Finite Group Databasebeta

← #74742

Subgroups — #74742

Order 384.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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×#939C211yesmaximal, characteristic
2C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C3×#386C211yesmaximal, characteristic, complemented
4#5220C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic, complemented
6#4958C211yesmaximal, characteristic, complemented
7#4957C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4650C212yesmaximal, complemented
12#2504—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C23×Dic3C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2212yes—
24#939—24no—
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×Dic3⋊C4—24no—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33C3×#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#121—31no—
39C22×C4⋊C4—31no—
40#397—32no—
41#397—32no—
42#390—32no—
43#386—31no—
44#386—31no—
45#386—31no—
46#386D311yescharacteristic, complemented
47#382—32no—
48C4×C22⋊C4—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C22×C3×C4—22no—
56C22×C3×C4—42no—
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2312yes—
59C22×C3×C4C2312yes—
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—24no—
65C22×Dic3—24no—
66C22×Dic3—22no—
67C22×Dic3—42no—
68C22×Dic3D414yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2312yes—
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2311yescharacteristic
74C3×C22⋊C4—22no—
75C3×C22⋊C4—42no—
76#217—22no—
77#217—22no—
78#217—42no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82Dic3⋊C4—24no—
83Dic3⋊C4—24no—
84Dic3⋊C4—22no—
85C4×Dic3—24no—
86C22×D4—31no—
87C22×D4—31no—
88C22×D4C2×D311yescharacteristic
89C23×C4—31no—
90C23×C4—31no—
91C23×C4C2×D311yescharacteristic
92#121—64no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—64no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D312yes—
105C2×C42—32no—
106#95—32no—
107#95—32no—
108#95—32no—
109#95—32no—
110#95C2×D312yes—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—21no—
114C23×C3—22no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—22no—
118C23×C3C2×D412yes—
119C23×C3#4112yes—
120C23×C3#4112yes—
121C23×C3C2411yescharacteristic
122C22×D3—21no—
123C22×D3—21no—
124C22×D3—21no—
125C22×D3—22no—
126C22×D3—22no—
127C3×D4—44no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—44no—
138#67—44no—
139#67—44no—
140C2×Dic3—22no—
141C2×Dic3—28no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—44no—
152C2×Dic3C2×D412yes—
153C2×Dic3C2×D412yes—
154C2×Dic3#4114yes—
155C24—31no—
156C24C22×D311yescharacteristic
157C24C22×D311yescharacteristic
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—62no—
162C2×D4—64no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—62no—
166C22×C4—22no—
167C22×C4—31no—
168C22×C4—34no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—32no—
172C22×C4—32no—
173C22×C4—42no—
174C22×C4—64no—
175C22×C4—62no—
176C22×C4—122no—
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D312yes—
179C22×C4C22×D312yes—
180C4⋊C4—64no—
181C4⋊C4—64no—
182C4⋊C4—62no—
183C22⋊C4—22no—
184C22⋊C4—42no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C42—64no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—24no—
195C22×C3—24no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3#12414yes—
202C22×C3C22×D411yescharacteristic
203C22×C3C2×#4111yescharacteristic
204C22×C3C2×#4111yescharacteristic
205C22×C3C2×#4112yes—
206C22×C3H(5,𝔽2)12yes—
207C2×D3—41no—
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—42no—
211C2×D3—42no—
212C3×C4—41no—
213C3×C4—42no—
214C3×C4—42no—
215Dic3—22no—
216Dic3—41no—
217Dic3—42no—
218Dic3—42no—
219Dic3#12314yes—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—61no—
228C23—61no—
229C23—61no—
230C23—62no—
231C23—62no—
232C23#23512yes—
233C23D3×D412yes—
234C23#23712yes—
235C23C23×D311yescharacteristic
236D4—44no—
237D4—124no—
238D4—124no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—32no—
249C2×C4—34no—
250C2×C4—32no—
251C2×C4—44no—
252C2×C4—68no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—62no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—124no—
264C2×C3—24no—
265C2×C3—81no—
266C2×C3C2×#12311yescharacteristic
267C2×C3C2×#12411yescharacteristic
268C2×C3#53511yescharacteristic
269C2×C3#55312yes—
270C2×C3#55312yes—
271D3—81no—
272C22—22no—
273C22—24no—
274C22—22no—
275C22—24no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—121no—
282C22—121no—
283C22—122no—
284C22—121no—
285C22—122no—
286C22#88514yes—
287C22C2×#23511yescharacteristic
288C22C2×D3×D411yescharacteristic
289C22C2×#23711yescharacteristic
290C22#100412yes—
291C22D3×#4112yes—
292C4—34no—
293C4—41no—
294C4—42no—
295C4—42no—
296C4—62no—
297C4—121no—
298C4—122no—
299C4—122no—
300C3#250411yescharacteristic, complemented, Sylow
301C2—24no—
302C2—81no—
303C2—241no—
304C2C2×#88511yescharacteristic
305C2#554012yes—
306C2#559211yescharacteristic
307C2#560112yes—
308C2D3×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#386C2
prime radicalC2×#939C2
hypercenter#386D3
hypocenterC3#2504
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×Dic3⋊C4C2semidirect
3. C3×#386C2semidirect
4. #5220C2semidirect
5. #5213C2semidirect
6. #4958C2semidirect
7. #4957C2semidirect
8. #4952C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4675C2semidirect
11. #4650C2semidirect
46. #386D3semidirect
300. C3#2504semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C22×Dic3⋊C42
C3×#3862
#52202
#52132
#49582
#49572
#49522
Dic3×C22⋊C42
#46752
#46502
#25043