Finite Group Databasebeta

← #10591525

Subgroups — #10591525

Order 576.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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#64297C211yesmaximal, characteristic, complemented
2#64141C211yesmaximal, characteristic, complemented
3C3×#936C211yesmaximal, characteristic, complemented
4C3×#932C211yesmaximal, characteristic, complemented
5#63570C211yesmaximal, characteristic, complemented
6#63569C211yesmaximal, characteristic, complemented
7#63560C211yesmaximal, characteristic, complemented
8#5201—31nomaximal
9#5062—31nomaximal
10C2×C32×D4C2211yescharacteristic
11#3828C2211yescharacteristic
12C3×#235C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14#3804—22no—
15#3799—21no—
16#3797—21no—
17#3796—21no—
18D3×Dic6—21no—
19#3754—21no—
20#3753—21no—
21#3748C2211yescharacteristic
22C3×#217—22no—
23C3×#214—21no—
24C3×#213—21no—
25C3×C4×Dic3C2211yescharacteristic
26C3×#208C2211yescharacteristic
27#1010—31no—
28#1007—31no—
29C3×AGL(1,ℤ/8)—31no—
30C3×#124—31no—
31#936D311yescharacteristic, complemented
32#932—31no—
33#932—31no—
34#932—61no—
35#932D311yescharacteristic, complemented
36#837—31no—
37#836—31no—
38#825—31no—
39#824—31no—
40#805—31no—
41#805—31no—
42C23×C32—21no—
43C32×D4—21no—
44C32×D4—21no—
45C2×C32×C4C2311yescharacteristic
46#638—21no—
47C4⋊(C3⋊D3)D411yescharacteristic
48C4×C3⋊D3—21no—
49#634D411yescharacteristic
50C3×#67—21no—
51C2×C3×Dic3—21no—
52C2×C3×Dic3D412yes—
53C3×D12D411yescharacteristic
54C3×C4×D3—21no—
55C3×Dic6—21no—
56C3×Dic6—21no—
57C3×Dic6D411yescharacteristic
58#627—22no—
59#626—22no—
60#624—22no—
61D3×Dic3—22no—
62C3⋊(C3⋊C8)—21no—
63C3×C3⋊C8—21no—
64#454—91noSylow
65C3×#41—31no—
66C2×C3×Q8—31no—
67C2×C3×D4—21no—
68C2×C3×D4C2×D311yescharacteristic
69C2×C3×D4C2×D311yescharacteristic
70#239—61no—
71#239—61no—
72D3×Q8—61no—
73D3×Q8—61no—
74D3×Q8—61no—
75#237—61no—
76#237—62no—
77#235—31no—
78#235—31no—
79#235—61no—
80#235—61no—
81#235—62no—
82#235C2×D311yescharacteristic
83C2×Dic6—61no—
84C2×Dic6C2×D311yescharacteristic
85C3×SD16—61no—
86C3×D8—61no—
87C3×M16—31no—
88C3×C22⋊C4—62no—
89C3×C42—31no—
90#217—22no—
91#214—21no—
92#214—61no—
93#214—61no—
94#214—62no—
95#213—21no—
96#213—61no—
97#213—61no—
98#213—62no—
99C4×Dic3C2×D311yescharacteristic
100#208—31no—
101#208—31no—
102#208—61no—
103#208C2×D311yescharacteristic
104C22×C32—41no—
105C22×C32—42no—
106C22×C32C2×D411yescharacteristic
107C2×C3⋊D3—21no—
108C2×C3×D3—21no—
109C32×C4C2×D411yescharacteristic
110C32×C4C2×D411yescharacteristic
111C3⋊Dic3—21no—
112C3×Dic3—22no—
113C3×Dic3—21no—
114C3×Dic3—41no—
1152−1+4—91no—
116AGL(1,ℤ/8)—31no—
117AGL(1,ℤ/8)—91no—
118#124—31no—
119#104—91no—
120#104—91no—
121#101—91no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—41no—
125C3×Q8—31no—
126C3×Q8—61no—
127C3×Q8—61no—
128C3×D4—22no—
129C3×D4—21no—
130C3×D4—21no—
131C3×D4—21no—
132C3×D4—21no—
133C3×D4—31no—
134C3×D4—41no—
135C3×D4—61no—
136C2×C3×C4—21no—
137C2×C3×C4—32no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4C22×D311yescharacteristic
141C2×C3×C4C22×D311yescharacteristic
142#67—21no—
143#67—62no—
144#67—61no—
145#67—61no—
146#67—121no—
147C2×Dic3—21no—
148C2×Dic3—62no—
149C2×Dic3#6712yes—
150D12—31no—
151D12—31no—
152D12—61no—
153D12—62no—
154D12#6711yescharacteristic
155C4×D3—21no—
156C4×D3—62no—
157C4×D3—62no—
158C4×D3—62no—
159C4×D3—61no—
160C4×D3—61no—
161C4×D3—121no—
162Dic6—21no—
163Dic6—21no—
164Dic6—31no—
165Dic6—31no—
166Dic6—61no—
167Dic6—62no—
168Dic6—62no—
169Dic6#6711yescharacteristic
170C3×C8—61no—
171C3⋊C8—21no—
172C3⋊C8—62no—
173C3⋊C8—61no—
174C3⋊C8—61no—
175C2×C32—21no—
176C2×C32—81no—
177C2×C32#12011yescharacteristic
178C3⋊D3—41no—
179C3×D3—41no—
180#41—31no—
181#41—91no—
182#41—182no—
183#41—181no—
184#41—181no—
185C2×Q8—31no—
186C2×Q8—181no—
187C2×Q8—181no—
188C2×D4D3211yescharacteristic
189SD16—61no—
190SD16—181no—
191D8—61no—
192D8—181no—
193M16—31no—
194M16—91no—
195C22⋊C4—62no—
196C42—31no—
197C22×C3—21no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3—61no—
204C22×C3—82no—
205C22×C3D3×D411yescharacteristic
206C22×C3C2×#6711yescharacteristic
207C2×D3—21no—
208C2×D3—61no—
209C2×D3—61no—
210C2×D3—121no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—61no—
214C3×C4—62no—
215C3×C4—121no—
216C3×C4D3×D411yescharacteristic
217C3×C4D3×D411yescharacteristic
218C3×C4D3×D411yescharacteristic
219C3×C4C2×#6711yescharacteristic
220Dic3—22no—
221Dic3—21no—
222Dic3—41no—
223Dic3—61no—
224Dic3—61no—
225Dic3—121no—
226C32#45411yescharacteristic, complemented, Sylow
227C23—21no—
228Q8—31no—
229Q8—61no—
230Q8—61no—
231Q8—91no—
232Q8—182no—
233D4—21no—
234D4—21no—
235D4—31no—
236D4—61no—
237D4—91no—
238D4—182no—
239D4—181no—
240C2×C4—32no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—182no—
244C2×C4—182no—
245C2×C4—181no—
246C2×C4C2×D3211yescharacteristic
247C8—61no—
248C8—181no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—41no—
253C2×C3—82no—
254C2×C3—81no—
255C2×C3—81no—
256C2×C3—121no—
257C2×C3#93711yescharacteristic
258C2×C3#93711yescharacteristic
259D3—41no—
260D3—121no—
261D3—121no—
262D3—241no—
263C22—42no—
264C22—41no—
265C22—61no—
266C22—181no—
267C22D3×#6711yescharacteristic
268C4—62no—
269C4—61no—
270C4—121no—
271C4—181no—
272C4D3×#6711yescharacteristic
273C4#381111yescharacteristic
274C3—21no—
275C3#506211yescharacteristic, complemented
276C3#520111yescharacteristic, complemented
277C2—21no—
278C2—81no—
279C2—121no—
280C2—361no—
281C2#6398311yescharacteristic
subgroupG/H
centerC2#63983
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC2×C32#120
hypercenterC2×D4D32
hypocenterC32#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64297C2semidirect
2. #64141C2semidirect
3. C3×#936C2semidirect
4. C3×#932C2semidirect
5. #63570C2semidirect
6. #63569C2semidirect
7. #63560C2semidirect
31. #936D3semidirect
35. #932D3semidirect
226. C32#454semidirect
275. C3#5062semidirect
276. C3#5201semidirect

Maximal subgroups

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

subgroupindex
#642972
#641412
C3×#9362
C3×#9322
#635702
#635692
#635602
#52013
#50623