Finite Group Databasebeta

← #82636

Subgroups — #82636

Order 384.

788 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
1C2×C3×#121C211yesmaximal, characteristic
2C2×D4×Dic3C212yesmaximal, complemented
3C2×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5220C212yesmaximal, complemented
7C4×#217C211yesmaximal, characteristic, complemented
8#4984C211yesmaximal, characteristic, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4951C212yesmaximal, complemented
11#4950C212yesmaximal, complemented
12#2361—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C23×Dic3C2212yes—
18C3×#121C418yescomplemented
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2211yescharacteristic
26D4×Dic3—28no—
27D4×Dic3—24no—
28C2×C4⋊Dic3—24no—
29C2×C4⋊Dic3C2212yes—
30C2×C4⋊Dic3C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4×Dic3C2212yes—
34C2×C4×Dic3C2211yescharacteristic
35C2×C4×Dic3C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C2×#121D311yescharacteristic
41C2×C4×D4—31no—
42C2×C4×D4—32no—
43C22×C4⋊C4—31no—
44#390—31no—
45#389—31no—
46#388—32no—
47#386—32no—
48#385—32no—
49C4×C4⋊C4—31no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—24no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4C2×C418yescomplemented
58C2×C3×D4C2×C414yescomplemented
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4C2×C414yescomplemented
62C22×C3×C4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×Dic3—24no—
66C22×Dic3—24no—
67C22×Dic3—24no—
68C22×Dic3—24no—
69C22×Dic3—42no—
70C22×Dic3—42no—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C22×Dic3C2311yescharacteristic
76C22×Dic3C2312yes—
77C3×C4⋊C4C2×C414yescomplemented
78C3×C22⋊C4C2×C418yescomplemented
79#217—22no—
80#217—24no—
81#217—24no—
82#217—42no—
83#217D414yescomplemented
84C4⋊Dic3—22no—
85C4⋊Dic3—24no—
86C4⋊Dic3—24no—
87C4⋊Dic3D414yescomplemented
88C4×Dic3—22no—
89C4×Dic3—22no—
90C4×Dic3—24no—
91C22×D4C2×D312yes—
92C22×D4C2×D311yescharacteristic
93C23×C4—32no—
94C23×C4—32no—
95C23×C4C2×D311yescharacteristic
96#121Dic318yescomplemented
97C4×D4—68no—
98C4×D4—64no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—64no—
104C2×C4⋊C4C2×D311yescharacteristic
105C2×C22⋊C4—31no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4C2×D312yes—
111C2×C42—31no—
112C2×C42—32no—
113C2×C42—31no—
114#95—31no—
115#95—31no—
116#95—32no—
117#95—32no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3—22no—
128C23×C3C22×C414yes—
129C23×C3C22×C412yes—
130C23×C3C2×D412yes—
131C23×C3#4112yes—
132C23×C3C2411yescharacteristic
133C3×D4—216no—
134C3×D4—28no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—28no—
140C2×C3×C4C22×C412yes—
141C2×C3×C4C22×C414yes—
142C2×C3×C4C22×C412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yes—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—21no—
160C2×Dic3—48no—
161C2×Dic3—48no—
162C2×Dic3C2×D412yes—
163C2×Dic3C2×D412yes—
164C2×Dic3C2×D414yes—
165C24C22×D312yes—
166C24C22×D311yescharacteristic
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4C2×Dic318yescomplemented
172C2×D4C2×Dic314yescomplemented
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—32no—
181C22×C4—64no—
182C22×C4—64no—
183C22×C4—64no—
184C22×C4—64no—
185C22×C4—122no—
186C22×C4—122no—
187C22×C4C2×Dic314yescomplemented
188C22×C4C22×D312yes—
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D311yescharacteristic
191C4⋊C4—34no—
192C4⋊C4—64no—
193C4⋊C4—64no—
194C4⋊C4—62no—
195C4⋊C4C2×Dic314yescomplemented
196C22⋊C4—34no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—62no—
200C22⋊C4—122no—
201C22⋊C4C2×Dic318yescomplemented
202C42—64no—
203C42—62no—
204C42—62no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—24no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3C4×D414yescomplemented
219C22×C3C23×C411yescharacteristic
220C22×C3C22×D411yescharacteristic
221C22×C3C22×D411yescharacteristic
222C22×C3C2×#4111yescharacteristic
223C22×C3C2×#4111yescharacteristic
224C22×C3H(5,𝔽2)11yescharacteristic
225C22×C32−1+411yescharacteristic
226C3×C4—22no—
227C3×C4—24no—
228C3×C4C4×D414yescomplemented
229Dic3—22no—
230Dic3—24no—
231Dic3—22no—
232Dic3—42no—
233Dic3—41no—
234Dic3—41no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23D3×D412yes—
246C23#23712yes—
247C23C22×Dic314yes—
248C23C22×Dic312yes—
249C23C23×D311yescharacteristic
250D4—216no—
251D4—28no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—28no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—34no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—128no—
276C2×C4—128no—
277C2×C4D3×D412yes—
278C2×C4#23712yes—
279C2×C4C22×Dic312yes—
280C2×C4C22×Dic314yes—
281C2×C4C22×Dic312yes—
282C2×C3—24no—
283C2×C3—44no—
284C2×C3C2×C4×D411yescharacteristic
285C2×C3C2×C4×D411yescharacteristic
286C2×C3#52011yescharacteristic
287C2×C3D4211yescharacteristic
288C2×C3#54711yescharacteristic
289C2×C3#54811yescharacteristic
290C2×C3#55311yescharacteristic
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—24no—
298C22—24no—
299C22—24no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22D4×Dic314yescomplemented
305C22C2×D3×D411yescharacteristic
306C22C2×D3×D411yescharacteristic
307C22C2×#23711yescharacteristic
308C22C2×#23711yescharacteristic
309C22#100911yescharacteristic
310C22#101011yescharacteristic
311C22C23×Dic311yescharacteristic
312C4—22no—
313C4—24no—
314C4—62no—
315C4—64no—
316C4—62no—
317C4—121no—
318C4—122no—
319C4—121no—
320C4D4×Dic314yescomplemented
321C3#236111yescharacteristic, complemented, Sylow
322C2—24no—
323C2—44no—
324C2#559811yescharacteristic
325C2#560911yescharacteristic
326C2#561111yescharacteristic
327C2#561811yescharacteristic
328C2C2×D4×Dic311yescharacteristic
329C2C2×D4×Dic311yescharacteristic
330C2#582711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
Fitting, prime radicalC2×C3×#121C2
socle, cosocleC23×C3C24
hypercenterC2×#121D3
hypocenterC3#2361
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic3C2semidirect
3. C2×D4×Dic3C2semidirect
4. C22×C4⋊Dic3C2semidirect
5. #5221C2semidirect
6. #5220C2semidirect
7. C4×#217C2semidirect
8. #4984C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4951C2semidirect
11. #4950C2semidirect
18. C3×#121C4semidirect
57. C2×C3×D4C2×C4semidirect
58. C2×C3×D4C2×C4semidirect
61. C22×C3×C4C2×C4semidirect
77. C3×C4⋊C4C2×C4semidirect
78. C3×C22⋊C4C2×C4semidirect
83. #217D4semidirect
87. C4⋊Dic3D4semidirect
96. #121Dic3semidirect
171. C2×D4C2×Dic3semidirect
172. C2×D4C2×Dic3semidirect
187. C22×C4C2×Dic3semidirect
195. C4⋊C4C2×Dic3semidirect
201. C22⋊C4C2×Dic3semidirect
218. C22×C3C4×D4semidirect
228. C3×C4C4×D4semidirect
304. C22D4×Dic3semidirect
320. C4D4×Dic3semidirect
321. C3#2361semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×D4×Dic32
C22×C4⋊Dic32
#52212
#52202
C4×#2172
#49842
Dic3×C4⋊C42
#49512
#49502
#23613