Finite Group Databasebeta

← #82573

Subgroups — #82573

Order 384.

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

normal subgroups only

This group has 386 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
1C3×#520C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic
3#5234C211yesmaximal, characteristic
4#5004C211yesmaximal, characteristic, complemented
5#4998C211yesmaximal, characteristic, complemented
6#4996C212yesmaximal, complemented
7#4996C212yesmaximal, complemented
8#4971C212yesmaximal, complemented
9#4970C212yesmaximal, complemented
10C2×#810C211yesmaximal, characteristic, complemented
11#4964C211yesmaximal, characteristic, complemented
12#3097—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C3×C4×Q8—21no—
16C3×C4×D4—22no—
17C3×C4×D4—21no—
18C3×#117C2212yes—
19C3×#117C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2212yes—
22#944—21no—
23#936—21no—
24#935—22no—
25#926C2211yescharacteristic
26#924C2212yes—
27#924C2212yes—
28C2×#208C2211yescharacteristic
29C22×C3⋊C8C2211yescharacteristic
30#830C2212yes—
31#830C2212yes—
32#810C2212yes—
33#810C2212yes—
34#810C2212yes—
35#810C2212yes—
36#808C2212yes—
37#808C2211yescharacteristic
38#808C2211yescharacteristic
39#807C2212yes—
40#807C2211yescharacteristic
41#807C2211yescharacteristic
42#536—31no—
43#520D311yescharacteristic
44#485—32no—
45#485—32no—
46#484—32no—
47#483—32no—
48#428—31no—
49#427—31no—
50#419—31no—
51C2×#103—31no—
52#408—31no—
53C3×#41—22no—
54C3×#41—22no—
55C2×C3×Q8D411yescharacteristic
56C2×C3×D4D412yes—
57C2×C3×D4D411yescharacteristic
58C22×C3×C4—22no—
59C22×C3×C4—21no—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×Dic3—21no—
63C2×Dic6—22no—
64C2×Dic6—22no—
65C2×Dic6—21no—
66C2×Dic6C2312yes—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4C2312yes—
70C3×C4⋊C4C2311yescharacteristic
71C3×C4⋊C4C2311yescharacteristic
72C3×C22⋊C4—22no—
73C3×C22⋊C4—21no—
74C3×C42—22no—
75C3×C42—21no—
76#217—22no—
77#217—22no—
78#217—21no—
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2311yescharacteristic
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83C4×Dic3—21no—
84#208—22no—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2312yes—
88C2×C3⋊C8C2311yescharacteristic
89C2×#41C2×D311yescharacteristic
90C22×Q8—31no—
91C2×M16—31no—
92C22×C8—31no—
93C4⋊Q8—61no—
94#124—61no—
95#123—62no—
96C22⋊Q8—32no—
97C22⋊Q8—32no—
98C4×Q8—21no—
99C4×D4—22no—
100C4×D4—21no—
101#117—31no—
102#117C2×D311yescharacteristic
103#117C2×D312yes—
104C2×C4⋊C4C2×D311yescharacteristic
105C2×C4⋊C4C2×D312yes—
106#107—31no—
107#107—31no—
108#107—32no—
109#106—31no—
110#106—31no—
111#106—32no—
112#103—32no—
113#103—32no—
114#103—32no—
115#103—32no—
116C22⋊C8—32no—
117C22⋊C8—32no—
118C23×C3C2×D412yes—
119C23×C3C2×D411yescharacteristic
120C3×Q8—22no—
121C3×D4—24no—
122C3×D4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—81no—
129C2×C3×C4—82no—
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4#4112yes—
133C2×C3×C4#4112yes—
134C2×C3×C4C2411yescharacteristic
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—42no—
139Dic6—22no—
140Dic6—22no—
141Dic6—42no—
142C3⋊C8—21no—
143C3⋊C8—21no—
144C3⋊C8—22no—
145#41—22no—
146#41—22no—
147C2×Q8—32no—
148C2×Q8—61no—
149C2×Q8—62no—
150C2×Q8—62no—
151C2×Q8#6711yescharacteristic
152C2×D4#6712yes—
153C2×D4#6711yescharacteristic
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—61no—
157C22×C4C22×D312yes—
158C22×C4C22×D311yescharacteristic
159M16—62no—
160C2×C8—31no—
161C2×C8—32no—
162C2×C8—31no—
163C2×C8—62no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4C22×D312yes—
171C4⋊C4C22×D311yescharacteristic
172C4⋊C4C22×D311yescharacteristic
173C22⋊C4—22no—
174C22⋊C4—21no—
175C22⋊C4—61no—
176C22⋊C4—62no—
177C22⋊C4—62no—
178C42—22no—
179C42—21no—
180C42—61no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—22no—
186C22×C3#12312yes—
187C22×C3C22×D411yescharacteristic
188C3×C4—22no—
189C3×C4—42no—
190C3×C4—41no—
191C3×C4—41no—
192C3×C4#12312yes—
193C3×C4C2×#4111yescharacteristic
194C3×C4C2×#4111yescharacteristic
195Dic3—42no—
196Dic3—41no—
197Dic3—41no—
198C23C2×#6712yes—
199C23C2×#6711yescharacteristic
200Q8—22no—
201Q8—62no—
202Q8—62no—
203Q8—122no—
204D4—24no—
205D4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—122no—
217C2×C4#23512yes—
218C2×C4#23512yes—
219C2×C4C2×#6712yes—
220C2×C4C2×#6711yescharacteristic
221C2×C4C23×D311yescharacteristic
222C8—61no—
223C8—61no—
224C8—62no—
225C2×C3—22no—
226C2×C3—42no—
227C2×C3C2×#12311yescharacteristic
228C2×C3#57711yescharacteristic
229C2×C3#57811yescharacteristic
230C22—21no—
231C22—21no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22#92912yes—
236C22C22×#6711yescharacteristic
237C4—22no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—122no—
242C4—121no—
243C4—121no—
244C4#92912yes—
245C4C2×#23511yescharacteristic
246C4C2×#23511yescharacteristic
247C3#309711yescharacteristic, complemented, Sylow
248C2—22no—
249C2—42no—
250C2C2×#92911yescharacteristic
251C2#583611yescharacteristic
252C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#520C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#520D3
hypocenterC3#3097
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
4. #5004C2semidirect
5. #4998C2semidirect
6. #4996C2semidirect
7. #4996C2semidirect
8. #4971C2semidirect
9. #4970C2semidirect
10. C2×#810C2semidirect
11. #4964C2semidirect
247. C3#3097semidirect

Maximal subgroups

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

subgroupindex
C3×#5202
#58252
#52342
#50042
#49982
#49962
#49712
#49702
C2×#8102
#49642
#30973