Finite Group Databasebeta

← #83183

Subgroups — #83183

Order 384.

612 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 141 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×#127C211yesmaximal, characteristic
2C2×#935C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal, complemented
4#5222C212yesmaximal, complemented
5#5222C212yesmaximal, complemented
6#5220C212yesmaximal, complemented
7#5220C212yesmaximal, complemented
8#5220C211yesmaximal, characteristic, complemented
9#4933C212yesmaximal, complemented
10#2709—31nomaximal, Sylow
11C22×C3×D4C2212yes—
12C22×C3×D4C2212yes—
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C23×Dic3C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C3×#127—24no—
18C2×C3×C42C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2211yescharacteristic
25C2×#217C2212yes—
26C2×#217C2212yes—
27#935—24no—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×#127D311yescharacteristic
35C2×#123—31no—
36#394—32no—
37#394—32no—
38#394—32no—
39#387—32no—
40#386—32no—
41#386—32no—
42#386—31no—
43C24×C3C2312yes—
44C24×C3C2311yescharacteristic
45C24×C3C2311yescharacteristic
46C2×C3×D4—24no—
47C2×C3×D4—24no—
48C2×C3×D4—22no—
49C2×C3×D4—24no—
50C2×C3×D4—24no—
51C2×C3×D4—41no—
52C2×C3×D4—41no—
53C2×C3×D4—42no—
54C2×C3×D4—42no—
55C2×C3×D4—41no—
56C2×C3×D4—41no—
57C2×C3×D4D414yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2312yes—
60C22×Dic3—22no—
61C22×Dic3—41no—
62C22×Dic3—41no—
63C22×Dic3—41no—
64C22×Dic3—42no—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C22×Dic3C2312yes—
69C3×C42—22no—
70#217—22no—
71#217—24no—
72#217—41no—
73#217—42no—
74#217—42no—
75#217—42no—
76#217—42no—
77#217—42no—
78Dic3⋊C4—24no—
79Dic3⋊C4—24no—
80C22×D4C2×D312yes—
81C22×D4C2×D312yes—
82C22×D4C2×D311yescharacteristic
83C22×D4C2×D311yescharacteristic
84C23×C4—31no—
85C23×C4—31no—
86#127—24no—
87#123—64no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—31no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C42C2×D311yescharacteristic
99#95—32no—
100#95—32no—
101#95—32no—
102#95—32no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3C2×D412yes—
119C23×C3C2×D412yes—
120C23×C3C2411yescharacteristic
121C3×D4—44no—
122C3×D4—48no—
123C3×D4—48no—
124C3×D4—44no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4C2×D412yes—
130C2×C3×C4#4114yes—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—44no—
144C2×Dic3—44no—
145C24C22×D312yes—
146C24C22×D311yescharacteristic
147C24C22×D311yescharacteristic
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4#6714yes—
160C22×C4—32no—
161C22×C4—32no—
162C22×C4—32no—
163C22×C4—32no—
164C22×C4—62no—
165C22×C4—121no—
166C22×C4—121no—
167C22×C4—121no—
168C22×C4—122no—
169C22×C4C22×D312yes—
170C22×C4C22×D311yescharacteristic
171C4⋊C4—64no—
172C4⋊C4—64no—
173C22⋊C4—62no—
174C22⋊C4—64no—
175C22⋊C4—121no—
176C22⋊C4—122no—
177C22⋊C4—122no—
178C22⋊C4—122no—
179C22⋊C4—122no—
180C22⋊C4—122no—
181C42—22no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—24no—
185C22×C3—24no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3C22×D411yescharacteristic
200C22×C3C2×#4112yes—
201C22×C3H(5,𝔽2)11yescharacteristic
202C22×C3H(5,𝔽2)11yescharacteristic
203C22×C3H(5,𝔽2)12yes—
204C3×C4—22no—
205C3×C4—24no—
206Dic3—42no—
207Dic3—42no—
208Dic3—42no—
209Dic3—42no—
210C23—22no—
211C23—22no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23C2×#6712yes—
226C23C2×#6712yes—
227C23C23×D311yescharacteristic
228D4—44no—
229D4—48no—
230D4—48no—
231D4—44no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—62no—
243C2×C4—62no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—124no—
249C2×C4—124no—
250C2×C4#23714yes—
251C2×C4C2×#6712yes—
252C2×C3—42no—
253C2×C3—42no—
254C2×C3—82no—
255C2×C3#53411yescharacteristic
256C2×C3#54012yes—
257C2×C3#54612yes—
258C2×C3#55911yescharacteristic
259C2×C3#56011yescharacteristic
260C22—22no—
261C22—22no—
262C22—24no—
263C22—24no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22C2×#23712yes—
278C22#100411yescharacteristic
279C22#100411yescharacteristic
280C22#100412yes—
281C22C22×#6711yescharacteristic
282C4—22no—
283C4—24no—
284C4—122no—
285C4—122no—
286C4—122no—
287C4—122no—
288C3#270911yescharacteristic, complemented, Sylow
289C2—42no—
290C2—42no—
291C2—82no—
292C2#571311yescharacteristic
293C2#571912yes—
294C2#572011yescharacteristic
295C2#580411yescharacteristic
296C2#580612yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
Fitting, prime radicalC2×C3×#127C2
hypercenterC2×#127D3
hypocenterC3#2709
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
2. C2×#935C2semidirect
3. #5222C2semidirect
4. #5222C2semidirect
5. #5222C2semidirect
6. #5220C2semidirect
7. #5220C2semidirect
8. #5220C2semidirect
9. #4933C2semidirect
288. C3#2709semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1272
C2×#9352
#52222
#52202
#49332
#27093