Finite Group Databasebeta

← #82662

Subgroups — #82662

Order 384.

632 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic
2C2×#936C212yesmaximal, complemented
3C2×#888C211yesmaximal, characteristic, complemented
4#5222C212yesmaximal, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5221C211yesmaximal, characteristic, complemented
7#5212C211yesmaximal, characteristic, complemented
8C4×#217C211yesmaximal, characteristic, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4947C212yesmaximal, complemented
11#4945C212yesmaximal, complemented
12#2491—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×C3×C4C2211yescharacteristic
16C22×Dic6C2212yes—
17C3×#121—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×#217C2211yescharacteristic
27#936—28no—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33C2×C4×Dic3C2211yescharacteristic
34#888—24no—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×C4⋊Q8—31no—
40C2×#124—32no—
41C2×#121D311yescharacteristic
42#394—32no—
43#393—32no—
44#390—31no—
45#390—31no—
46#388—32no—
47#380—31no—
48C4×C4⋊C4—31no—
49C4×C22⋊C4—31no—
50C24×C3C2312yes—
51C24×C3C2311yescharacteristic
52C2×C3×D4—24no—
53C2×C3×D4—24no—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—22no—
57C2×C3×D4—42no—
58C22×C3×C4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×Dic6—24no—
71C2×Dic6—24no—
72C2×Dic6—42no—
73C3×C4⋊C4D414yes—
74C3×C22⋊C4—24no—
75#217—24no—
76#217—24no—
77#217—24no—
78#217—22no—
79#217—24no—
80#217—42no—
81#217D414yescomplemented
82C4⋊Dic3—22no—
83C4⋊Dic3—22no—
84Dic3⋊C4—24no—
85C4×Dic3—22no—
86C4×Dic3—22no—
87C4×Dic3—24no—
88C22×Q8—32no—
89C22×D4C2×D311yescharacteristic
90C22×D4C2×D312yes—
91C23×C4C2×D311yescharacteristic
92C4⋊Q8—64no—
93#124—68no—
94#121—24no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4C2×D312yes—
107C2×C42—32no—
108C2×C42—31no—
109C2×C42—31no—
110#95—31no—
111#95—31no—
112#95—32no—
113#95—32no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—22no—
126C23×C3—22no—
127C23×C3C2×D412yes—
128C23×C3C2411yescharacteristic
129C3×D4—48no—
130C3×D4—44no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—44no—
137C2×C3×C4C2×D414yes—
138C2×C3×C4C2×D412yes—
139C2×C3×C4#4112yes—
140C2×C3×C4#4112yes—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156C2×Dic3C2×D412yes—
157C2×Dic3C2×D412yes—
158C2×Dic3#4112yes—
159Dic6—48no—
160C24C22×D312yes—
161C24C22×D311yescharacteristic
162C2×Q8—64no—
163C2×Q8—64no—
164C2×Q8—122no—
165C2×D4—24no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—42no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—32no—
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D312yes—
182C22×C4C22×D311yescharacteristic
183C4⋊C4—62no—
184C4⋊C4—64no—
185C4⋊C4—62no—
186C4⋊C4#6714yes—
187C22⋊C4—24no—
188C22⋊C4—34no—
189C22⋊C4—64no—
190C22⋊C4—64no—
191C22⋊C4—62no—
192C22⋊C4—64no—
193C22⋊C4—64no—
194C22⋊C4—122no—
195C42—64no—
196C42—62no—
197C42—62no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—22no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—41no—
204C22×C3—41no—
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×C3H(5,𝔽2)11yescharacteristic
218C22×C32−1+411yescharacteristic
219C3×C4—24no—
220C3×C4—41no—
221C3×C4#12414yescomplemented
222Dic3—22no—
223Dic3—22no—
224Dic3—22no—
225Dic3—42no—
226Dic3—42no—
227Dic3—41no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23D3×D412yes—
242C23C23×D311yescharacteristic
243Q8—128no—
244D4—48no—
245D4—44no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—44no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—61no—
269C2×C4—62no—
270C2×C4#23712yes—
271C2×C4#23712yes—
272C2×C4C2×#6714yes—
273C2×C4C2×#6712yes—
274C2×C3—42no—
275C2×C3—82no—
276C2×C3C2×#12411yescharacteristic
277C2×C3#53211yescharacteristic
278C2×C3#54211yescharacteristic
279C2×C3D4211yescharacteristic
280C2×C3#54711yescharacteristic
281C2×C3#54811yescharacteristic
282C2×C3#55511yescharacteristic
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22C2×D3×D411yescharacteristic
298C22C2×#23711yescharacteristic
299C22C2×#23711yescharacteristic
300C22#100411yescharacteristic
301C22D3×#4111yescharacteristic
302C22#101011yescharacteristic
303C22C22×#6711yescharacteristic
304C4—24no—
305C4—41no—
306C4—62no—
307C4—62no—
308C4—62no—
309C4—122no—
310C4—122no—
311C4—121no—
312C4#93614yescomplemented
313C3#249111yescharacteristic, complemented, Sylow
314C2—42no—
315C2—82no—
316C2#559611yescharacteristic
317C2#560311yescharacteristic
318C2#561111yescharacteristic
319C2#561611yescharacteristic
320C2C2×#93611yescharacteristic
321C2D4×#6711yescharacteristic
322C2#583111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2491
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#936C2semidirect
3. C2×#888C2semidirect
4. #5222C2semidirect
5. #5221C2semidirect
6. #5221C2semidirect
7. #5212C2semidirect
8. C4×#217C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4947C2semidirect
11. #4945C2semidirect
81. #217D4semidirect
221. C3×C4#124semidirect
312. C4#936semidirect
313. C3#2491semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9362
C2×#8882
#52222
#52212
#52122
C4×#2172
Dic3×C4⋊C42
#49472
#49452
#24913