Finite Group Databasebeta

← #81718

Subgroups — #81718

Order 384.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#5716C211yesmaximal, characteristic, complemented
2#5691C211yesmaximal, characteristic, complemented
3C3×#490C211yesmaximal, characteristic, complemented
4#5074C211yesmaximal, characteristic, complemented
5#5069C211yesmaximal, characteristic, complemented
6#4787C212yesmaximal, complemented
7#4783C212yesmaximal, complemented
8#4781C212yesmaximal, complemented
9#4767C212yesmaximal, complemented
10#4715C211yesmaximal, characteristic, complemented
11#4705C211yesmaximal, characteristic, complemented
12#3219—31nomaximal, Sylow
13C2×#237C2212yes—
14C3×#127C2211yescharacteristic
15C3×#125C2211yescharacteristic
16#938—21no—
17#938—21no—
18#936C2212yes—
19D4×Dic3—21no—
20D4×Dic3—21no—
21C2×#214C2212yes—
22C2×#213C2212yes—
23C2×Dic12C2212yes—
24C2×#204C2212yes—
25#898—21no—
26#897C2211yescharacteristic
27#895C2211yescharacteristic
28#894—21no—
29#893—21no—
30#893C2211yescharacteristic
31#892—21no—
32#887C2211yescharacteristic
33#875C2211yescharacteristic
34C42×D3C2211yescharacteristic
35#869C2211yescharacteristic
36C3×#102C2212yes—
37C3×#102C2212yes—
38C3×#97C2211yescharacteristic
39(C2×D3)⋊C8C2212yes—
40Dic3⋊C8C2212yes—
41#810C2212yes—
42#810C2212yes—
43#803C2211yescharacteristic
44#532—31no—
45#529—31no—
46#497—31no—
47#496—31no—
48#491—31no—
49#490D311yescharacteristic, complemented
50#464—32no—
51#463—32no—
52#452—32no—
53#449—32no—
54#432—31no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×#67—22no—
59C22×Dic3—22no—
60#237—42no—
61#237—42no—
62C2×D12—21no—
63C2×D12C2311yescharacteristic
64C2×C4×D3—21no—
65C2×C4×D3D411yescharacteristic
66C2×C4×D3D411yescharacteristic
67C2×C4×D3C2311yescharacteristic
68C2×Dic6C2312yes—
69C2×Dic6C2311yescharacteristic
70C2×C3×C8C2312yes—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—21no—
73C3×C4⋊C4C2311yescharacteristic
74C3×C4⋊C4C2311yescharacteristic
75C3×C42C2311yescharacteristic
76#217—22no—
77#217—22no—
78#214—42no—
79#213—42no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3—21no—
88Dic3⋊C4—21no—
89C4×Dic3—21no—
90C4×Dic3D411yescharacteristic
91C4×Dic3D411yescharacteristic
92C4×Dic3C2311yescharacteristic
93C2×C3⋊C8C2312yes—
94Dic12—42no—
95#204—42no—
96C2×#41—32no—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16—32no—
100C2×D8—32no—
101C4⋊Q8—31no—
102#127C2×D311yescharacteristic
103#126—61no—
104#125C2×D311yescharacteristic
105#124—32no—
106#124—31no—
107#123—61no—
108C22⋊Q8—31no—
109#121—31no—
110#121—61no—
111#121—61no—
112C4×Q8—31no—
113C4×D4—31no—
114C4×D4—61no—
115C4×D4—61no—
116C4×D4—61no—
117#117—61no—
118C2×C42—31no—
119C4⋊C8—32no—
120#103—32no—
121#103—32no—
122#102C2×D312yes—
123#102C2×D312yes—
124C22⋊C8—32no—
125#97—31no—
126#97C2×D311yescharacteristic
127C23×C3—22no—
128C22×D3—21no—
129C22×D3C2×D411yescharacteristic
130C3×D4—22no—
131C3×D4—22no—
132C3×D4—42no—
133C3×D4—42no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—44no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—44no—
143C2×Dic3C2×D411yescharacteristic
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146D12—22no—
147D12—42no—
148C4×D3—22no—
149C4×D3—41no—
150C4×D3—41no—
151C4×D3—42no—
152Dic6—22no—
153Dic6—22no—
154Dic6—22no—
155C3×C8—22no—
156C3⋊C8—22no—
157#41—122no—
158#41—122no—
159C2×Q8—32no—
160C2×Q8—31no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—31no—
164C2×D4—61no—
165C2×D4—62no—
166C2×D4C22×D312yes—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—61no—
171C22×C4—62no—
172Q16—122no—
173SD16—122no—
174SD16—122no—
175D8—122no—
176C2×C8—32no—
177C2×C8C22×D312yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4C22×D311yescharacteristic
185C4⋊C4C22×D311yescharacteristic
186C22⋊C4—61no—
187C22⋊C4—61no—
188C22⋊C4—61no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C42—31no—
194C42—31no—
195C42—31no—
196C42—61no—
197C42C22×D311yescharacteristic
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3C22×D411yescharacteristic
202C2×D3—21no—
203C2×D3—21no—
204C2×D3—21no—
205C2×D3—41no—
206C2×D3—41no—
207C2×D3—41no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4H(5,𝔽2)12yes—
213Dic3—21no—
214Dic3—21no—
215Dic3—21no—
216Dic3—41no—
217Dic3—42no—
218C23—22no—
219C23—31no—
220C23—61no—
221Q8—62no—
222Q8—62no—
223Q8—62no—
224D4—22no—
225D4—22no—
226D4—42no—
227D4—42no—
228D4—62no—
229D4—122no—
230D4—124no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—62no—
237C2×C4—61no—
238C2×C4—62no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—122no—
242C2×C4—124no—
243C2×C4D3×D411yescharacteristic
244C2×C4D3×D411yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—22no—
247C8—62no—
248C2×C3—82no—
249C2×C3#53511yescharacteristic
250C2×C3#57511yescharacteristic
251C2×C3#57511yescharacteristic
252D3—41no—
253D3—81no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—61no—
258C22—61no—
259C22—61no—
260C22—121no—
261C22—121no—
262C22—121no—
263C22C2×D3×D411yescharacteristic
264C4—21no—
265C4—21no—
266C4—41no—
267C4—41no—
268C4—61no—
269C4—61no—
270C4—61no—
271C4—122no—
272C4—121no—
273C4#100912yes—
274C3#321911yescharacteristic, complemented, Sylow
275C2—82no—
276C2—121no—
277C2—241no—
278C2#567411yescharacteristic
279C2#577311yescharacteristic
280C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#490C2
socleC22×C3C22×D4
hypercenter#490D3
hypocenterC3#3219
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. #5691C2semidirect
3. C3×#490C2semidirect
4. #5074C2semidirect
5. #5069C2semidirect
6. #4787C2semidirect
7. #4783C2semidirect
8. #4781C2semidirect
9. #4767C2semidirect
10. #4715C2semidirect
11. #4705C2semidirect
49. #490D3semidirect
274. C3#3219semidirect

Maximal subgroups

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

subgroupindex
#57162
#56912
C3×#4902
#50742
#50692
#47872
#47832
#47812
#47672
#47152
#47052
#32193