Finite Group Databasebeta

← #83397

Subgroups — #83397

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×#536C211yesmaximal, characteristic
2#5827C211yesmaximal, characteristic
3#5234C211yesmaximal, characteristic
4#5225C211yesmaximal, characteristic, complemented
5C2×#835C211yesmaximal, characteristic, complemented
6#5045C212yesmaximal, complemented
7#5045C212yesmaximal, complemented
8#5044C212yesmaximal, complemented
9#5043C212yesmaximal, complemented
10#4998C211yesmaximal, characteristic, complemented
11#4995C211yesmaximal, characteristic, complemented
12#3097—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C3×C4⋊Q8—21no—
16C3×#124—21no—
17C3×#123—22no—
18C3×C22⋊Q8C2212yes—
19C3×C22⋊Q8C2212yes—
20C3×#117C2211yescharacteristic
21Q8×Dic3—21no—
22D4×Dic3—22no—
23D4×Dic3—21no—
24#926C2211yescharacteristic
25#926C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2212yes—
28C2×#208C2211yescharacteristic
29C22×C3⋊C8C2211yescharacteristic
30#835C2212yes—
31#835C2212yes—
32#835C2212yes—
33#835C2212yes—
34#830C2212yes—
35#830C2212yes—
36#808C2212yes—
37#808C2211yescharacteristic
38#808C2211yescharacteristic
39#807C2212yes—
40#807C2211yescharacteristic
41#807C2211yescharacteristic
42#536D311yescharacteristic
43#520—31no—
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×Q8—21no—
56C2×C3×Q8—22no—
57C2×C3×Q8—22no—
58C2×C3×Q8D411yescharacteristic
59C2×C3×Q8C2312yes—
60C2×C3×D4D412yes—
61C2×C3×D4D411yescharacteristic
62C22×C3×C4—21no—
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C22×Dic3—22no—
66C22×Dic3—21no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4C2311yescharacteristic
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C22⋊C4—22no—
73C3×C22⋊C4—22no—
74C3×C42—21no—
75#217—22no—
76#217—21no—
77C4⋊Dic3—22no—
78C4⋊Dic3—21no—
79C4⋊Dic3C2312yes—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4×Dic3—22no—
83C4×Dic3—21no—
84#208—22no—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2312yes—
88C2×C3⋊C8C2311yescharacteristic
89C2×#41C2×D311yescharacteristic
90C22×Q8C2×D311yescharacteristic
91C2×M16—31no—
92C22×C8—31no—
93C4⋊Q8—21no—
94#124—21no—
95#123—22no—
96C22⋊Q8C2×D312yes—
97C22⋊Q8C2×D312yes—
98C4×Q8—61no—
99C4×D4—62no—
100C4×D4—61no—
101#117—31no—
102#117—32no—
103#117C2×D311yescharacteristic
104C2×C4⋊C4—31no—
105C2×C4⋊C4—32no—
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×Q8—22no—
122C3×Q8—22no—
123C3×Q8—42no—
124C3×D4—24no—
125C3×D4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—42no—
132C2×C3×C4C2×D412yes—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4#4112yes—
135C2×C3×C4#4112yes—
136C2×C3×C4C2411yescharacteristic
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—81no—
141C2×Dic3—82no—
142C3⋊C8—21no—
143C3⋊C8—21no—
144C3⋊C8—22no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8#6711yescharacteristic
151C2×Q8C22×D312yes—
152C2×D4#6712yes—
153C2×D4#6711yescharacteristic
154C22×C4—21no—
155C22×C4—62no—
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—22no—
166C4⋊C4—32no—
167C4⋊C4—31no—
168C4⋊C4—31no—
169C4⋊C4—62no—
170C4⋊C4—61no—
171C4⋊C4C22×D311yescharacteristic
172C4⋊C4C22×D311yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—62no—
177C22⋊C4—61no—
178C42—21no—
179C42—62no—
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—22no—
202Q8—22no—
203Q8—42no—
204D4—24no—
205D4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4—241no—
216C2×C4—242no—
217C2×C4#23712yes—
218C2×C4#23712yes—
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#93512yes—
236C22C22×#6711yescharacteristic
237C4—22no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—122no—
242C4—121no—
243C4—121no—
244C4#93512yes—
245C4C2×#23711yescharacteristic
246C4C2×#23711yescharacteristic
247C3#309711yescharacteristic, complemented, Sylow
248C2—22no—
249C2—42no—
250C2C2×#93511yescharacteristic
251C2#583711yescharacteristic
252C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#536D3
hypocenterC3#3097
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
4. #5225C2semidirect
5. C2×#835C2semidirect
6. #5045C2semidirect
7. #5045C2semidirect
8. #5044C2semidirect
9. #5043C2semidirect
10. #4998C2semidirect
11. #4995C2semidirect
247. C3#3097semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58272
#52342
#52252
C2×#8352
#50452
#50442
#50432
#49982
#49952
#30973