Finite Group Databasebeta

← #76745

Subgroups — #76745

Order 384.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#936C211yesmaximal, characteristic, complemented
2C2×#885C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5222C212yesmaximal, complemented
7#4959C212yesmaximal, complemented
8#4947C212yesmaximal, complemented
9#4947C211yesmaximal, characteristic, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4650C211yesmaximal, characteristic, complemented
12#2819—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×#67C2211yescharacteristic
16C22×#67C2212yes—
17C22×Dic6C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2212yes—
26#936—24no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#885—24no—
33C3×#95C2211yescharacteristic
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×#124—31no—
40#394—32no—
41#394—32no—
42#393—32no—
43#393—31no—
44#393—31no—
45#392—32no—
46#392—31no—
47#392D311yescharacteristic, complemented
48#382—31no—
49C24×C3C2312yes—
50C24×C3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C2×C3×D4—22no—
53C2×C3×D4—42no—
54C2×C3×D4—42no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—22no—
61C2×#67—42no—
62C2×#67—42no—
63C2×#67—42no—
64C2×#67—42no—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×Dic6—22no—
71C2×Dic6—22no—
72C2×Dic6—42no—
73C2×Dic6—42no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—42no—
76#217—22no—
77#217—24no—
78#217—24no—
79#217—42no—
80#217—42no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—42no—
84Dic3⋊C4—42no—
85C4×Dic3D414yescomplemented
86C22×Q8—31no—
87C22×Q8—31no—
88C22×D4—31no—
89C22×D4—32no—
90C22×D4C2×D311yescharacteristic
91C22×D4C2×D312yes—
92#124—64no—
93#124—64no—
94C2×C4⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
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—31no—
106#95—32no—
107#95—31no—
108#95—32no—
109#95—32no—
110#95C2×D311yescharacteristic
111C23×C3—22no—
112C23×C3—21no—
113C23×C3—22no—
114C23×C3—22no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—21no—
121C23×C3—22no—
122C23×C3—21no—
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—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4C2×D412yes—
138#67—44no—
139#67—48no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3C2×D412yes—
152C2×Dic3C2×D412yes—
153Dic6—44no—
154Dic6—44no—
155C24—31no—
156C24C22×D311yescharacteristic
157C24C22×D312yes—
158C2×Q8—62no—
159C2×Q8—62no—
160C2×Q8—122no—
161C2×Q8—122no—
162C2×D4—22no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—62no—
168C2×D4—122no—
169C2×D4—122no—
170C2×D4—122no—
171C2×D4—122no—
172C22×C4—31no—
173C22×C4—31no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4C22×D312yes—
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D311yescharacteristic
180C4⋊C4—122no—
181C22⋊C4—22no—
182C22⋊C4—42no—
183C22⋊C4—62no—
184C22⋊C4—62no—
185C22⋊C4—64no—
186C22⋊C4—64no—
187C22⋊C4—62no—
188C22⋊C4—122no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C42—34no—
192C22×C3—41no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—41no—
196C22×C3—42no—
197C22×C3—41no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3C22×D411yescharacteristic
205C22×C3H(5,𝔽2)12yes—
206C22×C3H(5,𝔽2)12yes—
207C22×C3H(5,𝔽2)11yescharacteristic
208C22×C32−1+411yescharacteristic
209C2×D3—41no—
210C2×D3—41no—
211C2×D3—41no—
212C2×D3—42no—
213C2×D3—42no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—42no—
217Dic3—22no—
218Dic3—22no—
219Dic3—41no—
220Dic3—42no—
221Dic3—42no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—21no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—21no—
233C23—22no—
234C23—61no—
235C23—62no—
236C23—62no—
237C23—61no—
238C23—61no—
239C23C23×D311yescharacteristic
240Q8—124no—
241Q8—124no—
242D4—44no—
243D4—48no—
244D4—124no—
245D4—128no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4D3×D412yes—
266C2×C3—81no—
267C2×C3—82no—
268C2×C3#53511yescharacteristic
269C2×C3#53511yescharacteristic
270C2×C3#53711yescharacteristic
271C2×C3H(3,𝔽4)12yes—
272C2×C3#56212yes—
273D3—81no—
274C22—41no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—121no—
287C22—121no—
288C22—122no—
289C22—121no—
290C22—122no—
291C22C2×D3×D411yescharacteristic
292C22#100412yes—
293C22#100412yes—
294C22#100411yescharacteristic
295C22#101011yescharacteristic
296C4—22no—
297C4—41no—
298C4—42no—
299C4—62no—
300C4—62no—
301C4—121no—
302C4—122no—
303C4—122no—
304C3#281911yescharacteristic, complemented, Sylow
305C2—81no—
306C2—82no—
307C2—241no—
308C2#559211yescharacteristic
309C2#559412yes—
310C2#560811yescharacteristic
311C2#562112yes—
312C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2819
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#936C2semidirect
2. C2×#885C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #4959C2semidirect
8. #4947C2semidirect
9. #4947C2semidirect
10. #4673C2semidirect
11. #4650C2semidirect
47. #392D3semidirect
85. C4×Dic3D4semidirect
304. C3#2819semidirect

Maximal subgroups

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

subgroupindex
C2×#9362
C2×#8852
C3×#3922
#52232
#52222
#49592
#49472
#46732
#46502
#28193