Finite Group Databasebeta

← #83626

Subgroups — #83626

Order 384.

342 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 123 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×#554C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic
3#5091C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5052C212yesmaximal, complemented
6#5045C212yesmaximal, complemented
7C4×#216C211yesmaximal, characteristic, complemented
8#5020C211yesmaximal, characteristic, complemented
9#5012C211yesmaximal, characteristic
10#4971C212yesmaximal, complemented
11#4964C212yesmaximal, complemented
12#3356—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×#125—21no—
15C3×C22⋊Q8—22no—
16C3×C22⋊Q8C2212yes—
17C3×C4×Q8C2211yescharacteristic
18C3×C4×D4—21no—
19C3×C4×D4C2211yescharacteristic
20C2×C3×C4⋊C4C2212yes—
21C2×#216C2211yescharacteristic
22#936—21no—
23#926C2212yes—
24#924C2212yes—
25C2×#208C2212yes—
26#887—21no—
27#879—22no—
28#869C2211yescharacteristic
29C4×Dic6C2211yescharacteristic
30#835C2212yes—
31#835C2211yescharacteristic
32#830C2212yes—
33#810C2211yescharacteristic
34#810C2212yes—
35#808C2212yes—
36#808C2212yes—
37#808C2212yes—
38#807C2211yescharacteristic
39#807C2212yes—
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#554D311yescharacteristic
44#498—31no—
45#485—32no—
46#483—32no—
47#477—31no—
48#475—31no—
49#470—32no—
50C4×Q16—31no—
51#436—31no—
52#428—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×D4D411yescharacteristic
56C22×C3×C4—22no—
57C22×C3×C4C2312yes—
58C2×Dic6—21no—
59C2×Dic6C2311yescharacteristic
60C3×C4⋊C4—22no—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4D411yescharacteristic
65C3×C4⋊C4C2311yescharacteristic
66C3×C4⋊C4C2312yes—
67C3×C22⋊C4—22no—
68C3×C22⋊C4D412yes—
69C3×C42—21no—
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—22no—
73#216—22no—
74C4⋊Dic3—21no—
75C4⋊Dic3C2311yescharacteristic
76C4⋊Dic3C2312yes—
77Dic3⋊C4—21no—
78Dic3⋊C4—22no—
79C4×Dic3—21no—
80C4×Dic3—22no—
81#208—24no—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2312yes—
84C2×C3⋊C8C2311yescharacteristic
85C2×Q16—31no—
86C2×M16—32no—
87C4⋊Q8—31no—
88C4⋊Q8C2×D311yescharacteristic
89#126—62no—
90#125—21no—
91#124—61no—
92C22⋊Q8—22no—
93C22⋊Q8—32no—
94C22⋊Q8C2×D312yes—
95C4×Q8—31no—
96C4×Q8—61no—
97C4×Q8C2×D311yescharacteristic
98C4×D4—21no—
99C4×D4C2×D311yescharacteristic
100#117—32no—
101C2×C4⋊C4C2×D312yes—
102#107—31no—
103#107—32no—
104#106—32no—
105#106—32no—
106#106—32no—
107C4⋊C8—31no—
108#103—31no—
109#103—32no—
110#103—31no—
111#103—32no—
112C22⋊C8—32no—
113C4×C8—31no—
114C23×C3C2×D412yes—
115C3×Q8—21no—
116C3×Q8—21no—
117C3×Q8—81no—
118C3×D4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—82no—
125C2×C3×C4C2×D412yes—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2411yescharacteristic
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132Dic6—21no—
133Dic6—21no—
134Dic6—42no—
135C3⋊C8—22no—
136C3⋊C8—21no—
137C3⋊C8#4112yes—
138C2×Q8—21no—
139C2×Q8—31no—
140C2×Q8—61no—
141C2×Q8C22×D311yescharacteristic
142C2×D4#6711yescharacteristic
143C22×C4—22no—
144C22×C4C22×D312yes—
145Q16—62no—
146M16—64no—
147C2×C8—31no—
148C2×C8—31no—
149C2×C8—32no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—32no—
155C4⋊C4—31no—
156C4⋊C4—61no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4#6711yescharacteristic
160C4⋊C4C22×D311yescharacteristic
161C4⋊C4C22×D312yes—
162C22⋊C4—22no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4#6712yes—
166C42—21no—
167C42—62no—
168C42—61no—
169C42C22×D311yescharacteristic
170C22×C3—22no—
171C22×C3—22no—
172C22×C3—22no—
173C22×C3C22×D411yescharacteristic
174C3×C4—22no—
175C3×C4—21no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—42no—
179C3×C4#13712yes—
180C3×C4C2×#4111yescharacteristic
181C3×C42−1+411yescharacteristic
182Dic3—41no—
183Dic3—42no—
184Dic3—41no—
185C23C2×#6712yes—
186Q8—21no—
187Q8—21no—
188Q8—61no—
189Q8—61no—
190Q8—81no—
191Q8—122no—
192D4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—61no—
199C2×C4—62no—
200C2×C4—61no—
201C2×C4—82no—
202C2×C4C2×#6712yes—
203C2×C4C2×#6711yescharacteristic
204C2×C4C2×#6711yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—32no—
207C8—62no—
208C8—61no—
209C2×C3—42no—
210C2×C3#54711yescharacteristic
211C2×C3C2×#13711yescharacteristic
212C2×C3#57711yescharacteristic
213C22—22no—
214C22—22no—
215C22—22no—
216C22C22×#6711yescharacteristic
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—42no—
222C4—122no—
223C4—121no—
224C4—121no—
225C4#94212yes—
226C4D3×#4111yescharacteristic
227C4#101011yescharacteristic
228C3#335611yescharacteristic, complemented, Sylow
229C2—42no—
230C2C2×#94211yescharacteristic
231C2#582411yescharacteristic
232C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#554C2
socleC22×C3C22×D4
prime radicalC2×C3×D4D4
hypercenter#554D3
hypocenterC3#3356
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
3. #5091C2semidirect
4. #5080C2semidirect
5. #5052C2semidirect
6. #5045C2semidirect
7. C4×#216C2semidirect
8. #5020C2semidirect
10. #4971C2semidirect
11. #4964C2semidirect
228. C3#3356semidirect

Maximal subgroups

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

subgroupindex
C3×#5542
#55422
#50912
#50802
#50522
#50452
C4×#2162
#50202
#50122
#49712
#49642
#33563