Finite Group Databasebeta

← #76286

Subgroups — #76286

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×#895C211yesmaximal, characteristic
2C2×#894C212yesmaximal, complemented
3C2×C42×D3C211yesmaximal, characteristic, complemented
4C3×#389C211yesmaximal, characteristic, complemented
5#4990C212yesmaximal, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4972C211yesmaximal, characteristic, complemented
9#4939C211yesmaximal, characteristic, complemented
10#4669C212yesmaximal, complemented
11#4665C212yesmaximal, complemented
12#2854—31nomaximal, Sylow
13C22×D12C2212yes—
14C22×D12C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33#895—24no—
34#894—28no—
35C42×D3—24no—
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#123—32no—
40C2×#121—31no—
41C22×C42—31no—
42#390—31no—
43#390—31no—
44#389—31no—
45#389D311yescharacteristic, complemented
46#388—32no—
47#388—32no—
48#386—32no—
49#380—31no—
50C23×D3C2312yes—
51C23×D3C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C2×D12—24no—
62C2×D12—24no—
63C2×D12—24no—
64C2×D12—22no—
65C2×D12—22no—
66C2×D12—22no—
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—24no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C42—22no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86C4⋊Dic3—22no—
87Dic3⋊C4—24no—
88C4×Dic3—22no—
89C4×Dic3—22no—
90C4×Dic3—22no—
91C22×D4—32no—
92C22×D4—31no—
93C23×C4—31no—
94C23×C4—31no—
95C23×C4—31no—
96#123—68no—
97#121—64no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4C2×D311yescharacteristic
101C2×C4⋊C4C2×D312yes—
102C2×C4⋊C4C2×D311yescharacteristic
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×C42—31no—
110C2×C42—31no—
111C2×C42—31no—
112C2×C42—64no—
113C2×C42C2×D311yescharacteristic
114#95—32no—
115#95—32no—
116#95C2×D312yes—
117C23×C3C2411yescharacteristic
118C22×D3—21no—
119C22×D3—21no—
120C22×D3—21no—
121C22×D3—21no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3C2×D412yes—
130C22×D3C2×D412yes—
131C22×D3#4112yes—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yes—
145C2×C3×C4#4112yes—
146C2×C3×C4#4112yes—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3C2×D412yes—
157C2×Dic3C2×D412yes—
158C2×Dic3#4112yes—
159D12—48no—
160D12—44no—
161C4×D3—28no—
162C4×D3—28no—
163C4×D3—28no—
164C24—31no—
165C24—32no—
166C2×D4—64no—
167C2×D4—64no—
168C2×D4—64no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—62no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—34no—
176C22×C4—34no—
177C22×C4—32no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4C22×D312yes—
186C22×C4C22×D312yes—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—64no—
194C4⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—64no—
201C42—22no—
202C42—62no—
203C42—62no—
204C42—62no—
205C22×C3C22×D411yescharacteristic
206C22×C3C22×D411yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3C2×#4111yescharacteristic
211C22×C3C2×#4111yescharacteristic
212C2×D3—22no—
213C2×D3—22no—
214C2×D3—22no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—22no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3#12114yes—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—42no—
229C3×C4—42no—
230C3×C4#12314yescomplemented
231Dic3—22no—
232Dic3—22no—
233Dic3—22no—
234Dic3—42no—
235C23—32no—
236C23—32no—
237C23—32no—
238C23—61no—
239C23—61no—
240C23—61no—
241C23—61no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23—62no—
246C23—62no—
247C23—62no—
248C23—62no—
249C23C23×D311yescharacteristic
250D4—128no—
251D4—124no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—32no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—68no—
266C2×C4—68no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—68no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4#23512yes—
278C2×C4D3×D412yes—
279C2×C4D3×D412yes—
280C2×C4#23912yes—
281C2×C4#23912yes—
282C2×C3C2×#12111yescharacteristic
283C2×C3C2×#12311yescharacteristic
284C2×C3#52511yescharacteristic
285C2×C3#52911yescharacteristic
286C2×C3#52911yescharacteristic
287C2×C3#53211yescharacteristic
288C2×C3#53211yescharacteristic
289D3—24no—
290D3—82no—
291C22—34no—
292C22—62no—
293C22—62no—
294C22—62no—
295C22—62no—
296C22—62no—
297C22—62no—
298C22—122no—
299C22—122no—
300C22—122no—
301C22—122no—
302C22—122no—
303C22—122no—
304C22—122no—
305C22C2×#23511yescharacteristic
306C22C2×D3×D411yescharacteristic
307C22C2×D3×D411yescharacteristic
308C22C2×#23911yescharacteristic
309C22C2×#23911yescharacteristic
310C22D3×#4111yescharacteristic
311C22D3×#4111yescharacteristic
312C4—22no—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—62no—
317C4—62no—
318C4—62no—
319C4—122no—
320C4#89414yescomplemented
321C3#285411yescharacteristic, complemented, Sylow
322C2—64no—
323C2—242no—
324C2C2×#89411yescharacteristic
325C2#554811yescharacteristic
326C2#558011yescharacteristic
327C2D3×#12111yescharacteristic
328C2#562711yescharacteristic
329C2#569111yescharacteristic
330C2#572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#389C2
prime radicalC2×#895C2
hypercenter#389D3
hypocenterC3#2854
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#894C2semidirect
3. C2×C42×D3C2semidirect
4. C3×#389C2semidirect
5. #4990C2semidirect
6. #4988C2semidirect
7. #4987C2semidirect
8. #4972C2semidirect
9. #4939C2semidirect
10. #4669C2semidirect
11. #4665C2semidirect
45. #389D3semidirect
75. C2×C4×D3D4semidirect
230. C3×C4#123semidirect
320. C4#894semidirect
321. C3#2854semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8942
C2×C42×D32
C3×#3892
#49902
#49882
#49872
#49722
#49392
#46692
#46652
#28543