Finite Group Databasebeta

← #73915

Subgroups — #73915

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×#930C211yesmaximal, characteristic
2C2×#888C211yesmaximal, characteristic, complemented
3C2×#885C212yesmaximal, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5C4×#217C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4988C211yesmaximal, characteristic, complemented
8Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
9#4959C212yesmaximal, complemented
10#4673C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2491—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C22×Dic6C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2211yescharacteristic
22#930—24no—
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×Dic3C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33#888—24no—
34#885—28no—
35C3×#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×C4⋊Q8—31no—
40C2×#124—32no—
41C2×#121—31no—
42#394—32no—
43#393—32no—
44#390—31no—
45#390—31no—
46#388—32no—
47#380D311yescharacteristic, complemented
48C4×C4⋊C4—31no—
49C4×C22⋊C4—31no—
50C24×C3C2311yescharacteristic
51C23×D3C2312yes—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—24no—
61C2×#67—42no—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2312yes—
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2311yescharacteristic
67C2×D12—22no—
68C2×D12—22no—
69C2×D12—22no—
70C2×Dic6—24no—
71C2×Dic6—24no—
72C2×Dic6—42no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—22no—
75C3×C22⋊C4—24no—
76#217—22no—
77#217D414yescomplemented
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—42no—
83C4⋊Dic3D414yes—
84Dic3⋊C4—24no—
85C4×Dic3—22no—
86C4×Dic3—22no—
87C4×Dic3—24no—
88C22×Q8—32no—
89C22×D4—32no—
90C22×D4—31no—
91C23×C4C2×D311yescharacteristic
92C4⋊Q8—64no—
93#124—68no—
94#121—64no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C4⋊C4C2×D311yescharacteristic
99C2×C22⋊C4—31no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
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#95C2×D312yes—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3C2×D412yes—
121C23×C3C2411yescharacteristic
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—44no—
139C2×C3×C4C2×D412yes—
140C2×C3×C4#4112yes—
141C2×C3×C4#4112yes—
142#67—48no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3C2×D412yes—
155C2×Dic3C2×D414yes—
156C2×Dic3C2×D412yes—
157C2×Dic3#4112yes—
158D12—44no—
159Dic6—48no—
160C24—32no—
161C24C22×D311yescharacteristic
162C2×Q8—64no—
163C2×Q8—64no—
164C2×Q8—122no—
165C2×D4—64no—
166C2×D4—64no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—122no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—31no—
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D312yes—
181C22×C4C22×D312yes—
182C22×C4C22×D311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—34no—
186C4⋊C4—64no—
187C22⋊C4—24no—
188C22⋊C4—34no—
189C22⋊C4—64no—
190C22⋊C4—64no—
191C22⋊C4—64no—
192C22⋊C4—64no—
193C22⋊C4—62no—
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×C3C22×D411yescharacteristic
206C22×C3C22×D411yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3H(5,𝔽2)11yescharacteristic
211C22×C32−1+411yescharacteristic
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C3×C4—41no—
220C3×C4—42no—
221C3×C4—42no—
222C3×C4#12414yescomplemented
223Dic3—22no—
224Dic3—24no—
225Dic3—22no—
226Dic3—22no—
227Dic3—41no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—62no—
235C23—62no—
236C23—62no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23D3×D412yes—
242C23C23×D311yescharacteristic
243Q8—128no—
244D4—128no—
245D4—124no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—32no—
256C2×C4—32no—
257C2×C4—34no—
258C2×C4—32no—
259C2×C4—44no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4#23512yes—
272C2×C4D3×D412yes—
273C2×C4#23712yes—
274C2×C3—42no—
275C2×C3C2×#12411yescharacteristic
276C2×C3#53211yescharacteristic
277C2×C3#54211yescharacteristic
278C2×C3D4211yescharacteristic
279C2×C3#54711yescharacteristic
280C2×C3#54811yescharacteristic
281C2×C3#55511yescharacteristic
282D3—82no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—122no—
291C22—122no—
292C22—122no—
293C22—122no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22C2×#23511yescharacteristic
298C22C2×D3×D411yescharacteristic
299C22C2×D3×D411yescharacteristic
300C22C2×#23711yescharacteristic
301C22#100711yescharacteristic
302C22D3×#4111yescharacteristic
303C22#100911yescharacteristic
304C4—41no—
305C4—42no—
306C4—42no—
307C4—62no—
308C4—62no—
309C4—64no—
310C4—62no—
311C4—121no—
312C4#88514yescomplemented
313C3#249111yescharacteristic, complemented, Sylow
314C2—42no—
315C2—242no—
316C2C2×#88511yescharacteristic
317C2#550711yescharacteristic
318C2#553211yescharacteristic
319C2#559611yescharacteristic
320C2#560911yescharacteristic
321C2#563311yescharacteristic
322C2#564311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#380C2
prime radicalC2×#930C2
hypercenter#380D3
hypocenterC3#2491
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#888C2semidirect
3. C2×#885C2semidirect
4. C3×#380C2semidirect
5. C4×#217C2semidirect
6. #4988C2semidirect
7. #4988C2semidirect
8. Dic3×C4⋊C4C2semidirect
9. #4959C2semidirect
10. #4673C2semidirect
11. #4669C2semidirect
47. #380D3semidirect
77. #217D4semidirect
222. C3×C4#124semidirect
312. C4#885semidirect
313. C3#2491semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#8882
C2×#8852
C3×#3802
C4×#2172
#49882
Dic3×C4⋊C42
#49592
#46732
#46692
#24913