Finite Group Databasebeta

← #82661

Subgroups — #82661

Order 384.

560 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic
2C2×#890C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal, complemented
4#5221C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5221C212yesmaximal, complemented
7#5212C211yesmaximal, characteristic, complemented
8C4×#217C211yesmaximal, characteristic, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4953C212yesmaximal, complemented
11#4945C212yesmaximal, complemented
12#2490—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×C3×C4C2211yescharacteristic
16C3×#121—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32C2×C4×Dic3C2212yes—
33#890—24no—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#121D311yescharacteristic
40#397—32no—
41#394—32no—
42#390—31no—
43#390—31no—
44#390—32no—
45#388—32no—
46#380—31no—
47C4×C4⋊C4—31no—
48C4×C22⋊C4—31no—
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C2×C3×D4—24no—
52C2×C3×D4—24no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—42no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C3×C4⋊C4D414yes—
70C3×C22⋊C4—24no—
71#217—22no—
72#217—22no—
73#217—24no—
74#217—42no—
75#217—42no—
76#217—42no—
77#217—42no—
78C4⋊Dic3—24no—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81Dic3⋊C4—24no—
82C4×Dic3—22no—
83C4×Dic3—22no—
84C4×Dic3—42no—
85C22×D4C2×D311yescharacteristic
86C22×D4C2×D312yes—
87C23×C4C2×D311yescharacteristic
88#125—64no—
89#121—24no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4C2×D312yes—
103C2×C42—31no—
104C2×C42—31no—
105C2×C42—32no—
106#95—31no—
107#95—31no—
108#95—32no—
109#95—32no—
110C23×C3—21no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3#4112yes—
124C23×C3C2411yescharacteristic
125C3×D4—48no—
126C3×D4—44no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—44no—
133C2×C3×C4C2×D414yes—
134C2×C3×C4C2×D412yes—
135C2×C3×C4#4112yes—
136C2×C3×C4#4112yes—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—22no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3#4112yes—
155C24C22×D312yes—
156C24C22×D311yescharacteristic
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—24no—
161C2×D4—22no—
162C2×D4—42no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—32no—
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C4⋊C4—62no—
176C4⋊C4—64no—
177C4⋊C4—64no—
178C4⋊C4—62no—
179C4⋊C4#6714yes—
180C22⋊C4—24no—
181C22⋊C4—62no—
182C22⋊C4—64no—
183C22⋊C4—62no—
184C22⋊C4—122no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C42—62no—
189C42—62no—
190C42—122no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3C22×D411yescharacteristic
206C22×C3C2×#4111yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3H(5,𝔽2)11yescharacteristic
211C22×C3H(5,𝔽2)11yescharacteristic
212C3×C4—24no—
213C3×C4—41no—
214C3×C4#12414yes—
215Dic3—22no—
216Dic3—42no—
217Dic3—41no—
218Dic3—41no—
219Dic3—42no—
220Dic3—41no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23#23712yes—
235C23C23×D311yescharacteristic
236D4—48no—
237D4—44no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—32no—
244C2×C4—44no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4#23712yes—
263C2×C4#23712yes—
264C2×C4C2×#6714yes—
265C2×C4C2×#6712yes—
266C2×C3—42no—
267C2×C3—82no—
268C2×C3C2×#12411yescharacteristic
269C2×C3#52911yescharacteristic
270C2×C3#54011yescharacteristic
271C2×C3#54611yescharacteristic
272C2×C3#55011yescharacteristic
273C2×C3#55311yescharacteristic
274C2×C3#55311yescharacteristic
275C22—22no—
276C22—22no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22C2×#23711yescharacteristic
290C22C2×#23711yescharacteristic
291C22C2×#23711yescharacteristic
292C22#100411yescharacteristic
293C22D3×#4111yescharacteristic
294C22#100911yescharacteristic
295C22C22×#6711yescharacteristic
296C4—24no—
297C4—41no—
298C4—62no—
299C4—121no—
300C4—121no—
301C4—122no—
302C4—121no—
303C4—122no—
304C4#93614yes—
305C3#249011yescharacteristic, complemented, Sylow
306C2—42no—
307C2—82no—
308C2#560011yescharacteristic
309C2#561411yescharacteristic
310C2#561811yescharacteristic
311C2#561911yescharacteristic
312C2C2×#93611yescharacteristic
313C2#580611yescharacteristic
314C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2490
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#890C2semidirect
3. #5222C2semidirect
4. #5221C2semidirect
5. #5221C2semidirect
6. #5221C2semidirect
7. #5212C2semidirect
8. C4×#217C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4953C2semidirect
11. #4945C2semidirect
305. C3#2490semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#8902
#52222
#52212
#52122
C4×#2172
Dic3×C4⋊C42
#49532
#49452
#24903