Finite Group Databasebeta

← #82643

Subgroups — #82643

Order 384.

596 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic
2C2×D4×Dic3C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal, complemented
4#5221C211yesmaximal, characteristic, complemented
5#5220C212yesmaximal, complemented
6#5212C211yesmaximal, characteristic, complemented
7C4×#217C211yesmaximal, characteristic, complemented
8#4977C211yesmaximal, characteristic, complemented
9#4972C211yesmaximal, characteristic, complemented
10#4952C212yesmaximal, complemented
11#4943C212yesmaximal, complemented
12#2420—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C3×#121—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26D4×Dic3—24no—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2212yes—
30C2×C4×Dic3C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#121D311yescharacteristic
39C2×C4×D4—31no—
40#397—32no—
41#394—32no—
42#390—31no—
43#389—31no—
44#386—32no—
45#382—31no—
46#381—32no—
47#380—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—42no—
56C2×C3×D4D414yes—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C22×Dic3—24no—
64C22×Dic3—24no—
65C22×Dic3—42no—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C3×C4⋊C4—22no—
73C3×C22⋊C4—24no—
74#217—24no—
75#217—22no—
76#217—22no—
77#217—42no—
78#217—42no—
79#217—42no—
80C4⋊Dic3—22no—
81C4⋊Dic3—22no—
82Dic3⋊C4—42no—
83C4×Dic3—22no—
84C4×Dic3—22no—
85C22×D4C2×D312yes—
86C22×D4C2×D311yescharacteristic
87C23×C4—32no—
88C23×C4C2×D311yescharacteristic
89#121—24no—
90C4×D4—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4C2×D312yes—
102C2×C42—31no—
103C2×C42—31no—
104#95—31no—
105#95—31no—
106#95—32no—
107#95—32no—
108#95—32no—
109#95—32no—
110C23×C3—21no—
111C23×C3—21no—
112C23×C3—22no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—21no—
122C23×C3C2×D414yes—
123C23×C3#4112yes—
124C23×C3C2411yescharacteristic
125C3×D4—28no—
126C3×D4—48no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—44no—
134C2×C3×C4C2×D412yes—
135C2×C3×C4#4112yes—
136C2×C3×C4#4112yes—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3—22no—
154C2×Dic3—48no—
155C2×Dic3#4112yes—
156C24C22×D311yescharacteristic
157C24C22×D312yes—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—42no—
163C2×D4#6714yes—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—32no—
171C22×C4—32no—
172C22×C4—31no—
173C22×C4—64no—
174C22×C4—64no—
175C22×C4—122no—
176C22×C4C22×D311yescharacteristic
177C22×C4C22×D312yes—
178C22×C4C22×D311yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—122no—
183C22⋊C4—24no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—64no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C42—62no—
191C42—62no—
192C22×C3—24no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—22no—
196C22×C3—24no—
197C22×C3—22no—
198C22×C3—41no—
199C22×C3—42no—
200C22×C3—41no—
201C22×C3—42no—
202C22×C3—41no—
203C22×C3—42no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3C22×D411yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3C2×#4111yescharacteristic
211C22×C3H(5,𝔽2)11yescharacteristic
212C22×C3H(5,𝔽2)11yescharacteristic
213C3×C4—41no—
214C3×C4—42no—
215C3×C4#12314yes—
216Dic3—22no—
217Dic3—42no—
218Dic3—41no—
219Dic3—42no—
220Dic3—41no—
221Dic3—41no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23#23712yes—
235C23C2×#6714yes—
236C23C23×D311yescharacteristic
237D4—28no—
238D4—48no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—32no—
246C2×C4—44no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—62no—
264C2×C4—128no—
265C2×C4#23712yes—
266C2×C4#23712yes—
267C2×C4C2×#6712yes—
268C2×C3—44no—
269C2×C3—42no—
270C2×C3C2×#12311yescharacteristic
271C2×C3#52911yescharacteristic
272C2×C3#54011yescharacteristic
273C2×C3#54611yescharacteristic
274C2×C3#54611yescharacteristic
275C2×C3#55311yescharacteristic
276C2×C3#55511yescharacteristic
277C22—24no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—22no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—42no—
291C22C2×#23711yescharacteristic
292C22C2×#23711yescharacteristic
293C22C2×#23711yescharacteristic
294C22#100411yescharacteristic
295C22D3×#4111yescharacteristic
296C22#100911yescharacteristic
297C22C22×#6711yescharacteristic
298C4—41no—
299C4—42no—
300C4—62no—
301C4—121no—
302C4—122no—
303C4—121no—
304C4—121no—
305C4—122no—
306C4#93514yes—
307C3#242011yescharacteristic, complemented, Sylow
308C2—44no—
309C2—42no—
310C2#560011yescharacteristic
311C2#561611yescharacteristic
312C2#561811yescharacteristic
313C2#561911yescharacteristic
314C2C2×#93511yescharacteristic
315C2#580611yescharacteristic
316C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2420
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic3C2semidirect
3. #5222C2semidirect
4. #5221C2semidirect
5. #5220C2semidirect
6. #5212C2semidirect
7. C4×#217C2semidirect
8. #4977C2semidirect
9. #4972C2semidirect
10. #4952C2semidirect
11. #4943C2semidirect
307. C3#2420semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×D4×Dic32
#52222
#52212
#52202
#52122
C4×#2172
#49772
#49722
#49522
#49432
#24203