Finite Group Databasebeta

← #70331

Subgroups — #70331

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#5805C211yesmaximal, characteristic, complemented
2C2×#932C211yesmaximal, characteristic, complemented
3#5748C211yesmaximal, characteristic, complemented
4C3×#418C211yesmaximal, characteristic, complemented
5#5006C211yesmaximal, characteristic, complemented
6#5005C211yesmaximal, characteristic, complemented
7#4556C211yesmaximal, characteristic, complemented
8#2008—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×#235C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×#117C2211yescharacteristic
14#953—22no—
15#940—21no—
16#938—22no—
17#936—21no—
18C2×#214—21no—
19#932—22no—
20#932—22no—
21C2×#213—21no—
22#926C2211yescharacteristic
23C2×#208C2211yescharacteristic
24#909—21no—
25#909—21no—
26#908—21no—
27#908—21no—
28#904—22no—
29C3×#102—22no—
30#806—22no—
31#805—22no—
32#575—31no—
33C2×AGL(1,ℤ/8)—31no—
34#535—31no—
35#421—31no—
36#418D311yescharacteristic, complemented
37#410—31no—
38#344—31no—
39C24×C3—21no—
40C2×C3×D4—21no—
41C2×C3×D4—21no—
42C2×C3×D4—21no—
43C2×C3×D4—21no—
44C2×C3×D4—21no—
45C2×C3×D4D412yes—
46C22×C3×C4C2311yescharacteristic
47C2×#67—21no—
48C2×#67—21no—
49C2×#67—24no—
50#235—21no—
51#235—21no—
52#235—21no—
53#235—21no—
54#235—21no—
55#235—21no—
56#235—21no—
57#235—21no—
58C2×D12D411yescharacteristic
59C2×D12D411yescharacteristic, complemented
60C2×C4×D3—21no—
61C2×C4×D3—21no—
62C2×Dic6D411yescharacteristic
63C2×Dic6D411yescharacteristic, complemented
64C3×M16—21no—
65C3×M16—21no—
66C2×C3×C8D412yescomplemented
67C3×C4⋊C4—21no—
68C3×C22⋊C4—21no—
69C3×C42—21no—
70#217—24no—
71#217D412yes—
72#214—24no—
73#213—24no—
74C4⋊Dic3—21no—
75C4×Dic3—21no—
76#208—21no—
77#208D412yescomplemented
78C2×C3⋊C8—21no—
79Dic12—22no—
80D24—22no—
81#204—22no—
82#204—22no—
83C2×#41—31no—
84C2×#41—31no—
85C22×D4C2×D311yescharacteristic
86#137—61no—
87#137—61no—
88AGL(1,ℤ/8)—61no—
89AGL(1,ℤ/8)—61no—
90AGL(1,ℤ/8)—62no—
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—62no—
104#102—22no—
105#99—62no—
106C23×C3—22no—
107C23×C3—41no—
108C23×C3—41no—
109C23×C3—82no—
110C23×C3C2×D411yescharacteristic
111C22×D3—21no—
112C22×D3—21no—
113C3×D4—22no—
114C3×D4—22no—
115C3×D4—41no—
116C3×D4—41no—
117C2×C3×C4—41no—
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2×D411yescharacteristic
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4#4111yescharacteristic
124#67—41no—
125#67—41no—
126#67—41no—
127#67—41no—
128#67—84no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132D12—21no—
133D12—21no—
134D12—21no—
135D12—21no—
136C4×D3—41no—
137C4×D3—41no—
138C4×D3—41no—
139C4×D3—41no—
140Dic6—21no—
141Dic6—21no—
142Dic6—21no—
143Dic6—21no—
144C3×C8—22no—
145C3⋊C8—22no—
146C24—21no—
147#41—61no—
148#41—61no—
149#41—61no—
150#41—61no—
151#41—61no—
152#41—61no—
153#41—61no—
154#41—61no—
155C2×Q8—31no—
156C2×Q8—31no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—31no—
163C2×D4—31no—
164C2×D4—61no—
165C2×D4—61no—
166C2×D4—64no—
167C2×D4D1212yes—
168C22×C4—61no—
169C22×C4—61no—
170C22×C4C22×D311yescharacteristic
171Q16—62no—
172SD16—62no—
173SD16—62no—
174SD16—64no—
175D8—62no—
176D8—64no—
177M16—21no—
178M16—21no—
179M16—32no—
180M16—61no—
181C2×C8—61no—
182C2×C8#6712yescomplemented
183C4⋊C4—21no—
184C4⋊C4—61no—
185C22⋊C4—21no—
186C22⋊C4—32no—
187C22⋊C4—64no—
188C42—21no—
189C42—61no—
190C22×C3—41no—
191C22×C3—44no—
192C22×C3—42no—
193C22×C3—82no—
194C22×C3—82no—
195C22×C3#12011yescharacteristic
196C22×C3#12111yescharacteristic
197C22×C3#12711yescharacteristic
198C2×D3—42no—
199C2×D3—41no—
200C2×D3—41no—
201C2×D3—81no—
202C3×C4—81no—
203C3×C4#12011yescharacteristic
204C3×C4#12011yescharacteristic
205C3×C4#12111yescharacteristic
206C3×C4#12111yescharacteristic
207Dic3—42no—
208Dic3—41no—
209Dic3—41no—
210C23—22no—
211C23—41no—
212C23—41no—
213C23—61no—
214C23—61no—
215C23—82no—
216C23D3×D411yescharacteristic
217Q8—61no—
218Q8—61no—
219Q8—61no—
220Q8—61no—
221D4—22no—
222D4—22no—
223D4—41no—
224D4—41no—
225D4—61no—
226D4—61no—
227D4—61no—
228D4—61no—
229D4—121no—
230D4—121no—
231D4—121no—
232D4—121no—
233D4—244no—
234C2×C4—41no—
235C2×C4—62no—
236C2×C4—61no—
237C2×C4—61no—
238C2×C4—121no—
239C2×C4—121no—
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—82no—
254C2×C3#39211yescharacteristic
255D3—81no—
256D3—81no—
257C22—41no—
258C22—44no—
259C22—42no—
260C22—82no—
261C22—82no—
262C22—122no—
263C22—121no—
264C22—121no—
265C22—241no—
266C22#88211yescharacteristic
267C22#88411yescharacteristic
268C22#94011yescharacteristic
269C4—81no—
270C4—122no—
271C4—121no—
272C4—121no—
273C4#88211yescharacteristic
274C4#88411yescharacteristic
275C4#93011yescharacteristic
276C4#93711yescharacteristic
277C3#200811yescharacteristic, complemented, Sylow
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—241no—
283C2—241no—
284C2#496011yescharacteristic
subgroupG/H
centerC2#4960
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#418C2
socleC2×C3#392
hypercenter#418D3
hypocenterC3#2008
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #5805C2semidirect
2. C2×#932C2semidirect
3. #5748C2semidirect
4. C3×#418C2semidirect
5. #5006C2semidirect
6. #5005C2semidirect
7. #4556C2semidirect
36. #418D3semidirect
59. C2×D12D4semidirect
63. C2×Dic6D4semidirect
66. C2×C3×C8D4semidirect
77. #208D4semidirect
182. C2×C8#67semidirect
277. C3#2008semidirect

Maximal subgroups

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

subgroupindex
#58052
C2×#9322
#57482
C3×#4182
#50062
#50052
#45562
#20083