Finite Group Databasebeta

← #81073

Subgroups — #81073

Order 384.

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

normal subgroups only

This group has 430 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
1#5725C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#476C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5015C211yesmaximal, characteristic, complemented
6#4870C212yesmaximal, complemented
7#4865C212yesmaximal, complemented
8#4843C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4786C212yesmaximal, complemented
11#4782C212yesmaximal, complemented
12#3253—31nomaximal, Sylow
13C2×#239—21no—
14C2×#235—21no—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#947—21no—
18#940C2211yescharacteristic
19#938C2211yescharacteristic
20C2×#213C2212yes—
21C4×#67—21no—
22C2×D24C2212yes—
23C2×ΓL(1,𝔽25)C2212yes—
24#895—21no—
25#895C2211yescharacteristic
26#895C2212yes—
27#893—21no—
28#893C2212yes—
29#888C2211yescharacteristic
30#885—21no—
31#884—21no—
32#874C2211yescharacteristic
33C4×D12C2211yescharacteristic
34C42×D3C2211yescharacteristic
35C3×#106C2212yes—
36C3×C4⋊C8C2211yescharacteristic
37C3×#102C2212yes—
38#821C2212yes—
39Dic3⋊C8C2212yes—
40#809C2212yes—
41#809C2212yes—
42#808C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#532—31no—
46#476—31no—
47#476—32no—
48#476D311yescharacteristic, complemented
49#469—32no—
50#469—32no—
51#459—31no—
52#459—31no—
53#459—32no—
54#423—31no—
55C2×C3×Q8—21no—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—21no—
59C2×#67—21no—
60C2×#67—21no—
61#239—44no—
62#235—44no—
63C2×D12—21no—
64C2×D12—22no—
65C2×D12—21no—
66C2×D12—21no—
67C2×D12C2311yescharacteristic
68C2×D12C2312yes—
69C2×C4×D3—21no—
70C2×C4×D3—22no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×Dic6—21no—
75C2×C3×C8C2312yes—
76C3×C4⋊C4—21no—
77C3×C4⋊C4C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C42C2311yescharacteristic
81#217—21no—
82#213—42no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87C4⋊Dic3C2311yescharacteristic
88Dic3⋊C4—21no—
89C4×Dic3D411yescharacteristic
90C4×Dic3D411yescharacteristic
91C4×Dic3C2311yescharacteristic
92C2×C3⋊C8C2312yes—
93D24—42no—
94ΓL(1,𝔽25)—42no—
95C2×#41—61no—
96C2×#41—61no—
97C2×D8—32no—
98C2×D8—32no—
99C2×M16—32no—
100C4⋊Q8—31no—
101C4⋊Q8C2×D311yescharacteristic
102#127—31no—
103#127—31no—
104#124—61no—
105#124—61no—
106#121—31no—
107#121—31no—
108#121—32no—
109#121—61no—
110#121—61no—
111C4×D4—32no—
112C4×D4—31no—
113C4×D4—61no—
114C4×D4—61no—
115C4×D4C2×D311yescharacteristic
116C2×C42—31no—
117#106—32no—
118#106C2×D312yes—
119C4⋊C8—32no—
120C4⋊C8—31no—
121C4⋊C8C2×D311yescharacteristic
122#102—32no—
123#102—32no—
124#102—32no—
125#102C2×D312yes—
126C23×C3—21no—
127C22×D3—22no—
128C22×D3—21no—
129C22×D3C2×D411yescharacteristic
130C3×Q8—42no—
131C3×D4—21no—
132C3×D4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—42no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—42no—
140#67—42no—
141#67—42no—
142C2×Dic3—21no—
143C2×Dic3C2×D411yescharacteristic
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146D12—21no—
147D12—22no—
148D12—21no—
149D12—22no—
150D12—42no—
151D12—42no—
152D12—44no—
153D12—42no—
154C4×D3—22no—
155C4×D3—21no—
156C4×D3—21no—
157C4×D3—22no—
158C4×D3—42no—
159C4×D3—44no—
160Dic6—42no—
161C3×C8—22no—
162C3⋊C8—22no—
163#41—124no—
164#41—124no—
165C2×Q8—21no—
166C2×Q8—61no—
167C2×D4—32no—
168C2×D4—31no—
169C2×D4—61no—
170C2×D4—62no—
171C2×D4—61no—
172C2×D4—61no—
173C2×D4—61no—
174C2×D4—61no—
175C2×D4—61no—
176C2×D4C22×D311yescharacteristic
177C22×C4—21no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—62no—
182C22×C4—61no—
183D8—122no—
184D8—122no—
185M16—122no—
186C2×C8—32no—
187C2×C8C22×D312yes—
188C4⋊C4—21no—
189C4⋊C4—31no—
190C4⋊C4—61no—
191C4⋊C4C22×D311yescharacteristic
192C4⋊C4C22×D312yes—
193C22⋊C4—21no—
194C22⋊C4—62no—
195C22⋊C4—61no—
196C22⋊C4—62no—
197C22⋊C4—61no—
198C22⋊C4—61no—
199C42—31no—
200C42—31no—
201C42—31no—
202C42C22×D311yescharacteristic
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3C22×D411yescharacteristic
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—41no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—41no—
214C2×D3—42no—
215C2×D3—41no—
216C3×C4—21no—
217C3×C4—41no—
218C3×C4—42no—
219C3×C4AGL(1,ℤ/8)12yes—
220C3×C4H(5,𝔽2)11yescharacteristic
221C3×C42−1+411yescharacteristic
222Dic3—21no—
223Dic3—21no—
224Dic3—41no—
225Dic3AGL(1,ℤ/8)12yes—
226C23—21no—
227C23—31no—
228C23—61no—
229C23—62no—
230Q8—42no—
231Q8—122no—
232D4—21no—
233D4—21no—
234D4—62no—
235D4—61no—
236D4—61no—
237D4—62no—
238D4—122no—
239D4—122no—
240D4—122no—
241D4—124no—
242D4—122no—
243D4—122no—
244D4—122no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—42no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—124no—
257C2×C4—122no—
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—22no—
262C8—62no—
263C2×C3—81no—
264C2×C3#53711yescharacteristic
265C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
266C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
267D3—41no—
268D3—82no—
269D3—81no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—61no—
274C22—61no—
275C22—61no—
276C22—121no—
277C22—122no—
278C22—122no—
279C22—121no—
280C22—121no—
281C22—122no—
282C22C2×D3×D411yescharacteristic
283C4—21no—
284C4—32no—
285C4—41no—
286C4—42no—
287C4—61no—
288C4—61no—
289C4—121no—
290C4#91412yes—
291C4#100711yescharacteristic
292C4#100911yescharacteristic
293C3#325311yescharacteristic, complemented, Sylow
294C2—81no—
295C2—121no—
296C2—241no—
297C2—242no—
298C2#562911yescharacteristic
299C2C2×#91411yescharacteristic
300C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#476C2
socleC22×C3C22×D4
hypercenter#476D3
hypocenterC3#3253
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #5725C2semidirect
2. #5548C2semidirect
3. C3×#476C2semidirect
4. #5084C2semidirect
5. #5015C2semidirect
6. #4870C2semidirect
7. #4865C2semidirect
8. #4843C2semidirect
9. #4837C2semidirect
10. #4786C2semidirect
11. #4782C2semidirect
48. #476D3semidirect
293. C3#3253semidirect

Maximal subgroups

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

subgroupindex
#57252
#55482
C3×#4762
#50842
#50152
#48702
#48652
#48432
#48372
#47862
#47822
#32533