Finite Group Databasebeta

← #82856

Subgroups — #82856

Order 384.

624 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic
2C2×#935C211yesmaximal, characteristic
3C2×D4×Dic3C211yesmaximal, characteristic
4C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#5211C211yesmaximal, characteristic, complemented
6#4982C212yesmaximal, complemented
7#4976C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4942C212yesmaximal, complemented
11#4942C211yesmaximal, characteristic, complemented
12#2486—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C3×#123—24no—
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2211yescharacteristic
25C2×#217C2211yescharacteristic
26#935—24no—
27D4×Dic3—24no—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4—24no—
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2212yes—
34C2×C4×Dic3C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#123—31no—
41C2×#123D311yescharacteristic
42C2×C4×D4—31no—
43C22×C4⋊C4—31no—
44#398—32no—
45#397—32no—
46#389—32no—
47#381—31no—
48#381—31no—
49#380—32no—
50#380—31no—
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4D414yes—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4C2312yes—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×Dic3—24no—
63C22×Dic3—22no—
64C22×Dic3—22no—
65C22×Dic3—22no—
66C22×Dic3—22no—
67C22×Dic3—24no—
68C22×Dic3D414yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2312yes—
72C22×Dic3C2312yes—
73C22×Dic3C2311yescharacteristic
74C22×Dic3C2311yescharacteristic
75C3×C4⋊C4—24no—
76C3×C22⋊C4—22no—
77C3×C22⋊C4—24no—
78#217—22no—
79#217—22no—
80#217—24no—
81#217—22no—
82C4⋊Dic3—22no—
83Dic3⋊C4—24no—
84Dic3⋊C4—24no—
85Dic3⋊C4—24no—
86Dic3⋊C4—24no—
87C4×Dic3—22no—
88C22×D4C2×D311yescharacteristic
89C23×C4—31no—
90C23×C4—31no—
91C23×C4—31no—
92C23×C4C2×D311yescharacteristic
93#123—24no—
94#123—64no—
95C4×D4—64no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—32no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—64no—
102C2×C4⋊C4C2×D312yes—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—31no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D311yescharacteristic
108C2×C22⋊C4C2×D312yes—
109C2×C42—31no—
110#95—32no—
111#95—32no—
112#95—32no—
113#95—32no—
114#95—32no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—21no—
118C23×C3—22no—
119C23×C3—21no—
120C23×C3—22no—
121C23×C3C2×D412yes—
122C23×C3C2×D412yes—
123C23×C3C2×D412yes—
124C23×C3#4112yes—
125C23×C3C2411yescharacteristic
126C3×D4—28no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—44no—
136C2×C3×C4C2×D412yes—
137C2×Dic3—22no—
138C2×Dic3—28no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—44no—
150C2×Dic3—44no—
151C2×Dic3C2×D412yes—
152C2×Dic3C2×D412yes—
153C2×Dic3#4114yes—
154C24C22×D311yescharacteristic
155C24C22×D311yescharacteristic
156C2×D4—22no—
157C2×D4—22no—
158C2×D4#6714yes—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—31no—
163C22×C4—34no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4—64no—
170C22×C4—62no—
171C22×C4—64no—
172C22×C4—62no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4C22×D312yes—
176C22×C4C22×D312yes—
177C22×C4C22×D311yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—64no—
180C4⋊C4—64no—
181C4⋊C4—62no—
182C4⋊C4—64no—
183C4⋊C4—64no—
184C22⋊C4—22no—
185C22⋊C4—24no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—64no—
190C42—62no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—24no—
194C22×C3—24no—
195C22×C3—22no—
196C22×C3—24no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—42no—
200C22×C3#12114yes—
201C22×C3C22×D411yescharacteristic
202C22×C3C22×D411yescharacteristic
203C22×C3C2×#4111yescharacteristic
204C22×C3C2×#4112yes—
205C22×C32−1+412yes—
206C3×C4—22no—
207C3×C4—42no—
208C3×C4—42no—
209Dic3—22no—
210Dic3—42no—
211Dic3—41no—
212Dic3—41no—
213Dic3—42no—
214Dic3#12314yescomplemented
215C23—22no—
216C23—22no—
217C23—21no—
218C23—21no—
219C23—22no—
220C23—22no—
221C23D3×D412yes—
222C23#23712yes—
223C23C2×#6712yes—
224C23C2×#6712yes—
225C23C23×D311yescharacteristic
226D4—28no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—32no—
236C2×C4—34no—
237C2×C4—32no—
238C2×C4—44no—
239C2×C4—68no—
240C2×C4—62no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—124no—
252C2×C4—124no—
253C2×C4C2×#6712yes—
254C2×C3—24no—
255C2×C3—42no—
256C2×C3C2×#12111yescharacteristic
257C2×C3C2×#12311yescharacteristic
258C2×C3#53611yescharacteristic
259C2×C3#54712yes—
260C2×C3#55212yes—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—24no—
267C22—41no—
268C22—42no—
269C22—41no—
270C22#93914yes—
271C22C2×D3×D411yescharacteristic
272C22C2×#23711yescharacteristic
273C22D3×#4112yes—
274C22#101012yes—
275C22C22×#6711yescharacteristic
276C4—22no—
277C4—34no—
278C4—42no—
279C4—42no—
280C4—62no—
281C4—121no—
282C4—121no—
283C4—122no—
284C4—122no—
285C3#248611yescharacteristic, complemented, Sylow
286C2—24no—
287C2—42no—
288C2#564811yescharacteristic
289C2#565012yes—
290C2D3×#12311yescharacteristic
291C2C2×#93911yescharacteristic
292C2#582412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radicalC22×C3×D4C22
hypercenterC2×#123D3
hypocenterC3#2486
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
4. C22×Dic3⋊C4C2semidirect
5. #5211C2semidirect
6. #4982C2semidirect
7. #4976C2semidirect
8. #4952C2semidirect
9. #4943C2semidirect
10. #4942C2semidirect
11. #4942C2semidirect
214. Dic3#123semidirect
285. C3#2486semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9352
C2×D4×Dic32
C22×Dic3⋊C42
#52112
#49822
#49762
#49522
#49432
#49422
#24863