Finite Group Databasebeta

← #74020

Subgroups — #74020

Order 384.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#67C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3C2×#897C211yesmaximal, characteristic, complemented
4C2×#892C211yesmaximal, characteristic, complemented
5C3×#380C211yesmaximal, characteristic, complemented
6#4973C211yesmaximal, characteristic, complemented
7#4972C211yesmaximal, characteristic, complemented
8#4959C212yesmaximal
9#4951C212yesmaximal, complemented
10#4675C212yesmaximal, complemented
11#4647C212yesmaximal, complemented
12#2415—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22C4×#67—24no—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28#924—24no—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34#897—24no—
35#892—24no—
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×C22⋊Q8—31no—
41C2×C22⋊Q8—31no—
42C2×C4×D4—31no—
43C2×#117—31no—
44#397—32no—
45#394—32no—
46#391—31no—
47#389—31no—
48#389—32no—
49#388—32no—
50#380D311yescharacteristic, complemented
51C24×C3C2311yescharacteristic
52C23×D3C2311yescharacteristic
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—22no—
56C22×C3×C4C2312yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—22no—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C2×C4×D3—22no—
69C2×C4×D3—22no—
70C2×C4×D3D414yes—
71C2×Dic6—22no—
72C2×Dic6—22no—
73C2×Dic6—22no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—22no—
76C3×C22⋊C4—24no—
77#217—24no—
78#217—22no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—24no—
82C4⋊Dic3—24no—
83C4⋊Dic3D414yescomplemented
84Dic3⋊C4—24no—
85Dic3⋊C4—22no—
86Dic3⋊C4—24no—
87C4×Dic3—22no—
88C4×Dic3—24no—
89C22×Q8—31no—
90C22×D4—31no—
91C23×C4—31no—
92C23×C4C2×D311yescharacteristic
93C22⋊Q8—64no—
94C22⋊Q8—64no—
95C4×D4—64no—
96#117—64no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C4⋊C4C2×D311yescharacteristic
104C2×C22⋊C4—31no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4C2×D312yes—
110C2×C42—32no—
111C2×C42—31no—
112#95—32no—
113#95—32no—
114#95—32no—
115#95C2×D312yes—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3#4112yes—
123C23×C3C2411yescharacteristic
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3C2×D412yes—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—22no—
140C2×C3×C4—44no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4#4112yes—
144#67—28no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3C2×D414yes—
158C2×Dic3C2×D412yes—
159C2×Dic3#4112yes—
160C4×D3—28no—
161Dic6—44no—
162C24—31no—
163C24C22×D311yescharacteristic
164C2×Q8—62no—
165C2×Q8—62no—
166C2×Q8—62no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—31no—
176C22×C4—34no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4C22×D312yes—
182C22×C4C22×D312yes—
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—34no—
188C4⋊C4—62no—
189C4⋊C4—64no—
190C4⋊C4—64no—
191C4⋊C4—64no—
192C22⋊C4—24no—
193C22⋊C4—64no—
194C22⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—62no—
197C22⋊C4—64no—
198C42—64no—
199C42—62no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3C22×D411yescharacteristic
208C22×C3C22×D411yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3C2×#4111yescharacteristic
211C22×C3C2×#4111yescharacteristic
212C22×C32−1+411yescharacteristic
213C22×C32−1+411yescharacteristic
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—41no—
220C2×D3—41no—
221C3×C4—41no—
222C3×C4—42no—
223C3×C4—42no—
224C3×C4#12114yes—
225Dic3—22no—
226Dic3—22no—
227Dic3—24no—
228Dic3—42no—
229Dic3—41no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—32no—
237C23—61no—
238C23—61no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23#23712yes—
244C23C23×D311yescharacteristic
245Q8—124no—
246D4—68no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—34no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—44no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—68no—
267C2×C4—62no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4#23512yes—
274C2×C4D3×D412yes—
275C2×C4D3×D412yes—
276C2×C3—42no—
277C2×C3C2×#12111yescharacteristic
278C2×C3#52511yescharacteristic
279C2×C3#53611yescharacteristic
280C2×C3#54711yescharacteristic
281C2×C3#54811yescharacteristic
282C2×C3#55211yescharacteristic
283C2×C3#55611yescharacteristic
284D3—42no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—62no—
293C22—62no—
294C22—62no—
295C22—121no—
296C22—121no—
297C22—121no—
298C22—121no—
299C22C2×#23511yescharacteristic
300C22C2×D3×D411yescharacteristic
301C22C2×D3×D411yescharacteristic
302C22C2×#23711yescharacteristic
303C22#100711yescharacteristic
304C22D3×#4111yescharacteristic
305C22#101011yescharacteristic
306C4—41no—
307C4—42no—
308C4—42no—
309C4—62no—
310C4—62no—
311C4—64no—
312C4—122no—
313C4—121no—
314C4#88414yes—
315C3#241511yescharacteristic, complemented, Sylow
316C2—42no—
317C2—122no—
318C2C2×#88411yescharacteristic
319C2#551311yescharacteristic
320C2#553311yescharacteristic
321C2#559811yescharacteristic
322C2#560611yescharacteristic
323C2#562811yescharacteristic
324C2#563111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
prime radicalC22×#67C22
hypercenter#380D3
hypocenterC3#2415
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C2×#897C2semidirect
4. C2×#892C2semidirect
5. C3×#380C2semidirect
6. #4973C2semidirect
7. #4972C2semidirect
9. #4951C2semidirect
10. #4675C2semidirect
11. #4647C2semidirect
50. #380D3semidirect
83. C4⋊Dic3D4semidirect
315. C3#2415semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#672
C2×#9242
C2×#8972
C2×#8922
C3×#3802
#49732
#49722
#49592
#49512
#46752
#46472
#24153