Finite Group Databasebeta

← #10951980

Subgroups — #10951980

Order 768.

668 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82019C211yesmaximal, characteristic, complemented
2#82013C211yesmaximal, characteristic, complemented
3#80941C212yesmaximal, complemented
4C3×#2221C211yesmaximal, characteristic, complemented
5#71464C211yesmaximal, characteristic, complemented
6#71452C211yesmaximal, characteristic, complemented
7#71094C212yesmaximal, complemented
8#69361C211yesmaximal, characteristic, complemented
9#69357C211yesmaximal, characteristic, complemented
10#69263C212yesmaximal, complemented
11#69258C212yesmaximal, complemented
12#33858—31nomaximal, Sylow
13#5774—22no—
14#5757C2212yescomplemented
15#5752C2212yescomplemented
16C2×C3×D16C2211yescharacteristic, complemented
17C3×#504C2211yescharacteristic, complemented
18C3×#472C2212yescomplemented
19#5198—22no—
20#5151C2212yescomplemented
21#5147C2212yescomplemented
22C2×#826C2212yescomplemented
23#5144C2211yescharacteristic
24#4912—21no—
25#4911—22no—
26D3×D16—21no—
27#4908—22no—
28#4908C2212yescomplemented
29#4904—21no—
30#4904C2211yescharacteristic
31C2×D48—21no—
32C2×D48C2211yescharacteristic, complemented
33#4901C2211yescharacteristic, complemented
34#4897C2212yescomplemented
35#4895C2212yescomplemented
36#4749—22no—
37#4747—22no—
38C3×#368C2211yescharacteristic, complemented
39C3×#361C2212yescomplemented
40#4561C2211yescharacteristic, complemented
41#4515C2211yescharacteristic, complemented
42#4511C2211yescharacteristic, complemented
43Dic3×C16C2211yescharacteristic, complemented
44#4497C2212yescomplemented
45#4495C2212yescomplemented
46#3415—31no—
47#3415—31no—
48#3292—32no—
49#2246—31no—
50#2221—31no—
51#2221D311yescharacteristic, complemented
52#2213—32no—
53#2192—32no—
54#2192—32no—
55#2178—31no—
56#2170—31no—
57#1009—22no—
58#1004—22no—
59C3×AGL(1,ℤ/8)—22no—
60C2×C3×D8C2312yes—
61C3×#131C2311yescharacteristic
62#950D411yescharacteristic
63#940—22no—
64#932—22no—
65C2×#213—22no—
66#916—42no—
67#914—42no—
68#912—22no—
69#911—42no—
70#911—42no—
71D3×D8—22no—
72D3×D8—42no—
73#908—22no—
74#908—22no—
75#908—42no—
76#907—22no—
77#904—22no—
78#904C2312yes—
79C2×D24—22no—
80C2×D24C2312yes—
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#826—22no—
93#826—42no—
94#825—22no—
95#825D411yescharacteristic
96#823—22no—
97#822D411yescharacteristic
98C8×Dic3C2311yescharacteristic
99#812C2311yescharacteristic
100C2×C3⋊C16C2311yescharacteristic
101Dic24D411yescharacteristic, complemented
102#800—22no—
103#800D412yescomplemented
104D48—22no—
105D48—21no—
106D48—21no—
107D48D411yescharacteristic, complemented
108#798—21no—
109D3×C16—21no—
110#576—32no—
111#576—32no—
112#576—62no—
113#509—32no—
114#509—32no—
115#509—62no—
116#509—62no—
117#508—31no—
118#508—61no—
119#508—61no—
120C2×D16—32no—
121C2×D16—31no—
122C2×D16—61no—
123C2×D16—61no—
124C2×D16C2×D311yescharacteristic, complemented
125#504—31no—
126#504C2×D311yescharacteristic, complemented
127#493—32no—
128#472—32no—
129#472—62no—
130#472C2×D312yescomplemented
131#453—62no—
132#453—62no—
133#443—31no—
134#443—32no—
135#368C2×D311yescharacteristic, complemented
136#364—31no—
137#361—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—
146C2×#67—82no—
147#239—42no—
148#237—42no—
149#237—42no—
150D3×D4—42no—
151D3×D4—42no—
152D3×D4—42no—
153D3×D4—82no—
154#235—22no—
155#235—42no—
156#235C2×D411yescharacteristic
157C2×D12—22no—
158C2×D12—42no—
159C3×SD16—42no—
160C3×D8—22no—
161C3×D8—22no—
162C3×D8—42no—
163C3×M16—21no—
164C3×M16—22no—
165C3×M16C2×D411yescharacteristic
166C2×C3×C8—21no—
167C2×C3×C8C2411yescharacteristic
168#216D811yescharacteristic
169#215D811yescharacteristic
170#214—42no—
171#214D811yescharacteristic
172#213—42no—
173#213—42no—
174#213—42no—
175#213D811yescharacteristic
176C4×Dic3C2×D411yescharacteristic
177#208C2×D411yescharacteristic
178C2×C3⋊C8C2×D411yescharacteristic
179Dic12C2×D412yes—
180D24—22no—
181D24—22no—
182D24—42no—
183D24—42no—
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—
197H(5,𝔽2)—62no—
198H(5,𝔽2)—62no—
199AGL(1,ℤ/8)—22no—
200AGL(1,ℤ/8)—62no—
201AGL(1,ℤ/8)—62no—
202AGL(1,ℤ/8)—62no—
203AGL(1,ℤ/8)—122no—
204AGL(1,ℤ/8)—122no—
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×D8—32no—
213C2×D8—62no—
214C2×D8—62no—
215C2×D8—62no—
216C2×D8—122no—
217C2×D8C22×D312yes—
218#131—31no—
219#131—62no—
220#131C22×D311yescharacteristic
221#127—62no—
222Q32—31no—
223SD32—32no—
224SD32—62no—
225SD32—62no—
226D16—21no—
227D16—21no—
228D16—31no—
229D16—62no—
230D16—62no—
231D16—61no—
232D16—61no—
233D16—122no—
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#100—22no—
248#100—62no—
249C4×C8—31no—
250C23×C3—42no—
251C22×D3—42no—
252C22×D3—82no—
253C22×D3—82no—
254C3×Q8C2×D811yescharacteristic
255C3×D4—21no—
256C3×D4—42no—
257C3×D4—42no—
258C3×D4C2×D811yescharacteristic
259C2×C3×C4—21no—
260C2×C3×C4C22×D411yescharacteristic
261#67—21no—
262#67—42no—
263#67—82no—
264#67—82no—
265#67—82no—
266#67—164no—
267C2×Dic3—21no—
268C2×Dic3—82no—
269D12—22no—
270D12—42no—
271D12—42no—
272D12—82no—
273D12C2×D811yescharacteristic
274C4×D3—21no—
275C4×D3—42no—
276C4×D3—82no—
277Dic6—22no—
278Dic6C2×D811yescharacteristic
279C3×C8—21no—
280C3×C8—42no—
281C3×C8C22×D411yescharacteristic
282C3×C8H(5,𝔽2)11yescharacteristic
283C3⋊C8—21no—
284C3⋊C8C2×D811yescharacteristic
285C3⋊C8C2×D811yescharacteristic
286#41—31no—
287#41—62no—
288#41—122no—
289#41—122no—
290#41—122no—
291#41—122no—
292#41D3×D411yescharacteristic
293C2×D4—22no—
294C2×D4—62no—
295C2×D4—122no—
296C2×D4—122no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—242no—
300C2×D4—242no—
301C2×D4—242no—
302Q16—31no—
303Q16—32no—
304SD16—31no—
305SD16—31no—
306SD16—42no—
307SD16—62no—
308SD16—122no—
309SD16—122no—
310SD16—122no—
311D8—22no—
312D8—22no—
313D8—31no—
314D8—32no—
315D8—42no—
316D8—62no—
317D8—62no—
318D8—122no—
319D8—122no—
320D8—122no—
321D8—122no—
322D8—122no—
323M16—21no—
324M16—22no—
325M16—31no—
326M16—61no—
327M16—122no—
328M16D3×D411yescharacteristic
329C2×C8—21no—
330C2×C8—31no—
331C2×C8—61no—
332C2×C8—122no—
333C2×C8C23×D311yescharacteristic
334C42—31no—
335C16—21no—
336C16—61no—
337C16—61no—
338C16D3×D411yescharacteristic
339C16D3×D411yescharacteristic
340C22×C3—21no—
341C22×C3—82no—
342C22×C3—162no—
343C22×C3#57611yescharacteristic
344C2×D3—21no—
345C2×D3—42no—
346C2×D3—82no—
347C2×D3—162no—
348C2×D3—322no—
349C2×D3—322no—
350C3×C4—21no—
351C3×C4D4211yescharacteristic
352C3×C4C22×D811yescharacteristic
353Dic3—22no—
354Dic3—21no—
355Dic3—42no—
356C23—42no—
357C23—122no—
358C23—242no—
359C23—242no—
360Q8—31no—
361Q8—62no—
362Q8D3×D811yescharacteristic
363D4—21no—
364D4—31no—
365D4—42no—
366D4—42no—
367D4—61no—
368D4—62no—
369D4—122no—
370D4—122no—
371D4—122no—
372D4—242no—
373D4—242no—
374D4—242no—
375D4—242no—
376D4—484no—
377D4D3×D811yescharacteristic
378C2×C4—21no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—122no—
382C2×C4—242no—
383C2×C4—242no—
384C2×C4C2×D3×D411yescharacteristic
385C8—21no—
386C8—31no—
387C8—31no—
388C8—42no—
389C8—61no—
390C8C2×D3×D411yescharacteristic
391C8#100911yescharacteristic
392C2×C3—21no—
393C2×C3—41no—
394C2×C3—162no—
395C2×C3D4×D811yescharacteristic
396D3—41no—
397D3—82no—
398D3—162no—
399C22—21no—
400C22—61no—
401C22—82no—
402C22—122no—
403C22—162no—
404C22—242no—
405C22—482no—
406C22—962no—
407C22—962no—
408C22#576911yescharacteristic
409C4—21no—
410C4—62no—
411C4—61no—
412C4—122no—
413C4#560911yescharacteristic
414C4C2×D3×D811yescharacteristic
415C3#3385811yescharacteristic, complemented, Sylow
416C2—21no—
417C2—41no—
418C2—121no—
419C2—162no—
420C2—242no—
421C2—482no—
422C2#8072911yescharacteristic
subgroupG/H
centerC2#80729
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2221C2
socleC2×C3D4×D8
hypercenter#2221D3
hypocenterC3#33858
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. #82013C2semidirect
3. #80941C2semidirect
4. C3×#2221C2semidirect
5. #71464C2semidirect
6. #71452C2semidirect
7. #71094C2semidirect
8. #69361C2semidirect
9. #69357C2semidirect
10. #69263C2semidirect
11. #69258C2semidirect
14. #5757C22semidirect
15. #5752C22semidirect
16. C2×C3×D16C22semidirect
17. C3×#504C22semidirect
18. C3×#472C22semidirect
20. #5151C22semidirect
21. #5147C22semidirect
22. C2×#826C22semidirect
28. #4908C22semidirect
32. C2×D48C22semidirect
33. #4901C22semidirect
34. #4897C22semidirect
35. #4895C22semidirect
38. C3×#368C22semidirect
39. C3×#361C22semidirect
40. #4561C22semidirect
41. #4515C22semidirect
42. #4511C22semidirect
43. Dic3×C16C22semidirect
44. #4497C22semidirect
45. #4495C22semidirect
51. #2221D3semidirect
101. Dic24D4semidirect
103. #800D4semidirect
107. D48D4semidirect
124. C2×D16C2×D3semidirect
126. #504C2×D3semidirect
130. #472C2×D3semidirect
135. #368C2×D3semidirect
138. #361C2×D3semidirect
415. C3#33858semidirect

Maximal subgroups

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

subgroupindex
#820192
#820132
#809412
C3×#22212
#714642
#714522
#710942
#693612
#693572
#692632
#692582
#338583