Finite Group Databasebeta

← #10951994

Subgroups — #10951994

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82019C211yesmaximal, characteristic, complemented
2#82015C211yesmaximal, characteristic, complemented
3#80940C212yesmaximal, complemented
4C3×#2221C211yesmaximal, characteristic, complemented
5#71465C211yesmaximal, characteristic, complemented
6#71452C211yesmaximal, characteristic, complemented
7#71095C212yesmaximal, complemented
8#69362C211yesmaximal, characteristic, complemented
9#69357C211yesmaximal, characteristic, complemented
10#69262C212yesmaximal, complemented
11#69259C212yesmaximal, complemented
12#33859—31nomaximal, Sylow
13#5775—22no—
14#5757C2212yes—
15#5753C2212yescomplemented
16C2×C3×D16C2211yescharacteristic
17C3×#504C2211yescharacteristic, complemented
18C3×#472C2212yescomplemented
19#5201—22no—
20#5155C2212yescomplemented
21C2×#827C2212yescomplemented
22#5147C2212yes—
23#5144C2211yescharacteristic
24#4912—21no—
25#4911—22no—
26D3×D16—21no—
27#4909—22no—
28#4909C2212yescomplemented
29C2×Dic24—21no—
30C2×Dic24C2211yescharacteristic, complemented
31#4904—21no—
32#4904C2211yescharacteristic
33#4901C2211yescharacteristic
34#4896C2212yes—
35#4895C2212yescomplemented
36#4748—22no—
37#4746—22no—
38C3×#368C2211yescharacteristic, complemented
39C3×#361C2212yescomplemented
40#4561C2211yescharacteristic, complemented
41#4515C2211yescharacteristic
42#4511C2211yescharacteristic, complemented
43Dic3×C16C2211yescharacteristic, complemented
44#4497C2212yescomplemented
45#4496C2212yescomplemented
46#3417—31no—
47#3415—31no—
48#3290—32no—
49#2249—31no—
50#2222—31no—
51#2221D311yescharacteristic, complemented
52#2212—32no—
53#2194—32no—
54#2193—32no—
55#2178—31no—
56#2170—31no—
57#1010—22no—
58#1004—22no—
59C3×AGL(1,ℤ/8)—22no—
60C2×C3×D8C2312yes—
61C3×#131C2311yescharacteristic
62#950D411yescharacteristic
63#936—22no—
64C2×#214—22no—
65#932—22no—
66#915—42no—
67D3×SD16—42no—
68#912—22no—
69#912—42no—
70#911—42no—
71#911—42no—
72D3×D8—22no—
73#909—22no—
74#909—22no—
75#909—42no—
76#907—22no—
77C2×Dic12—22no—
78C2×Dic12C2312yes—
79#904—22no—
80#904C2312yes—
81#901C2311yescharacteristic
82C3×D16—21no—
83C3×D16—21no—
84C3×M32—21no—
85C3×M32C2311yescharacteristic
86C2×C3×C16—21no—
87C2×C3×C16C2311yescharacteristic
88C3×#108C2312yes—
89C3×#100—22no—
90#837D411yescharacteristic
91#827—22no—
92#827—42no—
93#826—22no—
94#825—22no—
95#825D411yescharacteristic
96#824—22no—
97#822D411yescharacteristic
98C8×Dic3C2311yescharacteristic
99#812C2311yescharacteristic
100C2×C3⋊C16C2311yescharacteristic
101Dic24—22no—
102Dic24—21no—
103Dic24—21no—
104Dic24D411yescharacteristic
105#800—22no—
106#800D412yes—
107D48D411yescharacteristic
108#798—21no—
109D3×C16—21no—
110#578—32no—
111#577—62no—
112#576—32no—
113#510—32no—
114#510—62no—
115#509—32no—
116#509—62no—
117#508—31no—
118#508—61no—
119#508—61no—
120C2×Q32—31no—
121C2×Q32—61no—
122C2×SD32—32no—
123C2×D16—61no—
124C2×D16C2×D311yescharacteristic
125#504—31no—
126#504C2×D311yescharacteristic, complemented
127#495—32no—
128#473—32no—
129#472C2×D312yescomplemented
130#471—62no—
131#455—62no—
132#454—62no—
133#443—31no—
134#443—32no—
135#368C2×D311yescharacteristic, complemented
136#364—31no—
137#362—32no—
138#361C2×D312yescomplemented
139#359—32no—
140#350—31no—
141#350—31no—
142C4×C16—31no—
143C3×#41C2×D411yescharacteristic
144C2×C3×D4—22no—
145C2×#67—82no—
146D3×Q8—42no—
147#237—42no—
148#237—42no—
149#237—42no—
150#237—82no—
151D3×D4—42no—
152D3×D4—42no—
153#235—22no—
154#235—42no—
155#235C2×D411yescharacteristic
156C2×Dic6—22no—
157C2×Dic6—42no—
158C3×SD16—42no—
159C3×D8—22no—
160C3×D8—22no—
161C3×D8—42no—
162C3×M16—21no—
163C3×M16—22no—
164C3×M16C2×D411yescharacteristic
165C2×C3×C8—21no—
166C2×C3×C8C2411yescharacteristic
167#217—82no—
168#216D811yescharacteristic
169#215D811yescharacteristic
170#214—42no—
171#214—42no—
172#214—42no—
173#214D811yescharacteristic
174#213—42no—
175#213D811yescharacteristic
176C4×Dic3C2×D411yescharacteristic
177#208C2×D411yescharacteristic
178C2×C3⋊C8C2×D411yescharacteristic
179Dic12—22no—
180Dic12—22no—
181Dic12—42no—
182Dic12—42no—
183Dic12C2×D412yes—
184D24C2×D412yes—
185#204—22no—
186#204—42no—
187#204—42no—
188ΓL(1,𝔽25)—21no—
189ΓL(1,𝔽25)—42no—
190D3×C8—21no—
191D3×C8—42no—
192C3×C16—21no—
193C3×C16C2×D411yescharacteristic
194C3×C16C2×D411yescharacteristic
195C3⋊C16—21no—
196C3⋊C16—21no—
1972−1+4—62no—
198H(5,𝔽2)—62no—
199#137—62no—
200#137—62no—
201#137—122no—
202#137—122no—
203AGL(1,ℤ/8)—22no—
204AGL(1,ℤ/8)—62no—
205AGL(1,ℤ/8)—122no—
206AGL(1,ℤ/8)—122no—
207#135—31no—
208#135—32no—
209#135—62no—
210#135—62no—
211#135—122no—
212C2×Q16—32no—
213C2×Q16—62no—
214C2×SD16—62no—
215C2×SD16—122no—
216C2×D8—62no—
217C2×D8C22×D312yes—
218#131—31no—
219#131—62no—
220#131C22×D311yescharacteristic
221#124—62no—
222Q32—31no—
223Q32—62no—
224Q32—61no—
225Q32—61no—
226SD32—32no—
227SD32—62no—
228SD32—62no—
229SD32—122no—
230D16—21no—
231D16—21no—
232D16—31no—
233D16—62no—
234M32—21no—
235M32—31no—
236M32—61no—
237M32C22×D311yescharacteristic
238C2×C16—21no—
239C2×C16—31no—
240C2×C16—61no—
241C2×C16C22×D311yescharacteristic
242#108—31no—
243#108C22×D312yes—
244#104—31no—
245#104—31no—
246#104—62no—
247#101—62no—
248#100—22no—
249C4×C8—31no—
250C23×C3—42no—
251C22×D3—82no—
252C3×Q8C2×D811yescharacteristic
253C3×D4—21no—
254C3×D4—42no—
255C3×D4—42no—
256C3×D4C2×D811yescharacteristic
257C2×C3×C4—21no—
258C2×C3×C4C22×D411yescharacteristic
259#67—21no—
260#67—42no—
261#67—82no—
262#67—82no—
263#67—82no—
264C2×Dic3—21no—
265C2×Dic3—42no—
266C2×Dic3—82no—
267C2×Dic3—82no—
268D12—22no—
269D12C2×D811yescharacteristic
270C4×D3—21no—
271C4×D3—42no—
272C4×D3—82no—
273Dic6—22no—
274Dic6—42no—
275Dic6—42no—
276Dic6—82no—
277Dic6C2×D811yescharacteristic
278C3×C8—21no—
279C3×C8—42no—
280C3×C8C22×D411yescharacteristic
281C3×C8H(5,𝔽2)11yescharacteristic
282C3⋊C8—21no—
283C3⋊C8C2×D811yescharacteristic
284C3⋊C8C2×D811yescharacteristic
285#41—31no—
286#41—62no—
287#41—122no—
288#41—122no—
289#41—122no—
290#41—122no—
291#41—242no—
292#41D3×D411yescharacteristic
293C2×Q8—62no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×D4—22no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—242no—
300Q16—31no—
301Q16—32no—
302Q16—62no—
303Q16—62no—
304Q16—122no—
305Q16—122no—
306SD16—31no—
307SD16—31no—
308SD16—42no—
309SD16—62no—
310SD16—122no—
311SD16—122no—
312SD16—122no—
313SD16—122no—
314SD16—122no—
315D8—22no—
316D8—22no—
317D8—31no—
318D8—32no—
319D8—42no—
320D8—122no—
321M16—21no—
322M16—22no—
323M16—31no—
324M16—61no—
325M16—122no—
326M16D3×D411yescharacteristic
327C2×C8—21no—
328C2×C8—31no—
329C2×C8—61no—
330C2×C8—122no—
331C2×C8C23×D311yescharacteristic
332C22⋊C4—242no—
333C42—31no—
334C16—21no—
335C16—61no—
336C16—61no—
337C16D3×D411yescharacteristic
338C16D3×D411yescharacteristic
339C22×C3—21no—
340C22×C3—82no—
341C22×C3—162no—
342C22×C3#57611yescharacteristic
343C2×D3—21no—
344C2×D3—42no—
345C2×D3—322no—
346C3×C4—21no—
347C3×C4D4211yescharacteristic
348C3×C4C22×D811yescharacteristic
349Dic3—21no—
350Dic3—41no—
351Dic3—42no—
352Dic3—82no—
353C23—42no—
354C23—242no—
355Q8—31no—
356Q8—62no—
357Q8—122no—
358Q8—122no—
359Q8—242no—
360Q8D3×D811yescharacteristic
361D4—21no—
362D4—31no—
363D4—42no—
364D4—42no—
365D4—61no—
366D4—62no—
367D4—122no—
368D4—242no—
369D4—242no—
370D4—242no—
371D4D3×D811yescharacteristic
372C2×C4—21no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—242no—
378C2×C4—242no—
379C2×C4—242no—
380C2×C4C2×D3×D411yescharacteristic
381C8—21no—
382C8—31no—
383C8—31no—
384C8—42no—
385C8—61no—
386C8C2×D3×D411yescharacteristic
387C8#100911yescharacteristic
388C2×C3—21no—
389C2×C3—41no—
390C2×C3—162no—
391C2×C3D4×D811yescharacteristic
392D3—41no—
393D3—82no—
394C22—21no—
395C22—61no—
396C22—82no—
397C22—122no—
398C22—162no—
399C22—962no—
400C22#576911yescharacteristic
401C4—21no—
402C4—61no—
403C4—121no—
404C4—122no—
405C4—242no—
406C4#560911yescharacteristic
407C4C2×D3×D811yescharacteristic
408C3#3385911yescharacteristic, complemented, Sylow
409C2—21no—
410C2—41no—
411C2—121no—
412C2—162no—
413C2—242no—
414C2#8072911yescharacteristic
subgroupG/H
centerC2#80729
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2221C2
socleC2×C3D4×D8
hypercenter#2221D3
hypocenterC3#33859
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #2221

Complements

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

NH ≅ G/Ntype
1. #82019C2semidirect
2. #82015C2semidirect
3. #80940C2semidirect
4. C3×#2221C2semidirect
5. #71465C2semidirect
6. #71452C2semidirect
7. #71095C2semidirect
8. #69362C2semidirect
9. #69357C2semidirect
10. #69262C2semidirect
11. #69259C2semidirect
15. #5753C22semidirect
17. C3×#504C22semidirect
18. C3×#472C22semidirect
20. #5155C22semidirect
21. C2×#827C22semidirect
28. #4909C22semidirect
30. C2×Dic24C22semidirect
35. #4895C22semidirect
38. C3×#368C22semidirect
39. C3×#361C22semidirect
40. #4561C22semidirect
42. #4511C22semidirect
43. Dic3×C16C22semidirect
44. #4497C22semidirect
45. #4496C22semidirect
51. #2221D3semidirect
126. #504C2×D3semidirect
129. #472C2×D3semidirect
135. #368C2×D3semidirect
138. #361C2×D3semidirect
408. C3#33859semidirect

Maximal subgroups

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

subgroupindex
#820192
#820152
#809402
C3×#22212
#714652
#714522
#710952
#693622
#693572
#692622
#692592
#338593