Finite Group Databasebeta

← #76475

Subgroups — #76475

Order 384.

688 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×#938C211yesmaximal, characteristic
2C2×#883C212yesmaximal, complemented
3C2×C42×D3C211yesmaximal, characteristic, complemented
4C3×#390C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4984C211yesmaximal, characteristic, complemented
8#4956C212yesmaximal, complemented
9#4951C212yesmaximal, complemented
10#4945C212yesmaximal, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2854—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25#938—24no—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34#883—28no—
35C42×D3—24no—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#123—32no—
40C2×#121—31no—
41C22×C42—31no—
42#390—31no—
43#390D311yescharacteristic, complemented
44#389—31no—
45#389—31no—
46#388—32no—
47#388—32no—
48#386—32no—
49#380—31no—
50C24×C3C2312yes—
51C23×D3C2311yescharacteristic
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—24no—
62C22×Dic3C2312yes—
63C22×Dic3C2312yes—
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—22no—
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3—22no—
73C2×C4×D3—22no—
74C2×C4×D3D414yes—
75C2×C4×D3D414yescomplemented
76C3×C4⋊C4—22no—
77C3×C22⋊C4—24no—
78C3×C22⋊C4—24no—
79C3×C42—22no—
80#217—24no—
81#217—24no—
82#217—24no—
83(C2×D3)⋊C4—24no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—24no—
87Dic3⋊C4—24no—
88C4×Dic3—22no—
89C4×Dic3—22no—
90C4×Dic3—22no—
91C22×D4—32no—
92C22×D4C2×D311yescharacteristic
93C23×C4—31no—
94C23×C4—31no—
95C23×C4—31no—
96#123—68no—
97#121—64no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D312yes—
108C2×C22⋊C4C2×D312yes—
109C2×C42—31no—
110C2×C42—31no—
111C2×C42—31no—
112C2×C42—64no—
113C2×C42C2×D311yescharacteristic
114#95—32no—
115#95—32no—
116#95—32no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3C2411yescharacteristic
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3C2×D412yes—
130C22×D3C2×D412yes—
131C22×D3#4112yes—
132C3×D4—44no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2×D412yes—
142C2×C3×C4#4112yes—
143C2×C3×C4#4112yes—
144C2×C3×C4#4112yes—
145#67—48no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3C2×D412yes—
159C2×Dic3C2×D412yes—
160C2×Dic3#4112yes—
161C4×D3—28no—
162C4×D3—28no—
163C4×D3—28no—
164C24—31no—
165C24C22×D312yes—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—64no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—34no—
176C22×C4—34no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C4⋊C4—64no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C42—22no—
202C42—62no—
203C42—62no—
204C42—62no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3C22×D411yescharacteristic
213C22×C3C22×D411yescharacteristic
214C22×C3C2×#4111yescharacteristic
215C22×C3C2×#4111yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3C2×#4111yescharacteristic
218C22×C3C2×#4111yescharacteristic
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—22no—
224C2×D3—22no—
225C2×D3#12114yes—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—42no—
229C3×C4#12314yescomplemented
230Dic3—22no—
231Dic3—22no—
232Dic3—22no—
233Dic3—42no—
234Dic3—42no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—32no—
243C23—32no—
244C23—32no—
245C23—61no—
246C23—61no—
247C23—61no—
248C23—61no—
249C23C23×D311yescharacteristic
250D4—44no—
251D4—128no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—68no—
263C2×C4—68no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—68no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4#23512yes—
278C2×C4D3×D412yes—
279C2×C4D3×D412yes—
280C2×C4#23712yes—
281C2×C4#23712yes—
282C2×C3—82no—
283C2×C3C2×#12111yescharacteristic
284C2×C3C2×#12311yescharacteristic
285C2×C3#52511yescharacteristic
286C2×C3#52911yescharacteristic
287C2×C3#52911yescharacteristic
288C2×C3#53211yescharacteristic
289C2×C3#53211yescharacteristic
290D3—24no—
291C22—34no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—62no—
300C22—62no—
301C22—62no—
302C22—62no—
303C22—62no—
304C22—62no—
305C22C2×#23511yescharacteristic
306C22C2×D3×D411yescharacteristic
307C22C2×D3×D411yescharacteristic
308C22C2×#23711yescharacteristic
309C22C2×#23711yescharacteristic
310C22D3×#4111yescharacteristic
311C22D3×#4111yescharacteristic
312C4—22no—
313C4—22no—
314C4—42no—
315C4—62no—
316C4—62no—
317C4—62no—
318C4—122no—
319C4—122no—
320C4#88314yescomplemented
321C3#285411yescharacteristic, complemented, Sylow
322C2—64no—
323C2—82no—
324C2C2×#88311yescharacteristic
325C2#554811yescharacteristic
326C2#555711yescharacteristic
327C2D3×#12111yescharacteristic
328C2#560611yescharacteristic
329C2#568011yescharacteristic
330C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#390C2
prime radicalC2×#938C2
hypercenter#390D3
hypocenterC3#2854
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#883C2semidirect
3. C2×C42×D3C2semidirect
4. C3×#390C2semidirect
5. #5221C2semidirect
6. #4987C2semidirect
7. #4984C2semidirect
8. #4956C2semidirect
9. #4951C2semidirect
10. #4945C2semidirect
11. #4928C2semidirect
43. #390D3semidirect
75. C2×C4×D3D4semidirect
229. C3×C4#123semidirect
320. C4#883semidirect
321. C3#2854semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#8832
C2×C42×D32
C3×#3902
#52212
#49872
#49842
#49562
#49512
#49452
#49282
#28543