Finite Group Databasebeta

← #77121

Subgroups — #77121

Order 384.

596 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 139 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×#938C211yesmaximal, characteristic
2C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4#5222C211yesmaximal, characteristic, complemented
5#4986C211yesmaximal, characteristic, complemented
6#4956C212yesmaximal, complemented
7#4956C212yesmaximal, complemented
8#4951C212yesmaximal, complemented
9#4944C212yesmaximal, complemented
10#4666C211yesmaximal, characteristic, complemented
11#4659C211yesmaximal, characteristic, complemented
12#2591—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24#938—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×Dic3⋊C4C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2212yes—
31D3×C4⋊C4—24no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#121—31no—
39C22×C4⋊C4—31no—
40#400—31no—
41#400—31no—
42#394—31no—
43#394D311yescharacteristic, complemented
44#388—32no—
45#388—32no—
46#386—32no—
47#386—32no—
48C4×C22⋊C4—31no—
49C24×C3C2312yes—
50C23×D3C2311yescharacteristic
51C2×C3×D4—22no—
52C2×C3×D4—42no—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C2×#67—24no—
58C2×#67—24no—
59C2×#67—24no—
60C22×Dic3C2312yes—
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2311yescharacteristic
64C22×Dic3C2311yescharacteristic
65C2×C4×D3—22no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—42no—
70C2×C4×D3—42no—
71C2×C4×D3D414yes—
72C3×C4⋊C4—22no—
73C3×C22⋊C4—24no—
74C3×C22⋊C4—42no—
75#217—24no—
76#217—24no—
77#217—42no—
78(C2×D3)⋊C4—24no—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C4×Dic3—42no—
84C22×D4—32no—
85C22×D4C2×D311yescharacteristic
86C23×C4—31no—
87C23×C4—31no—
88C23×C4—31no—
89#121—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—64no—
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4C2×D312yes—
101C2×C22⋊C4C2×D312yes—
102C2×C42—32no—
103#95—31no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95C2×D311yescharacteristic
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3C2411yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3—22no—
120C22×D3C2×D412yes—
121C22×D3#4114yes—
122C3×D4—44no—
123C2×C3×C4—21no—
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×C4—22no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4#4112yes—
133#67—48no—
134C2×Dic3—21no—
135C2×Dic3—21no—
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—
146C2×Dic3#4112yes—
147C4×D3—28no—
148C4×D3—44no—
149C4×D3—44no—
150C24—31no—
151C24C22×D312yes—
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—64no—
155C2×D4—64no—
156C2×D4—64no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—34no—
160C22×C4—32no—
161C22×C4—32no—
162C22×C4—31no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—122no—
168C22×C4—122no—
169C22×C4C22×D312yes—
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C22⋊C4—24no—
179C22⋊C4—42no—
180C22⋊C4—64no—
181C22⋊C4—64no—
182C22⋊C4—64no—
183C22⋊C4—122no—
184C42—122no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3C22×D411yescharacteristic
193C22×C3C2×#4111yescharacteristic
194C22×C3C2×#4111yescharacteristic
195C22×C3C2×#4112yes—
196C22×C3H(5,𝔽2)12yes—
197C2×D3—22no—
198C2×D3—22no—
199C2×D3—24no—
200C2×D3—24no—
201C2×D3#12414yes—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—41no—
205C3×C4—42no—
206Dic3—22no—
207Dic3—22no—
208Dic3—42no—
209Dic3—41no—
210Dic3—42no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—32no—
219C23—34no—
220C23—61no—
221C23—62no—
222C23—61no—
223C23C23×D311yescharacteristic
224D4—44no—
225D4—128no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—68no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—61no—
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×C4D3×D412yes—
251C2×C4#23712yes—
252C2×C3—82no—
253C2×C3#52511yescharacteristic
254C2×C3C2×#12411yescharacteristic
255C2×C3#53411yescharacteristic
256C2×C3#55312yes—
257C2×C3#55312yes—
258D3—24no—
259C22—34no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—64no—
268C22—62no—
269C22—62no—
270C22—64no—
271C22C2×D3×D411yescharacteristic
272C22C2×#23711yescharacteristic
273C22#100412yes—
274C22D3×#4111yescharacteristic
275C22D3×#4112yes—
276C4—22no—
277C4—22no—
278C4—41no—
279C4—42no—
280C4—62no—
281C4—62no—
282C4—122no—
283C4—122no—
284C4—121no—
285C3#259111yescharacteristic, complemented, Sylow
286C2—64no—
287C2—82no—
288C2#558911yescharacteristic
289C2#560611yescharacteristic
290C2#561412yes—
291C2#566112yes—
292C2D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#938C2
hypercenter#394D3
hypocenterC3#2591
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊C4C2semidirect
3. C3×#394C2semidirect
4. #5222C2semidirect
5. #4986C2semidirect
6. #4956C2semidirect
7. #4956C2semidirect
8. #4951C2semidirect
9. #4944C2semidirect
10. #4666C2semidirect
11. #4659C2semidirect
43. #394D3semidirect
285. C3#2591semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×D3×C4⋊C42
C3×#3942
#52222
#49862
#49562
#49512
#49442
#46662
#46592
#25913