Finite Group Databasebeta

← #81018

Subgroups — #81018

Order 384.

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

normal subgroups only

This group has 362 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
1D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#5580C211yesmaximal, characteristic
3C3×#475C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5#5020C211yesmaximal, characteristic, complemented
6#4863C212yesmaximal
7#4856C212yesmaximal, complemented
8#4846C211yesmaximal, characteristic, complemented
9#4835C211yesmaximal, characteristic, complemented
10#4811C212yesmaximal, complemented
11#4794C212yesmaximal, complemented
12#3239—31nomaximal, Sylow
13C2×D3×Q8—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#947C2211yescharacteristic
17#946C2211yescharacteristic
18#944—21no—
19#898—21no—
20#895C2211yescharacteristic
21#894—21no—
22D3×C4⋊C4—21no—
23D3×C4⋊C4C2212yes—
24#890C2211yescharacteristic
25#888—21no—
26#888C2212yes—
27C4×D12—21no—
28C4×D12C2211yescharacteristic
29#872—21no—
30C42×D3C2211yescharacteristic
31#869C2211yescharacteristic
32C3×#106C2212yes—
33C3×C4⋊C8C2211yescharacteristic
34C3×#103C2212yes—
35#835C2212yes—
36#821C2212yes—
37(C2×D3)⋊C8C2212yes—
38#818C2212yes—
39#815C2212yes—
40#809C2212yes—
41#808C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊Q8—31no—
45#529—31no—
46#501—32no—
47#489—32no—
48#484—32no—
49#481—32no—
50#475—31no—
51#475D311yescharacteristic, complemented
52#474—31no—
53#474—31no—
54#432—31no—
55C2×C3×Q8—21no—
56C2×C3×Q8C2311yescharacteristic
57D3×Q8—44no—
58C2×D12—21no—
59C2×D12C2311yescharacteristic
60C2×C4×D3—22no—
61C2×C4×D3—21no—
62C2×C4×D3D411yescharacteristic
63C2×C4×D3D411yescharacteristic
64C2×C4×D3C2311yescharacteristic
65C2×Dic6—21no—
66C2×Dic6—22no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2312yes—
71C3×C4⋊C4C2311yescharacteristic
72C3×C42—21no—
73C3×C42C2311yescharacteristic
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3—21no—
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2312yes—
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—22no—
86Dic3⋊C4—22no—
87C4×Dic3D411yescharacteristic
88C4×Dic3D411yescharacteristic
89C4×Dic3C2311yescharacteristic
90C2×C3⋊C8C2312yes—
91C22×Q8—61no—
92C4⋊Q8—32no—
93C4⋊Q8—31no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96C4⋊Q8C2×D311yescharacteristic
97#126—61no—
98#125—31no—
99#124—31no—
100#123—61no—
101C22⋊Q8—31no—
102#121—31no—
103C4×Q8C2×D311yescharacteristic
104C4×D4—31no—
105C4×D4—61no—
106#117—61no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—61no—
109C2×C42—31no—
110#107—32no—
111#107—32no—
112#106—32no—
113#106C2×D312yes—
114C4⋊C8—31no—
115C4⋊C8C2×D311yescharacteristic
116#103—32no—
117#103C2×D312yes—
118#102—32no—
119#102—32no—
120C22⋊C8—32no—
121#97—32no—
122C22×D3—21no—
123C22×D3C2×D411yescharacteristic
124C3×Q8—21no—
125C3×Q8—21no—
126C3×Q8—42no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2411yescharacteristic
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3C2×D411yescharacteristic
136C2×Dic3C2×D411yescharacteristic
137C2×Dic3C2×D411yescharacteristic
138D12—21no—
139D12—21no—
140D12—42no—
141C4×D3—22no—
142C4×D3—22no—
143C4×D3—21no—
144C4×D3—21no—
145C4×D3—44no—
146C4×D3—42no—
147Dic6—44no—
148Dic6—42no—
149C3×C8—22no—
150C3⋊C8—22no—
151C2×Q8—21no—
152C2×Q8—62no—
153C2×Q8—61no—
154C2×Q8—124no—
155C2×Q8C22×D311yescharacteristic
156C2×D4—31no—
157C2×D4—61no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—62no—
162C22×C4—61no—
163C2×C8—32no—
164C2×C8C22×D312yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—32no—
168C4⋊C4—31no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4C22×D312yes—
176C4⋊C4C22×D311yescharacteristic
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C42—21no—
183C42—31no—
184C42—31no—
185C42—31no—
186C42C22×D311yescharacteristic
187C22×C3C22×D411yescharacteristic
188C2×D3—21no—
189C2×D3—21no—
190C2×D3—41no—
191C2×D3—41no—
192C2×D3—41no—
193C2×D3#13712yes—
194C3×C4—21no—
195C3×C4—41no—
196C3×C4—42no—
197C3×C4—41no—
198C3×C4AGL(1,ℤ/8)12yes—
199C3×C42−1+411yescharacteristic
200C3×C42−1+411yescharacteristic
201Dic3—21no—
202Dic3—21no—
203Dic3—21no—
204Dic3—42no—
205Dic3—41no—
206C23—31no—
207C23—61no—
208Q8—21no—
209Q8—21no—
210Q8—42no—
211Q8—124no—
212Q8—122no—
213D4—61no—
214D4—61no—
215D4—122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—124no—
229C2×C4—122no—
230C2×C4D3×D411yescharacteristic
231C2×C4D3×D411yescharacteristic
232C2×C4C23×D311yescharacteristic
233C8—22no—
234C8—62no—
235C2×C3#53611yescharacteristic
236C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
237C2×C3C2×#13711yescharacteristic
238D3—22no—
239D3—81no—
240C22—32no—
241C22—61no—
242C22—61no—
243C22—121no—
244C22—121no—
245C22—121no—
246C22C2×D3×D411yescharacteristic
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—61no—
252C4—61no—
253C4—61no—
254C4—122no—
255C4—121no—
256C4#91412yes—
257C4#100711yescharacteristic
258C4#101011yescharacteristic
259C3#323911yescharacteristic, complemented, Sylow
260C2—62no—
261C2—241no—
262C2#562811yescharacteristic
263C2C2×#91411yescharacteristic
264C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#475C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#475D3
hypocenterC3#3239
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. D3×C4⋊Q8C2semidirect
3. C3×#475C2semidirect
4. #5087C2semidirect
5. #5020C2semidirect
7. #4856C2semidirect
8. #4846C2semidirect
9. #4835C2semidirect
10. #4811C2semidirect
11. #4794C2semidirect
51. #475D3semidirect
259. C3#3239semidirect

Maximal subgroups

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

subgroupindex
D3×C4⋊Q82
#55802
C3×#4752
#50872
#50202
#48632
#48562
#48462
#48352
#48112
#47942
#32393