Finite Group Databasebeta

← #82981

Subgroups — #82981

Order 384.

534 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 137 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×C3×#124C211yesmaximal, characteristic, complemented
2C2×#876C211yesmaximal, characteristic, complemented
3#5231C211yesmaximal, characteristic, complemented
4#5230C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic
6#5222C211yesmaximal, characteristic, complemented
7#4961C212yesmaximal, complemented
8#4957C212yesmaximal, complemented
9#4948C212yesmaximal, complemented
10#4945C212yesmaximal, complemented
11C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#2662—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C3×#124—24no—
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33#876—24no—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×#124D311yescharacteristic, complemented
40#396—32no—
41#395—31no—
42#394—31no—
43#394—32no—
44#393—31no—
45#392—31no—
46#390—32no—
47#388—32no—
48C4×C4⋊C4—31no—
49C24×C3C2312yes—
50C23×D3C2311yescharacteristic
51C2×C3×Q8—22no—
52C2×C3×Q8—42no—
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C22×C3×C4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—24no—
60C2×#67—42no—
61C2×#67—42no—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C22×Dic3C2311yescharacteristic
66C22×Dic3C2312yes—
67C3×C22⋊C4—24no—
68C3×C22⋊C4—24no—
69C3×C42D414yescomplemented
70#217—24no—
71#217—42no—
72#217—42no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—42no—
77C4⋊Dic3—24no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4—42no—
82C4×Dic3—42no—
83C22×Q8C2×D311yescharacteristic
84C22×D4—32no—
85C22×D4C2×D311yescharacteristic
86#126—64no—
87#124—24no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4C2×D312yes—
101C2×C22⋊C4C2×D312yes—
102C2×C42—32no—
103C2×C42C2×D311yescharacteristic
104#95—31no—
105#95—31no—
106#95—32no—
107#95—32no—
108C23×C3—22no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3C2411yescharacteristic
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—21no—
120C22×D3—21no—
121C3×Q8—44no—
122C3×D4—44no—
123C2×C3×C4—22no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D412yes—
133#67—48no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3#4114yes—
147C24—31no—
148C24C22×D312yes—
149C2×Q8—22no—
150C2×Q8—42no—
151C2×D4—22no—
152C2×D4—42no—
153C2×D4—64no—
154C2×D4—122no—
155C2×D4—122no—
156C22×C4—32no—
157C22×C4—32no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4C22×D312yes—
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D311yescharacteristic
164C22×C4C22×D311yescharacteristic
165C4⋊C4—64no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—122no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—64no—
173C22⋊C4—62no—
174C22⋊C4—62no—
175C22⋊C4—62no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C42—122no—
180C42#6714yescomplemented
181C22×C3—42no—
182C22×C3—42no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3C22×D411yescharacteristic
189C22×C3C2×#4112yes—
190C22×C3H(5,𝔽2)12yes—
191C22×C3H(5,𝔽2)11yescharacteristic
192C22×C32−1+411yescharacteristic
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C2×D3—42no—
197C2×D3—42no—
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—42no—
202Dic3—24no—
203Dic3—41no—
204Dic3—41no—
205Dic3—41no—
206Dic3—42no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—62no—
215C23—62no—
216C23—61no—
217C23—61no—
218C23—61no—
219C23C23×D311yescharacteristic
220Q8—44no—
221D4—44no—
222D4—128no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—34no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4C2×#6712yes—
244C2×C4C2×#6712yes—
245C2×C4C2×#6712yes—
246C2×C3—82no—
247C2×C3#53211yescharacteristic
248C2×C3#53511yescharacteristic
249C2×C3#53711yescharacteristic
250C2×C3#53812yes—
251C2×C3#54212yes—
252D3—81no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—122no—
261C22—122no—
262C22—121no—
263C22—121no—
264C22—121no—
265C22#100412yes—
266C22D3×#4112yes—
267C22#100911yescharacteristic
268C22#101011yescharacteristic
269C22C22×#6711yescharacteristic
270C4—22no—
271C4—22no—
272C4—22no—
273C4—42no—
274C4—64no—
275C4—122no—
276C4—121no—
277C4—121no—
278C4—121no—
279C3#266211yescharacteristic, complemented, Sylow
280C2—82no—
281C2—241no—
282C2#566912yes—
283C2#567812yes—
284C2#580511yescharacteristic
285C2#583211yescharacteristic
286C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radical#5223C2
hypercenterC2×#124D3
hypocenterC3#2662
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×C3×#124C2semidirect
2. C2×#876C2semidirect
3. #5231C2semidirect
4. #5230C2semidirect
6. #5222C2semidirect
7. #4961C2semidirect
8. #4957C2semidirect
9. #4948C2semidirect
10. #4945C2semidirect
11. C4×C4⋊Dic3C2semidirect
39. C2×#124D3semidirect
69. C3×C42D4semidirect
180. C42#67semidirect
279. C3#2662semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#8762
#52312
#52302
#52232
#52222
#49612
#49572
#49482
#49452
C4×C4⋊Dic32
#26623