Finite Group Databasebeta

← #83574

Subgroups — #83574

Order 384.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#550C211yesmaximal, characteristic, complemented
2#5555C211yesmaximal, characteristic, complemented
3#5238C212yesmaximal, complemented
4#5232C212yesmaximal, complemented
5#5072C211yesmaximal, characteristic, complemented
6#5070C211yesmaximal, characteristic, complemented
7#5039C212yesmaximal, complemented
8#5034C212yesmaximal, complemented
9C4×#215C211yesmaximal, characteristic, complemented
10#5020C211yesmaximal, characteristic, complemented
11#5012C211yesmaximal, characteristic, complemented
12#3332—31nomaximal, Sylow
13C2×C3×#41C2212yes—
14C2×#237—21no—
15C3×#127—21no—
16C3×#127C2211yescharacteristic
17C3×#121—22no—
18C3×#121C2212yes—
19C3×C4×Q8C2211yescharacteristic
20C3×C4×D4—21no—
21C3×C4×D4C2211yescharacteristic
22C2×#215C2211yescharacteristic
23C2×#213C2212yes—
24#930C2212yes—
25C2×C4⋊Dic3C2212yes—
26C2×#208C2212yes—
27#897—21no—
28#886—22no—
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic
31#835C2211yescharacteristic
32#835C2212yes—
33#832C2212yes—
34#832C2212yes—
35#832C2212yes—
36#830C2212yes—
37#809C2211yescharacteristic
38#808C2212yes—
39#808C2211yescharacteristic
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#550D311yescharacteristic, complemented
43#547—31no—
44#486—31no—
45#481—32no—
46#475—31no—
47#469—32no—
48#459—31no—
49#449—32no—
50C4×SD16—31no—
51#436—31no—
52#417—32no—
53C3×#41—24no—
54C3×#41—42no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4D411yescharacteristic
61C2×C3×D4C2312yes—
62C22×C3×C4—22no—
63C22×C3×C4C2312yes—
64C2×#67—22no—
65C22×Dic3—22no—
66#237—44no—
67C2×D12C2311yescharacteristic
68C2×C4×D3—21no—
69C2×Dic6—21no—
70C3×C4⋊C4—21no—
71C3×C4⋊C4D411yescharacteristic
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—22no—
74C3×C22⋊C4D412yes—
75C3×C42—21no—
76C3×C42C2311yescharacteristic
77#215—22no—
78#213—42no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—22no—
81C4⋊Dic3—21no—
82C4⋊Dic3—22no—
83C4⋊Dic3C2312yes—
84C4⋊Dic3C2311yescharacteristic
85#208—42no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2312yes—
89C2×#41—61no—
90C2×#41C2×D312yes—
91C2×SD16—31no—
92C2×D8—32no—
93C2×M16—32no—
94C4⋊Q8—31no—
95#127—21no—
96#127C2×D311yescharacteristic
97#123—62no—
98C22⋊Q8—61no—
99#121—22no—
100#121—32no—
101#121C2×D312yes—
102C4×Q8C2×D311yescharacteristic
103C4×D4—21no—
104C4×D4—31no—
105C4×D4C2×D311yescharacteristic
106C2×C4⋊C4—32no—
107#106—31no—
108#106—32no—
109C4⋊C8—31no—
110#103—31no—
111#103—32no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102—31no—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—22no—
119C23×C3C2×D412yes—
120C22×D3—21no—
121C3×Q8—21no—
122C3×Q8#4112yescomplemented
123C3×D4—22no—
124C3×D4—22no—
125C3×D4—22no—
126C3×D4—44no—
127C3×D4—42no—
128C3×D4—81no—
129C3×D4—82no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—44no—
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D412yes—
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—44no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3—44no—
143D12—21no—
144D12—21no—
145C4×D3—42no—
146Dic6—42no—
147C3⋊C8—21no—
148C3⋊C8—21no—
149C3⋊C8—22no—
150#41—24no—
151#41—42no—
152#41—124no—
153C2×Q8—61no—
154C2×Q8C22×D311yescharacteristic
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—31no—
160C2×D4—62no—
161C2×D4#6711yescharacteristic
162C2×D4C22×D312yes—
163C22×C4—22no—
164C22×C4—62no—
165C22×C4—61no—
166C22×C4C22×D312yes—
167SD16—62no—
168D8—122no—
169M16—122no—
170C2×C8—31no—
171C2×C8—31no—
172C2×C8—32no—
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—32no—
176C4⋊C4—62no—
177C4⋊C4—61no—
178C4⋊C4#6711yescharacteristic
179C4⋊C4C22×D311yescharacteristic
180C22⋊C4—22no—
181C22⋊C4—61no—
182C22⋊C4—62no—
183C22⋊C4#6712yes—
184C42—21no—
185C42C22×D311yescharacteristic
186C22×C3—22no—
187C22×C3—22no—
188C22×C3—22no—
189C22×C3—42no—
190C22×C3—42no—
191C22×C3—42no—
192C22×C3C22×D411yescharacteristic
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—21no—
199C3×C4—41no—
200C3×C4AGL(1,ℤ/8)12yes—
201C3×C4C2×#4111yescharacteristic
202C3×C4H(5,𝔽2)11yescharacteristic
203Dic3—42no—
204Dic3—41no—
205C23—22no—
206C23—61no—
207C23C2×#6712yes—
208Q8—21no—
209Q8—122no—
210Q8#23712yescomplemented
211D4—22no—
212D4—22no—
213D4—22no—
214D4—44no—
215D4—42no—
216D4—61no—
217D4—61no—
218D4—81no—
219D4—82no—
220D4—124no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—44no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—124no—
229C2×C4—122no—
230C2×C4C2×#6711yescharacteristic
231C2×C4C2×#6712yes—
232C2×C4C2×#6711yescharacteristic
233C2×C4C23×D311yescharacteristic
234C8—61no—
235C8—61no—
236C8—62no—
237C2×C3—42no—
238C2×C3—82no—
239C2×C3#54611yescharacteristic
240C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C3#57711yescharacteristic
242D3—81no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—121no—
250C22—121no—
251C22—121no—
252C22C22×#6711yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—41no—
257C4—121no—
258C4—122no—
259C4#94912yes—
260C4C2×#23711yescharacteristic
261C4#100411yescharacteristic
262C3#333211yescharacteristic, complemented, Sylow
263C2—42no—
264C2—82no—
265C2—241no—
266C2#580611yescharacteristic
267C2C2×#94911yescharacteristic
268C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#550C2
socleC22×C3C22×D4
hypercenter#550D3
hypocenterC3#3332
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5555C2semidirect
3. #5238C2semidirect
4. #5232C2semidirect
5. #5072C2semidirect
6. #5070C2semidirect
7. #5039C2semidirect
8. #5034C2semidirect
9. C4×#215C2semidirect
10. #5020C2semidirect
11. #5012C2semidirect
42. #550D3semidirect
122. C3×Q8#41semidirect
210. Q8#237semidirect
262. C3#3332semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#55552
#52382
#52322
#50722
#50702
#50392
#50342
C4×#2152
#50202
#50122
#33323