Finite Group Databasebeta

← #70409

Subgroups — #70409

Order 384.

380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 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#5830C211yesmaximal, characteristic, complemented
2#5822C211yesmaximal, characteristic, complemented
3C2×#908C211yesmaximal, characteristic, complemented
4C3×#421C211yesmaximal, characteristic, complemented
5#5006C211yesmaximal, characteristic, complemented
6#5001C211yesmaximal, characteristic, complemented
7#4559C211yesmaximal, characteristic, complemented
8#2008—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C22×D12C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×#117C2211yescharacteristic
14#951—21no—
15#950—22no—
16#949—21no—
17#947—21no—
18#942—21no—
19#940—21no—
20#937—22no—
21#932—21no—
22#930—22no—
23#926C2211yescharacteristic
24C2×#208C2211yescharacteristic
25#908—22no—
26#908—22no—
27C2×D24—21no—
28C2×#204—21no—
29C3×#104—22no—
30#809—22no—
31#806—22no—
32#575—31no—
33C2×AGL(1,ℤ/8)—31no—
34#535—31no—
35#421D311yescharacteristic, complemented
36#418—31no—
37#410—31no—
38#344—31no—
39C23×D3—21no—
40C3×#41—21no—
41C3×#41—21no—
42C3×#41—21no—
43C3×#41—21no—
44C2×C3×Q8D411yescharacteristic
45C2×C3×D4—21no—
46C2×C3×D4D411yescharacteristic
47C22×C3×C4—21no—
48C22×C3×C4C2311yescharacteristic
49C2×#67—21no—
50C2×#67—24no—
51#235—21no—
52#235—21no—
53#235—21no—
54#235—21no—
55C2×D12—21no—
56C2×D12—21no—
57C2×D12—21no—
58C2×D12—21no—
59C2×D12—21no—
60C2×D12D412yes—
61C2×D12D411yescharacteristic, complemented
62C2×C4×D3—21no—
63C2×Dic6D411yescharacteristic, complemented
64C3×M16—21no—
65C3×M16D412yescomplemented
66C2×C3×C8—21no—
67C3×C4⋊C4—21no—
68C3×C22⋊C4—21no—
69C3×C42—21no—
70#217D412yes—
71#216—22no—
72#215—22no—
73#214—22no—
74#213—22no—
75(C2×D3)⋊C4—24no—
76C4⋊Dic3—21no—
77C4×Dic3—21no—
78#208—21no—
79#208—21no—
80C2×C3⋊C8D412yescomplemented
81D24—24no—
82#204—24no—
83C2×#41—31no—
84C2×#41C2×D311yescharacteristic
85C22×D4—31no—
86#137—61no—
87#137—61no—
88AGL(1,ℤ/8)—62no—
89AGL(1,ℤ/8)—61no—
90AGL(1,ℤ/8)—61no—
91AGL(1,ℤ/8)—62no—
92#135—62no—
93C2×SD16—61no—
94C2×D8—61no—
95C2×M16—31no—
96C2×M16C2×D311yescharacteristic
97#127—61no—
98#124—61no—
99#121—62no—
100#120—62no—
101#117—31no—
102#117C2×D311yescharacteristic
103#104—22no—
104#102—62no—
105#99—62no—
106C23×C3—21no—
107C23×C3C2×D411yescharacteristic
108C22×D3—22no—
109C22×D3—21no—
110C22×D3—41no—
111C22×D3—41no—
112C22×D3—82no—
113C3×Q8—21no—
114C3×Q8—21no—
115C3×D4—21no—
116C3×D4—21no—
117C3×D4—41no—
118C3×D4—41no—
119C2×C3×C4—21no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4—41no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4#4111yescharacteristic
129#67—41no—
130#67—41no—
131#67—84no—
132C2×Dic3—22no—
133C2×Dic3—21no—
134D12—22no—
135D12—21no—
136D12—21no—
137D12—22no—
138D12—41no—
139D12—41no—
140C4×D3—41no—
141C4×D3—41no—
142Dic6—21no—
143Dic6—21no—
144C3×C8—22no—
145C3⋊C8—22no—
146C24—61no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—61no—
152#41—61no—
153#41—61no—
154#41—61no—
155C2×Q8—31no—
156C2×Q8D1211yescharacteristic
157C2×D4—21no—
158C2×D4—32no—
159C2×D4—31no—
160C2×D4—61no—
161C2×D4—64no—
162C2×D4—61no—
163C2×D4—61no—
164C2×D4—61no—
165C2×D4—61no—
166C2×D4—61no—
167C2×D4D1211yescharacteristic
168C22×C4—21no—
169C22×C4—61no—
170C22×C4C22×D311yescharacteristic
171Q16—62no—
172SD16—64no—
173SD16—62no—
174SD16—62no—
175D8—64no—
176D8—62no—
177M16—21no—
178M16—61no—
179M16—61no—
180M16#6712yescomplemented
181C2×C8—21no—
182C2×C8—32no—
183C4⋊C4—21no—
184C4⋊C4—61no—
185C22⋊C4—21no—
186C22⋊C4—32no—
187C22⋊C4—64no—
188C42—21no—
189C42—61no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3—41no—
193C22×C3#12011yescharacteristic
194C22×C3#12111yescharacteristic
195C22×C3#12711yescharacteristic
196C2×D3—44no—
197C2×D3—41no—
198C2×D3—42no—
199C2×D3—82no—
200C2×D3—82no—
201C2×D3—81no—
202C3×C4—41no—
203C3×C4—81no—
204C3×C4#12011yescharacteristic
205C3×C4#12011yescharacteristic
206C3×C4#12111yescharacteristic
207C3×C4#12111yescharacteristic
208Dic3—42no—
209Dic3—41no—
210C23—21no—
211C23—62no—
212C23—61no—
213C23—121no—
214C23—121no—
215C23—242no—
216C23D3×D411yescharacteristic
217Q8—21no—
218Q8—21no—
219Q8—61no—
220Q8—61no—
221D4—21no—
222D4—21no—
223D4—41no—
224D4—41no—
225D4—61no—
226D4—62no—
227D4—61no—
228D4—62no—
229D4—121no—
230D4—121no—
231D4—121no—
232D4—121no—
233D4—244no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—62no—
239C2×C4—61no—
240C2×C4—121no—
241C2×C4—121no—
242C2×C4C2×D1211yescharacteristic
243C2×C4#23511yescharacteristic
244C2×C4D3×D411yescharacteristic
245C2×C4D3×D411yescharacteristic
246C2×C4D3×D411yescharacteristic
247C2×C4C2×#6711yescharacteristic
248C8—22no—
249C8—62no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—81no—
254C2×C3#39211yescharacteristic
255D3—82no—
256D3—81no—
257C22—41no—
258C22—42no—
259C22—41no—
260C22—124no—
261C22—121no—
262C22—122no—
263C22—241no—
264C22—242no—
265C22—242no—
266C22#88211yescharacteristic
267C22#88411yescharacteristic
268C22#94011yescharacteristic
269C4—41no—
270C4—81no—
271C4—122no—
272C4—121no—
273C4#88211yescharacteristic
274C4#88411yescharacteristic
275C4#93011yescharacteristic
276C4#93711yescharacteristic
277C3#200811yescharacteristic, complemented, Sylow
278C2—21no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—242no—
283C2—241no—
284C2#496011yescharacteristic
subgroupG/H
centerC2#4960
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#392
hypercenter#421D3
hypocenterC3#2008
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #5830C2semidirect
2. #5822C2semidirect
3. C2×#908C2semidirect
4. C3×#421C2semidirect
5. #5006C2semidirect
6. #5001C2semidirect
7. #4559C2semidirect
35. #421D3semidirect
61. C2×D12D4semidirect
63. C2×Dic6D4semidirect
65. C3×M16D4semidirect
80. C2×C3⋊C8D4semidirect
180. M16#67semidirect
277. C3#2008semidirect

Maximal subgroups

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

subgroupindex
#58302
#58222
C2×#9082
C3×#4212
#50062
#50012
#45592
#20083