Finite Group Databasebeta

← #82721

Subgroups — #82721

Order 384.

578 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 133 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×#121C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic, complemented
3#5223C212yesmaximal, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#5222C212yesmaximal, complemented
7#5212C211yesmaximal, characteristic, complemented
8#4993C211yesmaximal, characteristic, complemented
9#4985C211yesmaximal, characteristic, complemented
10#4961C212yesmaximal, complemented
11#4948C212yesmaximal, complemented
12#2813—31nomaximal, Sylow
13C22×C3×D4C2212yes—
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×#67C2212yes—
18C3×#121—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26#929—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2212yes—
31C2×Dic3⋊C4C2211yescharacteristic
32C2×Dic3⋊C4C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2212yes—
38C2×#123—31no—
39C2×#121D311yescharacteristic, complemented
40#400—31no—
41#399—31no—
42#396—32no—
43#394—31no—
44#394—32no—
45#394—32no—
46#392—31no—
47#392—32no—
48#380—31no—
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C2×C3×D4—24no—
53C2×C3×D4—22no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C22×C3×C4—42no—
58C22×C3×C4D414yescomplemented
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—42no—
64C2×#67—42no—
65C2×#67—42no—
66C2×#67—42no—
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2312yes—
71C22×Dic3C2311yescharacteristic
72C3×C4⋊C4—22no—
73C3×C22⋊C4—24no—
74#217—22no—
75#217—42no—
76#217—42no—
77#217—42no—
78#217—42no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—42no—
82C4⋊Dic3—42no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85Dic3⋊C4—42no—
86C22×D4—31no—
87C22×D4—32no—
88C22×D4C2×D312yes—
89C22×D4C2×D311yescharacteristic
90C23×C4C2×D311yescharacteristic
91#123—64no—
92#121—24no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—32no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C22⋊C4—31no—
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—
107#95—31no—
108#95—31no—
109#95—32no—
110#95—32no—
111C23×C3—21no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—21no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3C2×D412yes—
123C23×C3C2411yescharacteristic
124C22×D3—21no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—21no—
128C22×D3—21no—
129C3×D4—44no—
130C3×D4—48no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—44no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D412yes—
139#67—44no—
140#67—48no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C24—31no—
155C24C22×D312yes—
156C24C22×D311yescharacteristic
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—62no—
163C2×D4—122no—
164C2×D4—122no—
165C2×D4—122no—
166C2×D4—122no—
167C22×C4—32no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—32no—
171C22×C4—31no—
172C22×C4—42no—
173C22×C4#6714yescomplemented
174C22×C4C22×D312yes—
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C22⋊C4—24no—
183C22⋊C4—62no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3C22×D411yescharacteristic
204C22×C3H(5,𝔽2)12yes—
205C22×C3H(5,𝔽2)12yes—
206C22×C3H(5,𝔽2)11yescharacteristic
207C22×C32−1+411yescharacteristic
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—41no—
211C2×D3—42no—
212C2×D3—42no—
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—42no—
216Dic3—41no—
217Dic3—41no—
218Dic3—41no—
219Dic3—42no—
220Dic3—42no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—62no—
233C23—62no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23C2×#6712yes—
238C23C23×D311yescharacteristic
239D4—44no—
240D4—48no—
241D4—124no—
242D4—128no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4C2×#6712yes—
263C2×C4C2×#6712yes—
264C2×C3—42no—
265C2×C3—82no—
266C2×C3#53411yescharacteristic
267C2×C3#53511yescharacteristic
268C2×C3#53711yescharacteristic
269C2×C3#56012yes—
270C2×C3#56212yes—
271D3—81no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—121no—
285C22—121no—
286C22—121no—
287C22—122no—
288C22—122no—
289C22#100412yes—
290C22#100412yes—
291C22#100911yescharacteristic
292C22#101011yescharacteristic
293C22C22×#6711yescharacteristic
294C4—22no—
295C4—22no—
296C4—42no—
297C4—121no—
298C4—121no—
299C4—121no—
300C4—122no—
301C4—122no—
302C3#281311yescharacteristic, complemented, Sylow
303C2—42no—
304C2—82no—
305C2—241no—
306C2#562012yes—
307C2#562112yes—
308C2#580411yescharacteristic
309C2#580511yescharacteristic
310C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2813
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#929C2semidirect
3. #5223C2semidirect
4. #5223C2semidirect
5. #5222C2semidirect
6. #5222C2semidirect
7. #5212C2semidirect
8. #4993C2semidirect
9. #4985C2semidirect
10. #4961C2semidirect
11. #4948C2semidirect
39. C2×#121D3semidirect
58. C22×C3×C4D4semidirect
173. C22×C4#67semidirect
302. C3#2813semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9292
#52232
#52222
#52122
#49932
#49852
#49612
#49482
#28133