Finite Group Databasebeta

← #77687

Subgroups — #77687

Order 384.

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

normal subgroups only

This group has 576 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×#897C211yesmaximal, characteristic, complemented
2C2×#884C211yesmaximal, characteristic
3C3×#397C211yesmaximal, characteristic, complemented
4#4992C211yesmaximal, characteristic, complemented
5#4961C212yesmaximal, complemented
6#4956C212yesmaximal, complemented
7#4947C211yesmaximal, characteristic, complemented
8#4673C211yesmaximal, characteristic, complemented
9#4665C212yesmaximal, complemented
10D3×#95C211yesmaximal, characteristic, complemented
11#4660C212yesmaximal, complemented
12#2639—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C22×C4×D3C2212yes—
17C22×C4×D3C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2212yes—
22C2×#217C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2212yes—
28C2×Dic3⋊C4C2211yescharacteristic
29#897—24no—
30#884—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C2×#121—31no—
40#400—32no—
41#397—31no—
42#397D311yescharacteristic, complemented
43#394—32no—
44#393—31no—
45#393—31no—
46#386—32no—
47#386—32no—
48C2×#95—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C23×D3C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C2×#67—22no—
57C2×#67—24no—
58C2×#67—22no—
59C2×#67—42no—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2312yes—
63C22×Dic3C2311yescharacteristic
64C2×D12—22no—
65C2×D12—24no—
66C2×C4×D3—24no—
67C2×C4×D3—42no—
68C2×C4×D3—42no—
69C2×C4×D3—42no—
70C2×C4×D3D414yes—
71C2×Dic6—22no—
72C2×Dic6—42no—
73C3×C4⋊C4—22no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—42no—
76#217—42no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81C4⋊Dic3—24no—
82Dic3⋊C4—22no—
83C22×Q8—31no—
84C22×D4—31no—
85C22×D4—31no—
86C22×D4—31no—
87C23×C4—31no—
88C23×C4—32no—
89C22⋊Q8—64no—
90#121—64no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4C2×D311yescharacteristic
100C2×C22⋊C4C2×D312yes—
101#95—31no—
102#95—32no—
103#95—32no—
104#95—32no—
105#95—32no—
106#95C2×D311yescharacteristic
107#95C2×D312yes—
108C23×C3—21no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3C2411yescharacteristic
114C22×D3—21no—
115C22×D3—22no—
116C22×D3—21no—
117C22×D3—21no—
118C22×D3—22no—
119C22×D3—21no—
120C22×D3—22no—
121C22×D3—21no—
122C22×D3C2×D412yes—
123C22×D3#4114yes—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134#67—44no—
135#67—44no—
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—
146D12—44no—
147C4×D3—44no—
148C4×D3—48no—
149Dic6—44no—
150C24—31no—
151C24—31no—
152C24C22×D311yescharacteristic
153C2×Q8—62no—
154C2×Q8—122no—
155C2×D4—62no—
156C2×D4—64no—
157C2×D4—62no—
158C2×D4—62no—
159C2×D4—64no—
160C2×D4—122no—
161C22×C4—31no—
162C22×C4—34no—
163C22×C4—32no—
164C22×C4—31no—
165C22×C4—32no—
166C22×C4—64no—
167C22×C4—122no—
168C22×C4—122no—
169C22×C4—122no—
170C22×C4C22×D312yes—
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D312yes—
173C22×C4C22×D311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—62no—
176C4⋊C4—64no—
177C22⋊C4—22no—
178C22⋊C4—42no—
179C22⋊C4—64no—
180C22⋊C4—62no—
181C22⋊C4—122no—
182C22⋊C4—122no—
183C22⋊C4—122no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C22×C3C2×#4112yes—
191C22×C3H(5,𝔽2)11yescharacteristic
192C22×C3H(5,𝔽2)12yes—
193C22×C32−1+411yescharacteristic
194C2×D3—22no—
195C2×D3—22no—
196C2×D3—24no—
197C2×D3—24no—
198C2×D3—41no—
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—42no—
202C2×D3—42no—
203C2×D3#12414yes—
204C3×C4—22no—
205C3×C4—41no—
206C3×C4—42no—
207C3×C4—42no—
208Dic3—22no—
209Dic3—41no—
210Dic3—42no—
211Dic3—42no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—32no—
218C23—34no—
219C23—61no—
220C23—62no—
221C23—61no—
222C23—61no—
223C23—61no—
224C23—62no—
225C23—62no—
226C23—61no—
227C23C23×D311yescharacteristic
228Q8—124no—
229D4—124no—
230D4—124no—
231D4—124no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—32no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—124no—
252C2×C4—128no—
253C2×C4D3×D412yes—
254C2×C3—81no—
255C2×C3C2×#12411yescharacteristic
256C2×C3#53411yescharacteristic
257C2×C3#53711yescharacteristic
258C2×C3#53812yes—
259C2×C3#53912yes—
260D3—24no—
261D3—81no—
262C22—34no—
263C22—41no—
264C22—42no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—62no—
269C22—64no—
270C22—64no—
271C22—62no—
272C22—121no—
273C22—122no—
274C22—121no—
275C22—122no—
276C22—121no—
277C22C2×D3×D411yescharacteristic
278C22#100411yescharacteristic
279C22D3×#4112yes—
280C22#100912yes—
281C22#101011yescharacteristic
282C4—22no—
283C4—42no—
284C4—41no—
285C4—42no—
286C4—62no—
287C4—122no—
288C4—122no—
289C4—121no—
290C3#263911yescharacteristic, complemented, Sylow
291C2—64no—
292C2—81no—
293C2—241no—
294C2#560811yescharacteristic
295C2#565311yescharacteristic
296C2#565712yes—
297C2D3×#12411yescharacteristic
298C2#570712yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC2×#884C2
hypercenter#397D3
hypocenterC3#2639
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
3. C3×#397C2semidirect
4. #4992C2semidirect
5. #4961C2semidirect
6. #4956C2semidirect
7. #4947C2semidirect
8. #4673C2semidirect
9. #4665C2semidirect
10. D3×#95C2semidirect
11. #4660C2semidirect
42. #397D3semidirect
290. C3#2639semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8842
C3×#3972
#49922
#49612
#49562
#49472
#46732
#46652
D3×#952
#46602
#26393