Finite Group Databasebeta

← #10951990

Subgroups — #10951990

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#82028C211yesmaximal, characteristic, complemented
2#82019C211yesmaximal, characteristic, complemented
3#81835C212yesmaximal, complemented
4C3×#2249C211yesmaximal, characteristic, complemented
5#71507C211yesmaximal, characteristic, complemented
6#71490C212yesmaximal, complemented
7#71489C212yesmaximal, complemented
8#71472C211yesmaximal, characteristic, complemented
9#71325C212yesmaximal, complemented
10#69067C211yesmaximal, characteristic, complemented
11#69056C211yesmaximal, characteristic, complemented
12#33859—31nomaximal, Sylow
13#5766C2212yescomplemented
14#5762—22no—
15#5757C2212yes—
16C2×C3×Q32C2211yescharacteristic, complemented
17C2×C3×SD32C2212yescomplemented
18C2×C3×D16C2211yescharacteristic
19C3×#495C2212yescomplemented
20#5192C2212yes—
21#5179C2212yescomplemented
22#5177—22no—
23#5160C2212yescomplemented
24#5147C2212yes—
25#5062—22no—
26#5061—22no—
27#4919—21no—
28#4918—22no—
29D3×Q32—21no—
30#4912—21no—
31#4911—22no—
32D3×D16—21no—
33#4904C2211yescharacteristic
34#4901C2211yescharacteristic
35#4901C2211yescharacteristic, complemented
36#4700C2211yescharacteristic
37#4700C2212yescomplemented
38C3×C4×C16C2211yescharacteristic, complemented
39#4565C2212yescomplemented
40#4563C2212yescomplemented
41#4510C2212yescomplemented
42#4508C2211yescharacteristic
43#4508C2211yescharacteristic, complemented
44#4506C2211yescharacteristic, complemented
45#4463C2211yescharacteristic, complemented
46#3417—31no—
47#3415—31no—
48#3290—32no—
49#2249D311yescharacteristic, complemented
50#2222—31no—
51#2221—31no—
52#2212—32no—
53#2194—32no—
54#2193—32no—
55#2178—31no—
56#2170—31no—
57#1007—22no—
58#1004—22no—
59C2×C3×Q16C2312yes—
60C2×C3×SD16—22no—
61C2×C3×D8C2312yes—
62C3×#124—22no—
63#942—22no—
64#942—22no—
65#932—22no—
66#932—22no—
67#919—22no—
68#918—42no—
69D3×Q16—22no—
70#916—42no—
71#915—42no—
72#914—42no—
73D3×SD16—42no—
74#912—22no—
75#911—42no—
76D3×D8—22no—
77#904D411yescharacteristic
78#904C2312yes—
79#901—22no—
80#901C2311yescharacteristic
81#901C2311yescharacteristic
82C3×Q32—21no—
83C3×Q32—21no—
84C3×SD32—42no—
85C3×D16—21no—
86C3×D16—21no—
87C2×C3×C16C2311yescharacteristic
88C2×C3×C16C2311yescharacteristic
89C3×C4×C8C2311yescharacteristic
90#836—22no—
91#833—22no—
92#829—22no—
93#828—22no—
94#827—22no—
95#826—22no—
96#819D411yescharacteristic
97#819C2312yes—
98#812C2311yescharacteristic
99#812C2311yescharacteristic
100#805—22no—
101#805D411yescharacteristic
102#805D411yescharacteristic
103Dic24D411yescharacteristic
104#800D412yes—
105D48D411yescharacteristic
106#798—21no—
107#798—21no—
108D3×C16—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—61no—
121C2×Q32C2×D311yescharacteristic, complemented
122C2×SD32C2×D312yescomplemented
123C2×D16—61no—
124C2×D16C2×D311yescharacteristic
125#504—31no—
126#504—31no—
127#495C2×D312yescomplemented
128#473—32no—
129#472—32no—
130#471—62no—
131#455—62no—
132#454—62no—
133#443—31no—
134#443—32no—
135#368—31no—
136#364—31no—
137#362—32no—
138#361—32no—
139#359—32no—
140#350—31no—
141#350—31no—
142C4×C16C2×D311yescharacteristic, complemented
143C2×C3×Q8—22no—
144C2×C3×D4—22no—
145C2×#67—82no—
146#239—42no—
147#239—42no—
148D3×Q8—42no—
149D3×Q8—42no—
150#237—42no—
151#237—42no—
152D3×D4—42no—
153D3×D4—42no—
154#235—22no—
155#235—82no—
156#235C2×D411yescharacteristic
157#235C2×D411yescharacteristic
158C3×Q16—22no—
159C3×Q16—22no—
160C3×SD16—42no—
161C3×SD16—42no—
162C3×D8—22no—
163C3×D8—22no—
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2411yescharacteristic
166C3×C22⋊C4—82no—
167C3×C42C2×D411yescharacteristic
168#216—42no—
169#216—42no—
170#215—42no—
171#215—42no—
172#214—42no—
173#214—42no—
174#213—42no—
175#213—42no—
176#208—22no—
177#208C2×D411yescharacteristic
178#208C2×D411yescharacteristic
179Dic12D811yescharacteristic
180Dic12C2×D412yes—
181D24D811yescharacteristic
182D24C2×D412yes—
183#204—22no—
184#204D811yescharacteristic
185#204D811yescharacteristic
186ΓL(1,𝔽25)—21no—
187ΓL(1,𝔽25)—21no—
188ΓL(1,𝔽25)—42no—
189D3×C8—21no—
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)—62no—
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—62no—
213C2×Q16C22×D312yes—
214C2×SD16—22no—
215C2×SD16—122no—
216C2×D8—62no—
217C2×D8C22×D312yes—
218#131—31no—
219#131—31no—
220#131—62no—
221#124—22no—
222Q32—21no—
223Q32—21no—
224Q32—31no—
225Q32—62no—
226SD32—32no—
227SD32—42no—
228SD32—62no—
229SD32—62no—
230D16—21no—
231D16—21no—
232D16—31no—
233D16—62no—
234M32—31no—
235M32—31no—
236M32—61no—
237M32—61no—
238C2×C16—61no—
239C2×C16—61no—
240C2×C16C22×D311yescharacteristic
241C2×C16C22×D311yescharacteristic
242#108—31no—
243#108—32no—
244#104—31no—
245#104—31no—
246#104—62no—
247#101—62no—
248#100—62no—
249C4×C8C22×D311yescharacteristic
250C23×C3—42no—
251C22×D3—82no—
252C3×Q8—42no—
253C3×Q8—42no—
254C3×D4—42no—
255C3×D4—42no—
256C2×C3×C4—21no—
257C2×C3×C4—42no—
258C2×C3×C4C22×D411yescharacteristic
259#67—21no—
260#67—21no—
261#67—42no—
262#67—82no—
263#67—82no—
264C2×Dic3—82no—
265D12—22no—
266D12—82no—
267D12C2×D811yescharacteristic
268D12C2×D811yescharacteristic
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—42no—
272C4×D3—82no—
273C4×D3—82no—
274Dic6—22no—
275Dic6—82no—
276Dic6C2×D811yescharacteristic
277Dic6C2×D811yescharacteristic
278C3×C8C2×D811yescharacteristic
279C3×C8C2×D811yescharacteristic
280C3×C8C22×D411yescharacteristic
281C3×C8H(5,𝔽2)11yescharacteristic
282C3⋊C8—21no—
283C3⋊C8—21no—
284C3⋊C8—42no—
285#41—31no—
286#41—31no—
287#41—62no—
288#41—122no—
289#41—122no—
290#41—122no—
291#41—122no—
292#41—242no—
293C2×Q8—22no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×D4—22no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—242no—
300Q16—22no—
301Q16—22no—
302Q16—31no—
303Q16—32no—
304Q16—122no—
305Q16—122no—
306SD16—31no—
307SD16—31no—
308SD16—42no—
309SD16—42no—
310SD16—62no—
311SD16—122no—
312SD16—122no—
313SD16—122no—
314SD16—122no—
315D8—22no—
316D8—22no—
317D8—31no—
318D8—32no—
319D8—122no—
320D8—122no—
321M16—31no—
322M16—31no—
323M16—61no—
324M16—61no—
325M16—62no—
326M16—122no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—122no—
330C2×C8D3×D411yescharacteristic
331C2×C8C23×D311yescharacteristic
332C22⋊C4—82no—
333C42D3×D411yescharacteristic
334C16—21no—
335C16—61no—
336C16—61no—
337C16D3×D411yescharacteristic
338C16D3×D411yescharacteristic
339C22×C3—82no—
340C22×C3—162no—
341C22×C3#57611yescharacteristic
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—42no—
345C2×D3—322no—
346C3×C4—41no—
347C3×C4—82no—
348C3×C4D4211yescharacteristic
349C3×C4C22×D811yescharacteristic
350Dic3—21no—
351Dic3—21no—
352Dic3—42no—
353C23—42no—
354C23—242no—
355Q8—31no—
356Q8—31no—
357Q8—42no—
358Q8—42no—
359Q8—62no—
360Q8—242no—
361D4—31no—
362D4—31no—
363D4—42no—
364D4—42no—
365D4—61no—
366D4—61no—
367D4—62no—
368D4—122no—
369D4—242no—
370D4—242no—
371D4—242no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—122no—
377C2×C4—242no—
378C2×C4—242no—
379C2×C4—242no—
380C2×C4C2×D3×D411yescharacteristic
381C8—61no—
382C8—61no—
383C8—122no—
384C8D3×D811yescharacteristic
385C8D3×D811yescharacteristic
386C8C2×D3×D411yescharacteristic
387C8#100411yescharacteristic
388C2×C3—21no—
389C2×C3—162no—
390C2×C3D4×D811yescharacteristic
391D3—41no—
392D3—41no—
393D3—82no—
394C22—61no—
395C22—61no—
396C22—82no—
397C22—122no—
398C22—162no—
399C22—962no—
400C22#575711yescharacteristic
401C4—41no—
402C4—61no—
403C4—61no—
404C4—82no—
405C4—122no—
406C4#571711yescharacteristic
407C4C2×D3×D811yescharacteristic
408C3#3385911yescharacteristic, complemented, Sylow
409C2—21no—
410C2—121no—
411C2—121no—
412C2—162no—
413C2—242no—
414C2#8178911yescharacteristic
subgroupG/H
centerC2#81789
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2249C2
socleC2×C3D4×D8
hypercenter#2249D3
hypocenterC3#33859
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2249

Complements

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

NH ≅ G/Ntype
1. #82028C2semidirect
2. #82019C2semidirect
3. #81835C2semidirect
4. C3×#2249C2semidirect
5. #71507C2semidirect
6. #71490C2semidirect
7. #71489C2semidirect
8. #71472C2semidirect
9. #71325C2semidirect
10. #69067C2semidirect
11. #69056C2semidirect
13. #5766C22semidirect
16. C2×C3×Q32C22semidirect
17. C2×C3×SD32C22semidirect
19. C3×#495C22semidirect
21. #5179C22semidirect
23. #5160C22semidirect
35. #4901C22semidirect
37. #4700C22semidirect
38. C3×C4×C16C22semidirect
39. #4565C22semidirect
40. #4563C22semidirect
41. #4510C22semidirect
43. #4508C22semidirect
44. #4506C22semidirect
45. #4463C22semidirect
49. #2249D3semidirect
121. C2×Q32C2×D3semidirect
122. C2×SD32C2×D3semidirect
127. #495C2×D3semidirect
142. C4×C16C2×D3semidirect
408. C3#33859semidirect

Maximal subgroups

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

subgroupindex
#820282
#820192
#818352
C3×#22492
#715072
#714902
#714892
#714722
#713252
#690672
#690562
#338593