C22C22×#6712yes—
313C22C22×#6711yescharacteristic
314C4—22no—
315C4—24no—
316C4—22no—
317C4—64no—
318C4—122no—
319C4—121no—
320C4—121no—
321C4—121no—
322C3#243011yescharacteristic, complemented, Sylow
323C2—24no—
324C2—44no—
325C2—241no—
326C2#582412yes—
327C2#582912yes—
328C2#583211yescharacteristic
329C2#583311yescharacteristic
330C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×#41C2
hypercenterC22×#41D3
hypocenterC3#2430
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41C2semidirect
2. C2×#939C2semidirect
3. C2×#929C2semidirect
4. #5231C2semidirect
5. #5230C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #5220C2semidirect
9. #5213C2semidirect
10. #5210C2semidirect
11. C2×C3⋊#95C2semidirect
40. C22×#41D3semidirect
61. C2×C3×D4D4semidirect
66. C22×C3×C4D4semidirect
165. C2×Dic3#41semidirect
181. C2×D4#67semidirect
194. C22×C4#67semidirect
322. C3#2430semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×#9392
C2×#9292
#52312
#52302
#52232
#52222
#52202
#52132
#52102
C2×C3⋊#952
#24303