Finite Group Databasebeta

← #82995

Subgroups — #82995

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×C3×#124C211yesmaximal, characteristic, complemented
2C2×#946C211yesmaximal, characteristic, complemented
3C2×Q8×Dic3C211yesmaximal, characteristic, complemented
4C2×#938C211yesmaximal, characteristic
5#5221C211yesmaximal, characteristic, complemented
6#4961C212yesmaximal, complemented
7#4957C212yesmaximal, complemented
8#4948C212yesmaximal, complemented
9#4945C212yesmaximal, complemented
10C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2426—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C22×C4×D3C2211yescharacteristic
17C3×#124—24no—
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C42C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24#946—24no—
25Q8×Dic3—24no—
26#938—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2212yes—
32C2×C4⋊Dic3C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×Dic3⋊C4C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36C2×C4×Dic3C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2211yescharacteristic
40C2×#124D311yescharacteristic, complemented
41C2×C22⋊Q8—31no—
42C2×#121—31no—
43C2×C4×Q8—31no—
44#396—32no—
45#394—32no—
46#390—31no—
47#390—32no—
48#389—31no—
49#388—32no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×Q8—22no—
54C2×C3×Q8—22no—
55C2×C3×Q8D414yescomplemented
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67—24no—
64C2×#67—24no—
65C2×#67—42no—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C2×C4×D3—22no—
72C2×C4×D3—22no—
73C2×C4×D3—22no—
74C3×C22⋊C4—24no—
75C3×C22⋊C4—24no—
76C3×C42—22no—
77#217—24no—
78#217—24no—
79#217—42no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—22no—
83C4⋊Dic3—22no—
84C4⋊Dic3—24no—
85C4⋊Dic3D414yes—
86Dic3⋊C4—24no—
87Dic3⋊C4—42no—
88C4×Dic3—22no—
89C4×Dic3—24no—
90C22×Q8C2×D311yescharacteristic
91C22×D4—32no—
92C22×D4C2×D311yescharacteristic
93C23×C4—31no—
94#124—24no—
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—31no—
107C2×C22⋊C4—31no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4C2×D312yes—
110C2×C22⋊C4C2×D312yes—
111C2×C42—31no—
112C2×C42—32no—
113C2×C42C2×D311yescharacteristic
114#95—31no—
115#95—31no—
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×D3—21no—
130C22×D3—21no—
131C22×D3#4112yes—
132C3×Q8—28no—
133C3×D4—44no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
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×C4C2×D412yes—
143C2×C3×C4C2×D414yes—
144C2×C3×C4C2×D412yes—
145C2×C3×C4#4112yes—
146#67—48no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—22no—
157C2×Dic3—21no—
158C2×Dic3—22no—
159C2×Dic3—21no—
160C2×Dic3C2×D414yes—
161C2×Dic3#4112yes—
162C4×D3—44no—
163C24—31no—
164C24C22×D312yes—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8#6714yescomplemented
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—64no—
172C2×D4—64no—
173C2×D4—122no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D311yescharacteristic
184C22×C4C22×D312yes—
185C22×C4C22×D311yescharacteristic
186C4⋊C4—34no—
187C4⋊C4—64no—
188C4⋊C4—64no—
189C4⋊C4—62no—
190C4⋊C4—122no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—122no—
199C42—22no—
200C42—64no—
201C42—62no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—42no—
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—22no—
217C2×D3—22no—
218C2×D3—22no—
219C2×D3—41no—
220C2×D3—41no—
221C2×D3—41no—
222C2×D3—41no—
223C3×C4—24no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4#12114yes—
227Dic3—24no—
228Dic3—22no—
229Dic3—41no—
230Dic3—41no—
231Dic3—42no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—32no—
240C23—61no—
241C23—61no—
242C23—61no—
243C23—61no—
244C23—61no—
245C23—61no—
246C23C23×D311yescharacteristic
247Q8—28no—
248D4—44no—
249D4—128no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—34no—
259C2×C4—32no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—124no—
274C2×C4D3×D412yes—
275C2×C4#23712yes—
276C2×C4C2×#6714yes—
277C2×C4C2×#6712yes—
278C2×C3—82no—
279C2×C3C2×#12111yescharacteristic
280C2×C3#52911yescharacteristic
281C2×C3#54211yescharacteristic
282C2×C3#54611yescharacteristic
283C2×C3#54811yescharacteristic
284C2×C3#54811yescharacteristic
285C2×C3#55011yescharacteristic
286D3—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—62no—
295C22—62no—
296C22—62no—
297C22—121no—
298C22—121no—
299C22—121no—
300C22—121no—
301C22C2×D3×D411yescharacteristic
302C22C2×#23711yescharacteristic
303C22D3×#4111yescharacteristic
304C22D3×#4111yescharacteristic
305C22#100911yescharacteristic
306C22#101011yescharacteristic
307C22C22×#6711yescharacteristic
308C4—24no—
309C4—41no—
310C4—41no—
311C4—64no—
312C4—62no—
313C4—121no—
314C4—121no—
315C4—122no—
316C4#93914yes—
317C3#242611yescharacteristic, complemented, Sylow
318C2—82no—
319C2—122no—
320C2#567611yescharacteristic
321C2#567711yescharacteristic
322C2#568011yescharacteristic
323C2#568211yescharacteristic
324C2C2×#93911yescharacteristic
325C2#583111yescharacteristic
326C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC2×#938C2
hypercenterC2×#124D3
hypocenterC3#2426
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
2. C2×#946C2semidirect
3. C2×Q8×Dic3C2semidirect
5. #5221C2semidirect
6. #4961C2semidirect
7. #4957C2semidirect
8. #4948C2semidirect
9. #4945C2semidirect
10. C4×(C2×D3)⋊C4C2semidirect
11. #4928C2semidirect
40. C2×#124D3semidirect
55. C2×C3×Q8D4semidirect
167. C2×Q8#67semidirect
317. C3#2426semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#9462
C2×Q8×Dic32
C2×#9382
#52212
#49612
#49572
#49482
#49452
C4×(C2×D3)⋊C42
#49282
#24263