Finite Group Databasebeta

← #78090

Subgroups — #78090

Order 384.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#897C211yesmaximal, characteristic, complemented
2C2×#884C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4991C212yesmaximal, complemented
6#4982C212yesmaximal, complemented
7#4961C212yesmaximal, complemented
8#4959C212yesmaximal, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4664C211yesmaximal, characteristic, complemented
12#2816—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×Dic6C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31#897—24no—
32#884—24no—
33C3×#95C2211yescharacteristic
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C22⋊Q8—31no—
39C2×#121—31no—
40#399—31no—
41#399D311yescharacteristic, complemented
42#398—32no—
43#394—32no—
44#394—32no—
45#394—32no—
46#393—31no—
47#393—31no—
48#381—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—22no—
57C2×#67—22no—
58C2×#67—42no—
59C2×#67—42no—
60C22×Dic3C2312yes—
61C22×Dic3C2311yescharacteristic
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C2×D12—22no—
65C2×D12—42no—
66C2×C4×D3—42no—
67C2×C4×D3D414yes—
68C2×Dic6—22no—
69C2×Dic6—42no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—42no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—42no—
74#217—42no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4—42no—
80C4⋊Dic3—24no—
81Dic3⋊C4—22no—
82Dic3⋊C4—42no—
83C22×Q8—31no—
84C22×D4—31no—
85C22×D4—31no—
86C22×D4—31no—
87C23×C4—31no—
88C22⋊Q8—64no—
89#121—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D312yes—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4C2×D311yescharacteristic
102C2×C22⋊C4C2×D312yes—
103#95—31no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95C2×D311yescharacteristic
108C23×C3—21no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3C2411yescharacteristic
114C22×D3—22no—
115C22×D3—22no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—21no—
122C22×D3—21no—
123C22×D3C2×D412yes—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134#67—44no—
135#67—44no—
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3C2×D412yes—
146D12—44no—
147C4×D3—44no—
148Dic6—44no—
149C24—31no—
150C24—31no—
151C24C22×D311yescharacteristic
152C2×Q8—62no—
153C2×Q8—122no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4—62no—
157C2×D4—122no—
158C2×D4—122no—
159C2×D4—122no—
160C22×C4—31no—
161C22×C4—34no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—31no—
165C22×C4—122no—
166C22×C4C22×D312yes—
167C22×C4C22×D311yescharacteristic
168C22×C4C22×D312yes—
169C22×C4C22×D311yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—62no—
173C4⋊C4—64no—
174C4⋊C4—122no—
175C22⋊C4—22no—
176C22⋊C4—42no—
177C22⋊C4—64no—
178C22⋊C4—62no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C22⋊C4—122no—
182C22⋊C4—122no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3C22×D411yescharacteristic
189C22×C3H(5,𝔽2)12yes—
190C22×C3H(5,𝔽2)12yes—
191C22×C32−1+411yescharacteristic
192C22×C32−1+411yescharacteristic
193C2×D3—22no—
194C2×D3—22no—
195C2×D3—22no—
196C2×D3—42no—
197C2×D3—42no—
198C2×D3—41no—
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—42no—
202C2×D3—42no—
203C3×C4—22no—
204C3×C4—41no—
205C3×C4—42no—
206C3×C4—42no—
207Dic3—22no—
208Dic3—42no—
209Dic3—41no—
210Dic3—42no—
211C23—21no—
212C23—22no—
213C23—21no—
214C23—22no—
215C23—21no—
216C23—32no—
217C23—62no—
218C23—61no—
219C23—62no—
220C23—61no—
221C23—61no—
222C23—62no—
223C23—61no—
224C23—62no—
225C23—61no—
226C23C23×D311yescharacteristic
227Q8—124no—
228D4—124no—
229D4—124no—
230D4—124no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—32no—
241C2×C4—62no—
242C2×C4—61no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—124no—
251C2×C4D3×D412yes—
252C2×C3—81no—
253C2×C3#53411yescharacteristic
254C2×C3#53511yescharacteristic
255C2×C3#53611yescharacteristic
256C2×C3#56212yes—
257C2×C3#56212yes—
258D3—42no—
259D3—81no—
260C22—42no—
261C22—41no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—62no—
266C22—62no—
267C22—62no—
268C22—122no—
269C22—122no—
270C22—122no—
271C22—121no—
272C22—122no—
273C22—121no—
274C22—121no—
275C22C2×D3×D411yescharacteristic
276C22#100412yes—
277C22#100711yescharacteristic
278C22#100912yes—
279C22#101011yescharacteristic
280C4—22no—
281C4—42no—
282C4—41no—
283C4—42no—
284C4—62no—
285C4—122no—
286C4—122no—
287C4—121no—
288C3#281611yescharacteristic, complemented, Sylow
289C2—81no—
290C2—122no—
291C2—241no—
292C2#562811yescharacteristic
293C2#564712yes—
294C2#565311yescharacteristic
295C2#566712yes—
296C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC2×#884C2
hypercenter#399D3
hypocenterC3#2816
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
3. C3×#399C2semidirect
4. #4993C2semidirect
5. #4991C2semidirect
6. #4982C2semidirect
7. #4961C2semidirect
8. #4959C2semidirect
9. #4947C2semidirect
10. #4673C2semidirect
11. #4664C2semidirect
41. #399D3semidirect
288. C3#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8842
C3×#3992
#49932
#49912
#49822
#49612
#49592
#49472
#46732
#46642
#28163