Finite Group Databasebeta

← #80326

Subgroups — #80326

Order 384.

406 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 115 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#5718C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5154C212yesmaximal, complemented
5#5153C212yesmaximal, complemented
6#5076C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4762C212yesmaximal, complemented
11#4757C212yesmaximal, complemented
12#3397—31nomaximal, Sylow
13C2×#237—21no—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#940C2212yes—
18#939—22no—
19#938—21no—
20#936C2212yes—
21D4×Dic3C2212yes—
22C2×#214C2212yes—
23C2×#214C2212yes—
24C2×#213C2211yescharacteristic
25#926—21no—
26#924—21no—
27C2×#204C2212yes—
28#887C2212yes—
29#879—22no—
30C4×D12C2211yescharacteristic
31#869C2211yescharacteristic
32C4×Dic6C2211yescharacteristic
33C3×C4⋊C8C2211yescharacteristic
34C3×#102C2212yes—
35#832C2211yescharacteristic
36#816C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#556—31no—
43#550—31no—
44#496—32no—
45#492—31no—
46#463—32no—
47#461—31no—
48#460—32no—
49#459D311yescharacteristic, complemented
50#446—31no—
51#440—32no—
52C4×D8—31no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2311yescharacteristic
56C2×C3×D4C2312yes—
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60C22×Dic3—22no—
61#237—44no—
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×Dic6D411yescharacteristic
65C2×Dic6C2312yes—
66C3×D8—42no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4C2311yescharacteristic
69C3×C22⋊C4—21no—
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—22no—
73#217—22no—
74#214—24no—
75#214—42no—
76#213—22no—
77(C2×D3)⋊C4—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—22no—
83Dic3⋊C4D412yes—
84C4×Dic3—22no—
85C4×Dic3C2312yes—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88#204—42no—
89C2×#41—61no—
90C2×SD16—32no—
91C2×SD16—32no—
92C2×SD16—32no—
93C2×D8—31no—
94C2×D8C2×D312yes—
95C4⋊Q8—31no—
96#127—32no—
97#127C2×D311yescharacteristic
98#126—62no—
99#124—32no—
100C22⋊Q8—61no—
101#121—61no—
102#121—62no—
103C4×Q8—31no—
104C4×Q8—32no—
105C4×D4—32no—
106C4×D4—31no—
107C4×D4C2×D311yescharacteristic
108#117—61no—
109#107—31no—
110C4⋊C8—32no—
111C4⋊C8C2×D311yescharacteristic
112#103—32no—
113#102—31no—
114#102—31no—
115#102C2×D312yes—
116#97—32no—
117C4×C8—31no—
118C23×C3—21no—
119C23×C3—22no—
120C22×D3—21no—
121C3×D4—21no—
122C3×D4—22no—
123C3×D4—22no—
124C3×D4—21no—
125C3×D4—42no—
126C3×D4—81no—
127C2×C3×C4—21no—
128C2×C3×C4—42no—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—44no—
133#67—82no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—44no—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×D412yes—
139D12—21no—
140D12—21no—
141C4×D3—42no—
142Dic6—22no—
143Dic6—22no—
144Dic6—22no—
145C3×C8—22no—
146C3⋊C8—21no—
147C3⋊C8#4112yescomplemented
148#41—124no—
149C2×Q8—31no—
150C2×Q8—32no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—31no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4C22×D311yescharacteristic
157C2×D4C22×D312yes—
158C22×C4—21no—
159C22×C4—62no—
160C22×C4—61no—
161SD16—64no—
162SD16—122no—
163SD16—122no—
164D8—42no—
165D8—62no—
166C2×C8—31no—
167C2×C8—31no—
168C2×C8C22×D312yes—
169C4⋊C4—31no—
170C4⋊C4—32no—
171C4⋊C4—31no—
172C4⋊C4—62no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4C22×D311yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—61no—
181C42—32no—
182C42—62no—
183C42C22×D311yescharacteristic
184C22×C3—41no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3—41no—
190C22×C3C22×D411yescharacteristic
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—41no—
194C3×C4—21no—
195C3×C4—41no—
196C3×C4AGL(1,ℤ/8)12yes—
197C3×C4C2×#4111yescharacteristic
198C3×C4H(5,𝔽2)11yescharacteristic
199Dic3—22no—
200Dic3—22no—
201Dic3—42no—
202Dic3—41no—
203C23—21no—
204C23—22no—
205C23—61no—
206Q8—62no—
207Q8—62no—
208Q8—62no—
209D4—21no—
210D4—22no—
211D4—21no—
212D4—22no—
213D4—42no—
214D4—61no—
215D4—61no—
216D4—81no—
217D4—124no—
218D4—242no—
219C2×C4—21no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—42no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—124no—
226C2×C4—122no—
227C2×C4D3×D411yescharacteristic
228C2×C4D3×D411yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—22no—
231C8—32no—
232C8—61no—
233C2×C3—82no—
234C2×C3—81no—
235C2×C3#55011yescharacteristic
236C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
237C2×C3#57711yescharacteristic
238D3—81no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—62no—
252C4—62no—
253C4—121no—
254C4—122no—
255C4#91112yes—
256C4#100411yescharacteristic
257C4D3×#4111yescharacteristic
258C3#339711yescharacteristic, complemented, Sylow
259C2—81no—
260C2—82no—
261C2—241no—
262C2#559911yescharacteristic
263C2C2×#91111yescharacteristic
264C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3397
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5542C2semidirect
3. C3×#459C2semidirect
4. #5154C2semidirect
5. #5153C2semidirect
6. #5076C2semidirect
7. C4×#213C2semidirect
8. #4841C2semidirect
9. #4831C2semidirect
10. #4762C2semidirect
11. #4757C2semidirect
49. #459D3semidirect
147. C3⋊C8#41semidirect
258. C3#3397semidirect

Maximal subgroups

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

subgroupindex
#57182
#55422
C3×#4592
#51542
#51532
#50762
C4×#2132
#48412
#48312
#47622
#47572
#33973