Finite Group Databasebeta

← #11715796

Subgroups — #11715796

Order 768.

5274 conjugacy classes of subgroups; only the 509 normal subgroups are shown.

normal subgroups only

This group has 5274 conjugacy classes of subgroups — too many to draw. Only its 509 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#87536C211yesmaximal, characteristic, complemented
2#87498C2110yesmaximal, complemented
3#87497C2110yesmaximal, complemented
4#87493C215yesmaximal, complemented
5C2×#5925C215yesmaximal, complemented
6#63091—31nomaximal, Sylow
7#5968—41nomaximal
8#5945C22110yescomplemented
9C2×#994C2215yescomplemented
10#5927C22110yescomplemented
11#5926C22130yescomplemented
12#5925C22110yescomplemented
13#5925C22120yescomplemented
14#5924C22110yescomplemented
15#5923C22130yescomplemented
16C2×#986C22115yescomplemented
17C2×#985C22115yescomplemented
18H(7,𝔽2)—31no—
19H(7,𝔽2)D311yescharacteristic, complemented
20#3585—310no—
21#3585—35no—
22#3585—310no—
23#3585—35no—
24#3584—35no—
25#3584—310no—
26C2×#576—310no—
27C2×#576—35no—
28#3573—31no—
29C3×2−1+4—41no—
30#1009—410no—
31D3×#41—410no—
32#1007—45no—
33C2×#239—45no—
342−1+4⋊C3C23110yescomplemented
35#994C23110yescomplemented
36C2×#41⋊C3C23115yescomplemented
37#986C23130yescomplemented
38#985C23110yescomplemented
39#985C23130yescomplemented
40#984C23110yescomplemented
41#983C23120yescomplemented
42C2×GL(2,𝔽3)C23115yescomplemented
43C2×#226C2315yescomplemented
44#585—31no—
45#585—35no—
46#585—310no—
47#585—35no—
48#585—65no—
49#585—61no—
50#585C2×D3110yescomplemented
51C2×H(5,𝔽2)—310no—
52C2×H(5,𝔽2)—310no—
53C2×H(5,𝔽2)—35no—
54C2×H(5,𝔽2)—610no—
55C2×H(5,𝔽2)C2×D315yescomplemented
56#578—310no—
57#578—310no—
58#577—330no—
59#577—320no—
60#577—320no—
61#577—310no—
62#577—330no—
63#577—310no—
64#576—330no—
65#576—320no—
66#576—360no—
67#576—320no—
68#576—320no—
69#576—310no—
70#576—320no—
71#575—330no—
72#575—330no—
73#575—330no—
74C2×AGL(1,ℤ/8)—315no—
75C2×AGL(1,ℤ/8)—330no—
76C2×AGL(1,ℤ/8)—330no—
77C2×AGL(1,ℤ/8)—315no—
78C2×#135—330no—
79C2×#135—315no—
80C22×D8—315no—
81#568—310no—
82#568—35no—
83C2×#131—310no—
84C2×#131—35no—
85C3×#41—410no—
86C2×C3×Q8—45no—
87#239—410no—
88#239—420no—
89D3×Q8—410no—
90#237—410no—
91D3×D4—430no—
92#235—430no—
93C2×D12—415no—
94C2×C4×D3—415no—
95#41⋊C3C24110yes—
96C2×SL(2,𝔽3)C2415yes—
97GL(2,𝔽3)C24110yes—
98#226C2411yescharacteristic
99#226C2415yes—
1002−1+4—35no—
1012−1+4—310no—
1022−1+4—310no—
1032−1+4—310no—
1042−1+4—610no—
1052−1+4—65no—
1062−1+4S411yescharacteristic
1072−1+4C22×D3110yes—
108H(5,𝔽2)—310no—
109H(5,𝔽2)—330no—
110H(5,𝔽2)—320no—
111H(5,𝔽2)—320no—
112H(5,𝔽2)—310no—
113H(5,𝔽2)—330no—
114H(5,𝔽2)—615no—
115H(5,𝔽2)—630no—
116H(5,𝔽2)—615no—
117H(5,𝔽2)—620no—
118H(5,𝔽2)—630no—
119H(5,𝔽2)—610no—
120H(5,𝔽2)C22×D3110yes—
121C2×#41—310no—
122C2×#41—35no—
123C2×#41—330no—
124C2×#41—330no—
125C2×#41—315no—
126C2×#41—610no—
127C2×#41—610no—
128C2×#41—620no—
129C2×#41—630no—
130C2×#41—610no—
131C2×#41—65no—
132C2×#41C22×D3115yes—
133C22×D4—315no—
134C22×D4—315no—
135C22×D4—610no—
136C22×D4—630no—
137C22×D4—615no—
138#137—310no—
139#137—330no—
140#137—330no—
141#137—320no—
142#137—310no—
143#137—320no—
144AGL(1,ℤ/8)—330no—
145AGL(1,ℤ/8)—360no—
146AGL(1,ℤ/8)—330no—
147AGL(1,ℤ/8)—360no—
148AGL(1,ℤ/8)—330no—
149AGL(1,ℤ/8)—360no—
150AGL(1,ℤ/8)—360no—
151AGL(1,ℤ/8)—330no—
152#135—330no—
153#135—320no—
154#135—360no—
155#135—330no—
156#135—360no—
157#135—310no—
158#135—330no—
159C2×Q16—310no—
160C2×Q16—35no—
161C2×SD16—330no—
162C2×SD16—315no—
163C2×SD16—330no—
164C2×SD16—315no—
165C2×D8—330no—
166C2×D8—315no—
167C2×D8—360no—
168C2×D8—330no—
169#131—310no—
170#131—310no—
171#131—310no—
172#131—320no—
173#131—330no—
174C2×M16—330no—
175C2×M16—315no—
176C22×C8—315no—
177C22×D3—415no—
178C3×Q8—410no—
179C3×D4—410no—
180C2×C3×C4—415no—
181#67—420no—
182C2×Dic3—45no—
183D12—430no—
184C4×D3—410no—
185C4×D3—430no—
186Dic6—410no—
187SL(2,𝔽3)C2511yescharacteristic
188C24—65no—
189C24—615no—
190#41—310no—
191#41—320no—
192#41—310no—
193#41—330no—
194#41—330no—
195#41—320no—
196#41—320no—
197#41—330no—
198#41—310no—
199#41—630no—
200#41—630no—
201#41—630no—
202#41—620no—
203#41—630no—
204#41—630no—
205#41—620no—
206#41—610no—
207#41—610no—
208#41—630no—
209#41C2×S4110yes—
210#41C23×D3110yes—
211C2×Q8—310no—
212C2×Q8—310no—
213C2×Q8—35no—
214C2×Q8—310no—
215C2×Q8—610no—
216C2×Q8—610no—
217C2×Q8C2×S415yes—
218C2×Q8C23×D315yes—
219C2×D4—330no—
220C2×D4—330no—
221C2×D4—315no—
222C2×D4—315no—
223C2×D4—330no—
224C2×D4—315no—
225C2×D4—630no—
226C2×D4—630no—
227C2×D4—630no—
228C2×D4—660no—
229C2×D4—660no—
230C2×D4—630no—
231C2×D4—630no—
232C2×D4—615no—
233C2×D4—630no—
234C2×D4—615no—
235C22×C4—315no—
236C22×C4—610no—
237C22×C4—630no—
238C22×C4—615no—
239Q16—35no—
240Q16—320no—
241Q16—35no—
242Q16—31no—
243Q16—35no—
244SD16—320no—
245SD16—330no—
246SD16—310no—
247SD16—320no—
248SD16—330no—
249SD16—310no—
250D8—310no—
251D8—360no—
252D8—330no—
253M16—320no—
254M16—310no—
255M16—330no—
256C2×C8—35no—
257C2×C8—310no—
258C2×C8—315no—
259C2×C8—330no—
260C22×C3—45no—
261C2×D3—410no—
262C2×D3—1615no—
263C3×C4—410no—
264Dic3—45no—
265Dic3—41no—
266C23—315no—
267C23—610no—
268C23—630no—
269C23—615no—
270C23—485no—
271C23—4815no—
272Q8—35no—
273Q8—310no—
274Q8—35no—
275Q8—31no—
276Q8—65no—
277Q8—610no—
278Q8C22×S4110yes—
279Q8C24×D311yescharacteristic
280D4—310no—
281D4—320no—
282D4—310no—
283D4—330no—
284D4—615no—
285D4—615no—
286D4—630no—
287D4—610no—
288D4—620no—
289D4—630no—
290D4C22×S4110yes—
291C2×C4—35no—
292C2×C4—310no—
293C2×C4—330no—
294C2×C4—610no—
295C2×C4—620no—
296C2×C4—610no—
297C2×C4—65no—
298C2×C4—630no—
299C2×C4—610no—
300C2×C4C22×S4115yes—
301C8—31no—
302C8—35no—
303C8—310no—
304C2×C3—41no—
305C2×C3—85no—
306D3—810no—
307C22—310no—
308C22—610no—
309C22—1215no—
310C22—2415no—
311C22—2410no—
312C22—2430no—
313C22C23×S415yes—
314C4—31no—
315C4—35no—
316C4—65no—
317C4—61no—
318C4C23×S4110yes—
319C3—41noSylow
320C2—25no—
321C2—610no—
322C2—1210no—
323C2C24×S411yescharacteristic
subgroupG/H
center, Frattini, socleC2C24×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C25
FittingH(7,𝔽2)D3
hypercenter2−1+4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #87536C2semidirect
2. #87498C2semidirect
3. #87497C2semidirect
4. #87493C2semidirect
5. C2×#5925C2semidirect
8. #5945C22semidirect
9. C2×#994C22semidirect
10. #5927C22semidirect
11. #5926C22semidirect
12. #5925C22semidirect
13. #5925C22semidirect
14. #5924C22semidirect
15. #5923C22semidirect
16. C2×#986C22semidirect
17. C2×#985C22semidirect
19. H(7,𝔽2)D3semidirect
34. 2−1+4⋊C3C23semidirect
35. #994C23semidirect
36. C2×#41⋊C3C23semidirect
37. #986C23semidirect
38. #985C23semidirect
39. #985C23semidirect
40. #984C23semidirect
41. #983C23semidirect
42. C2×GL(2,𝔽3)C23semidirect
43. C2×#226C23semidirect
50. #585C2×D3semidirect
55. C2×H(5,𝔽2)C2×D3semidirect

Maximal subgroups

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

subgroupindex
#875362
#874982
#874972
#874932
C2×#59252
#630913
#59684