Finite Group Databasebeta

← #73485

Subgroups — #73485

Order 384.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#930C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3C2×#875C211yesmaximal, characteristic, complemented
4C2×C4×Dic6C211yesmaximal, characteristic, complemented
5C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
6#4959C212yesmaximal, complemented
7#4957C212yesmaximal, complemented
8#4939C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4676C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2426—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×#217C2212yes—
22#930—24no—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30#924—24no—
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2212yes—
34C2×C4×Dic3C2212yes—
35#875—24no—
36C4×Dic6—24no—
37C3×#95C2211yescharacteristic
38C3×#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C2×#124—31no—
41C2×C22⋊Q8—31no—
42C2×#121—31no—
43C2×C4×Q8—31no—
44#396—32no—
45#394—32no—
46#390—32no—
47#390—31no—
48#389—31no—
49#388—32no—
50C4×C22⋊C4D311yescharacteristic, complemented
51C24×C3C2311yescharacteristic
52C23×D3C2312yes—
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—22no—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C2×#67—24no—
63C2×#67—24no—
64C2×#67—42no—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C2×D12—22no—
69C2×D12—22no—
70C2×D12—22no—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6D414yescomplemented
74C3×C22⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C42—22no—
77C3×C42—22no—
78#217—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—42no—
84C4⋊Dic3—22no—
85C4⋊Dic3D414yes—
86Dic3⋊C4—24no—
87Dic3⋊C4—24no—
88Dic3⋊C4—42no—
89C4×Dic3—24no—
90C22×Q8—31no—
91C22×D4—32no—
92C22×D4—31no—
93C23×C4C2×D311yescharacteristic
94#124—64no—
95C22⋊Q8—64no—
96#121—64no—
97C4×Q8—64no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4C2×D311yescharacteristic
110C2×C22⋊C4C2×D311yescharacteristic
111C2×C42—32no—
112C2×C42C2×D311yescharacteristic
113C2×C42C2×D311yescharacteristic
114#95—32no—
115#95C2×D311yescharacteristic
116#95C2×D311yescharacteristic
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3#4112yes—
124C23×C3C2411yescharacteristic
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—44no—
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D412yes—
149C2×C3×C4#4112yes—
150C2×C3×C4#4112yes—
151#67—48no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3C2×D414yes—
160C2×Dic3C2×D414yes—
161D12—44no—
162Dic6—28no—
163C24—32no—
164C24C22×D311yescharacteristic
165C2×Q8—34no—
166C2×Q8—62no—
167C2×Q8—62no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—122no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—32no—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C22×C4C22×D311yescharacteristic
186C4⋊C4—34no—
187C4⋊C4—64no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C4⋊C4—122no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—64no—
194C22⋊C4—64no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—122no—
199C42—22no—
200C42—22no—
201C42—64no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3C22×D411yescharacteristic
210C22×C3C22×D411yescharacteristic
211C22×C3C2×#4111yescharacteristic
212C22×C3C2×#4111yescharacteristic
213C22×C3C2×#4111yescharacteristic
214C22×C3H(5,𝔽2)11yescharacteristic
215C22×C32−1+411yescharacteristic
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C3×C4—22no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—41no—
228C3×C4#12114yes—
229Dic3—24no—
230Dic3—24no—
231Dic3—42no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23#23512yes—
246C23C23×D311yescharacteristic
247Q8—68no—
248D4—128no—
249D4—124no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—34no—
265C2×C4—34no—
266C2×C4—44no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4#23512yes—
275C2×C4#23512yes—
276C2×C4D3×D412yes—
277C2×C4D3×D412yes—
278C2×C3—42no—
279C2×C3C2×#12111yescharacteristic
280C2×C3#52911yescharacteristic
281C2×C3#54211yescharacteristic
282C2×C3#54611yescharacteristic
283C2×C3#54811yescharacteristic
284C2×C3#54811yescharacteristic
285C2×C3#55011yescharacteristic
286D3—82no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22—122no—
300C22—122no—
301C22C2×#23511yescharacteristic
302C22C2×#23511yescharacteristic
303C22C2×#23511yescharacteristic
304C22C2×D3×D411yescharacteristic
305C22C2×D3×D411yescharacteristic
306C22#100911yescharacteristic
307C22#101011yescharacteristic
308C4—22no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—64no—
314C4—64no—
315C4—122no—
316C4#88414yes—
317C3#242611yescharacteristic, complemented, Sylow
318C2—42no—
319C2—242no—
320C2#547911yescharacteristic
321C2C2×#88411yescharacteristic
322C2#553411yescharacteristic
323C2#555011yescharacteristic
324C2#555211yescharacteristic
325C2#555211yescharacteristic
326C2#555311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
prime radicalC2×#930C2
hypercenterC4×C22⋊C4D3
hypocenterC3#2426
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C2×#875C2semidirect
4. C2×C4×Dic6C2semidirect
5. C3×C4×C22⋊C4C2semidirect
6. #4959C2semidirect
7. #4957C2semidirect
8. #4939C2semidirect
9. #4928C2semidirect
10. #4676C2semidirect
11. #4669C2semidirect
50. C4×C22⋊C4D3semidirect
73. C2×Dic6D4semidirect
317. C3#2426semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#9242
C2×#8752
C2×C4×Dic62
C3×C4×C22⋊C42
#49592
#49572
#49392
#49282
#46762
#46692
#24263