Finite Group Databasebeta

← #83510

Subgroups — #83510

Order 384.

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

normal subgroups only

This group has 386 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
1C3×#547C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic, complemented
3#5239C212yesmaximal, complemented
4#5233C212yesmaximal, complemented
5#5090C211yesmaximal, characteristic, complemented
6#5079C211yesmaximal, characteristic, complemented
7#5051C212yesmaximal, complemented
8#5045C212yesmaximal, complemented
9C4×#214C211yesmaximal, characteristic, complemented
10D4×C3⋊C8C211yesmaximal, characteristic, complemented
11#5007C211yesmaximal, characteristic, complemented
12#3302—31nomaximal, Sylow
13C2×C3×#41C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#123—22no—
16C3×C22⋊Q8C2212yes—
17C3×#121—21no—
18C3×C4×D4C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20C2×C3×C4⋊C4—21no—
21C2×#216C2212yes—
22#936—21no—
23C2×#214C2211yescharacteristic
24#926C2212yes—
25#924C2212yes—
26C22×C3⋊C8C2212yes—
27#887—21no—
28#879—22no—
29#869C2211yescharacteristic
30C4×Dic6C2211yescharacteristic
31#835C2212yes—
32#835C2212yes—
33#835C2212yes—
34#832C2211yescharacteristic
35#832C2212yes—
36#830C2212yes—
37#810C2211yescharacteristic
38#808C2211yescharacteristic
39#807C2212yes—
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#547D311yescharacteristic
44#487—31no—
45#485—32no—
46#474—31no—
47#467—32no—
48#462—31no—
49#452—32no—
50C4×SD16—31no—
51C8×D4—31no—
52#416—32no—
53C3×#41—24no—
54C3×#41—42no—
55C2×C3×Q8C2312yes—
56C2×C3×D4—22no—
57C2×C3×D4D411yescharacteristic
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4—22no—
61C22×C3×C4C2312yes—
62C2×Dic6—21no—
63C2×Dic6C2311yescharacteristic
64C3×C4⋊C4—21no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—21no—
67C3×C4⋊C4D411yescharacteristic
68C3×C4⋊C4C2311yescharacteristic
69C3×C22⋊C4—22no—
70C3×C22⋊C4—21no—
71C3×C22⋊C4D412yes—
72C3×C42C2311yescharacteristic
73#217—22no—
74#217—22no—
75#216—42no—
76#214—22no—
77C4⋊Dic3—21no—
78C4⋊Dic3C2312yes—
79C4⋊Dic3C2311yescharacteristic
80Dic3⋊C4—21no—
81Dic3⋊C4—22no—
82C4×Dic3—21no—
83C4×Dic3—22no—
84C2×C3⋊C8—42no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2312yes—
88C2×#41C2×D312yes—
89C2×Q16—32no—
90C2×SD16—31no—
91C22×C8—32no—
92C4⋊Q8—31no—
93C4⋊Q8C2×D311yescharacteristic
94#126—62no—
95#124—61no—
96#123—22no—
97C22⋊Q8—32no—
98C22⋊Q8C2×D312yes—
99#121—21no—
100C4×Q8—31no—
101C4×Q8—61no—
102C4×D4C2×D311yescharacteristic
103C4×D4C2×D311yescharacteristic
104#117—32no—
105C2×C4⋊C4—21no—
106#107—32no—
107#106—31no—
108C4⋊C8—31no—
109#103—32no—
110#103—32no—
111#103—32no—
112#103—31no—
113#102—31no—
114#102—32no—
115C22⋊C8—32no—
116C4×C8—31no—
117C23×C3—21no—
118C23×C3C2×D412yes—
119C3×Q8—22no—
120C3×Q8—22no—
121C3×D4—22no—
122C3×D4—21no—
123C3×D4—44no—
124C3×D4#4112yes—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—44no—
130C2×C3×C4—42no—
131C2×C3×C4C2×D412yes—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138Dic6—21no—
139Dic6—21no—
140Dic6—42no—
141C3⋊C8—21no—
142C3⋊C8—21no—
143C3⋊C8—22no—
144#41—24no—
145#41—42no—
146C2×Q8—31no—
147C2×Q8—61no—
148C2×Q8C22×D312yes—
149C2×D4—22no—
150C2×D4#6711yescharacteristic
151C2×D4C22×D311yescharacteristic
152C22×C4—21no—
153C22×C4—22no—
154C22×C4C22×D312yes—
155Q16—122no—
156SD16—62no—
157C2×C8—31no—
158C2×C8—31no—
159C2×C8—32no—
160C2×C8—122no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—31no—
165C4⋊C4—32no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4#6711yescharacteristic
170C4⋊C4C22×D311yescharacteristic
171C22⋊C4—22no—
172C22⋊C4—21no—
173C22⋊C4—62no—
174C22⋊C4—62no—
175C22⋊C4#6712yes—
176C42—62no—
177C42—61no—
178C42C22×D311yescharacteristic
179C22×C3—22no—
180C22×C3—22no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3C22×D411yescharacteristic
186C3×C4—22no—
187C3×C4—21no—
188C3×C4—42no—
189C3×C4—41no—
190C3×C4#13512yes—
191C3×C4C2×#4111yescharacteristic
192C3×C4H(5,𝔽2)11yescharacteristic
193Dic3—41no—
194Dic3—42no—
195Dic3—41no—
196C23—21no—
197C23C2×#6712yes—
198Q8—22no—
199Q8—22no—
200Q8—61no—
201Q8—61no—
202Q8—122no—
203D4—22no—
204D4—21no—
205D4—44no—
206D4#23712yes—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4—61no—
216C2×C4C2×#6712yes—
217C2×C4C2×#6711yescharacteristic
218C2×C4C2×#6711yescharacteristic
219C2×C4C23×D311yescharacteristic
220C8—61no—
221C8—61no—
222C8—62no—
223C2×C3—42no—
224C2×C3—42no—
225C2×C3#54611yescharacteristic
226C2×C3C2×#13511yescharacteristic
227C2×C3#57811yescharacteristic
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22C22×#6711yescharacteristic
235C4—22no—
236C4—21no—
237C4—42no—
238C4—41no—
239C4—121no—
240C4—122no—
241C4—121no—
242C4#95012yes—
243C4C2×#23711yescharacteristic
244C4#100411yescharacteristic
245C3#330211yescharacteristic, complemented, Sylow
246C2—42no—
247C2—42no—
248C2#580611yescharacteristic
249C2C2×#95011yescharacteristic
250C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#547C2
socleC22×C3C22×D4
hypercenter#547D3
hypocenterC3#3302
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
2. #5542C2semidirect
3. #5239C2semidirect
4. #5233C2semidirect
5. #5090C2semidirect
6. #5079C2semidirect
7. #5051C2semidirect
8. #5045C2semidirect
9. C4×#214C2semidirect
10. D4×C3⋊C8C2semidirect
11. #5007C2semidirect
245. C3#3302semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#55422
#52392
#52332
#50902
#50792
#50512
#50452
C4×#2142
D4×C3⋊C82
#50072
#33023