Finite Group Databasebeta

← #68305

Subgroups — #68305

Order 384.

286 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 63 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
1#5606C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic, complemented
3#4667C211yesmaximal, characteristic, complemented
4C3×#327C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4467C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1513—31nomaximal, Sylow
9C2×#237C411yescharacteristic, complemented
10C22×C4×D3C2211yescharacteristic
11C3×#121C2211yescharacteristic
12#938C411yescharacteristic, complemented
13#937—21no—
14D4×Dic3—21no—
15C2×(C2×D3)⋊C4—21no—
16#924C2211yescharacteristic
17C2×Dic3⋊C4—21no—
18C2×ΓL(1,𝔽25)—21no—
19#897C411yescharacteristic, complemented
20#892C411yescharacteristic, complemented
21#883—21no—
22#67⋊C4—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×#95C2211yescharacteristic
25C3⋊#95C2211yescharacteristic
26#830C2211yescharacteristic
27(C2×D3)⋊C8—21no—
28#525—31no—
29#407—31no—
30#380—31no—
31#328—31no—
32#328—31no—
33#327—31no—
34#327D311yescharacteristic, complemented
35C23×D3D411yescharacteristic
36C2×C3×D4—21no—
37C2×C3×D4C2×C411yescharacteristic
38C22×C3×C4—21no—
39C22×C3×C4C2311yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C22×Dic3—21no—
43C22×Dic3—21no—
44C22×Dic3D411yescharacteristic
45#237—42no—
46C2×C4×D3—42no—
47C2×C4×D3C2×C411yescharacteristic
48C2×C4×D3C2×C411yescharacteristic
49C2×C4×D3D411yescharacteristic
50C2×C4×D3D411yescharacteristic
51C2×Dic6C2×C411yescharacteristic
52C2×C3×C8—21no—
53C3×C4⋊C4C2×C411yescharacteristic
54C3×C22⋊C4—21no—
55#217—21no—
56#217—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—42no—
60C4⋊Dic3—21no—
61C4⋊Dic3C2×C411yescharacteristic
62Dic3⋊C4—21no—
63Dic3⋊C4—42no—
64C4×Dic3—21no—
65C2×C3⋊C8—21no—
66ΓL(1,𝔽25)—42no—
67C2×#41—31no—
68C23×C4—31no—
69C2×M16—61no—
70#123—61no—
71C22⋊Q8—31no—
72C22⋊Q8—31no—
73#121—31no—
74#121C2×D311yescharacteristic
75#120—61no—
76C4×D4—61no—
77C4×D4—61no—
78#117—31no—
79C2×C4⋊C4—61no—
80C2×C22⋊C4—61no—
81C22⋊C8—31no—
82C22⋊C8—61no—
83C22⋊C8C2×D311yescharacteristic
84#95—31no—
85#95C2×D311yescharacteristic
86C23×C3—21no—
87C23×C3C2×D411yescharacteristic
88C22×D3—21no—
89C22×D3—21no—
90C22×D3—41no—
91C22×D3—41no—
92C22×D3C22⋊C411yescharacteristic
93C22×D3C22⋊C411yescharacteristic
94C3×D4—41no—
95C3×D4—41no—
96C3×D4—41no—
97C2×C3×C4—21no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4C22×C411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104#67—42no—
105#67—42no—
106C2×Dic3—22no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—42no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3C22⋊C411yescharacteristic
114C2×Dic3C22⋊C411yescharacteristic
115C4×D3—22no—
116C4×D3—22no—
117C4×D3—41no—
118C4×D3—41no—
119Dic6—41no—
120C3×C8—41no—
121C3⋊C8—41no—
122C24—31no—
123#41—122no—
124C2×Q8—31no—
125C2×D4—21no—
126C2×D4—61no—
127C2×D4—61no—
128C2×D4C4×D311yescharacteristic
129C22×C4—21no—
130C22×C4—31no—
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—61no—
136C22×C4—61no—
137C22×C4—122no—
138C22×C4C22×D311yescharacteristic
139M16—122no—
140C2×C8—21no—
141C2×C8—61no—
142C4⋊C4—31no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4—122no—
146C4⋊C4C4×D311yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C22⋊C4—122no—
153C42—61no—
154C22×C3—21no—
155C22×C3—21no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3C2×C22⋊C411yescharacteristic
160C22×C3AGL(1,ℤ/8)11yescharacteristic
161C22×C3#13711yescharacteristic
162C2×D3—21no—
163C2×D3—21no—
164C2×D3—21no—
165C2×D3—21no—
166C2×D3—21no—
167C2×D3—21no—
168C2×D3—42no—
169C2×D3—42no—
170C3×C4—21no—
171C3×C4—21no—
172C3×C4—41no—
173C3×C4—81no—
174Dic3—21no—
175Dic3—41no—
176Dic3—41no—
177Dic3—81no—
178Dic3#9912yes—
179C23—21no—
180C23—31no—
181C23—31no—
182C23—61no—
183C23—61no—
184C23—121no—
185C23—121no—
186C23D3×D411yescharacteristic
187Q8—121no—
188D4—41no—
189D4—41no—
190D4—41no—
191D4—122no—
192D4—122no—
193C2×C4—21no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—121no—
206C2×C4—121no—
207C2×C4—122no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4—121no—
211C2×C4C2×C4×D311yescharacteristic
212C2×C4D3×D411yescharacteristic
213C8—41no—
214C8—121no—
215C2×C3—21no—
216C2×C3—21no—
217C2×C3—81no—
218C2×C3C2×#9911yescharacteristic
219C2×C3#41711yescharacteristic
220C2×C3#42111yescharacteristic
221D3—41no—
222D3—41no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—61no—
229C22—61no—
230C22—61no—
231C22—61no—
232C22—61no—
233C22—61no—
234C22—122no—
235C22—122no—
236C22D3×C22⋊C411yescharacteristic
237C22#91111yescharacteristic
238C22#91511yescharacteristic
239C4—21no—
240C4—21no—
241C4—32no—
242C4—41no—
243C4—61no—
244C4—81no—
245C4—121no—
246C4—121no—
247C4—241no—
248C3#151311yescharacteristic, complemented, Sylow
249C2—21no—
250C2—21no—
251C2—81no—
252C2—121no—
253C2—121no—
254C2D3×#9911yescharacteristic
255C2#477111yescharacteristic
256C2#482111yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3C2×C22⋊C4
prime radical#5606C2
cosocleC22×C3×C4C23
hypercenter#327D3
hypocenterC3#1513
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #4726C2semidirect
3. #4667C2semidirect
4. C3×#327C2semidirect
5. #4537C2semidirect
6. #4467C2semidirect
7. #4452C2semidirect
9. C2×#237C4semidirect
12. #938C4semidirect
19. #897C4semidirect
20. #892C4semidirect
34. #327D3semidirect
248. C3#1513semidirect

Maximal subgroups

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

subgroupindex
#56062
#47262
#46672
C3×#3272
#45372
#44672
#44522
#15133