Finite Group Databasebeta

← #10951992

Subgroups — #10951992

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#82033C211yesmaximal, characteristic, complemented
2#82031C211yesmaximal, characteristic, complemented
3#79892C212yesmaximal, complemented
4C3×#2178C211yesmaximal, characteristic, complemented
5#70925C212yesmaximal, complemented
6#69362C211yesmaximal, characteristic, complemented
7#69361C211yesmaximal, characteristic, complemented
8#69244C212yesmaximal, complemented
9#69243C212yesmaximal, complemented
10#69062C211yesmaximal, characteristic, complemented
11#69056C211yesmaximal, characteristic, complemented
12#33859—31nomaximal, Sylow
13#5771C2212yescomplemented
14#5769C2212yes—
15#5751—22no—
16C3×#508C2211yescharacteristic
17C3×#443C2212yescomplemented
18#5195C2212yescomplemented
19#5193C2212yes—
20#5141—22no—
21#4919—21no—
22D3×Q32—21no—
23#4915—22no—
24#4914—22no—
25#4912—21no—
26D3×D16—21no—
27C2×Dic24C2211yescharacteristic, complemented
28C2×#800C2212yescomplemented
29C2×D48C2211yescharacteristic
30#4901C2211yescharacteristic
31#4901C2211yescharacteristic, complemented
32#4897C2212yescomplemented
33#4896C2212yes—
34#4827—22no—
35#4825—22no—
36#4700C2211yescharacteristic
37#4696C2212yescomplemented
38C3×#368C2211yescharacteristic, complemented
39C3×#359C2212yescomplemented
40C3×C4×C16C2211yescharacteristic, complemented
41#4508C2211yescharacteristic
42#4508C2211yescharacteristic, complemented
43#4496C2212yescomplemented
44#4495C2212yescomplemented
45#4463C2211yescharacteristic, complemented
46#3417—31no—
47#3415—31no—
48#3290—32no—
49#2249—31no—
50#2222—31no—
51#2221—31no—
52#2212—32no—
53#2194—32no—
54#2193—32no—
55#2178D311yescharacteristic, complemented
56#2170—31no—
57#1010—22no—
58#1009—22no—
59C3×#135C2312yes—
60C3×#131—22no—
61#951—22no—
62#949—22no—
63#919—22no—
64D3×Q16—22no—
65#915—42no—
66#914—42no—
67#912—22no—
68D3×D8—22no—
69#909—22no—
70#909—42no—
71#908—22no—
72#908—42no—
73C2×Dic12C2312yes—
74#904—42no—
75#904D411yescharacteristic
76C2×D24C2312yes—
77C2×#204—22no—
78C2×#204—42no—
79#901C2311yescharacteristic
80#901C2311yescharacteristic
81#875—22no—
82C3×Q32D411yescharacteristic
83C3×SD32D412yes—
84C3×D16D411yescharacteristic
85C2×C3×C16C2311yescharacteristic
86C2×C3×C16C2311yescharacteristic
87C3×#108C2312yes—
88C3×#104—22no—
89C3×C4×C8C2311yescharacteristic
90#829—22no—
91#828—22no—
92#827—22no—
93#826—22no—
94#824—22no—
95#823—22no—
96#819D411yescharacteristic
97#812C2311yescharacteristic
98#812C2311yescharacteristic
99#805D411yescharacteristic
100#805D411yescharacteristic
101Dic24—21no—
102Dic24—21no—
103#800—42no—
104D48—21no—
105D48—21no—
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—61no—
118#508—61no—
119#508C2×D311yescharacteristic
120C2×Q32—31no—
121C2×Q32—61no—
122C2×SD32—32no—
123C2×D16—31no—
124C2×D16—61no—
125#504—31no—
126#504—31no—
127#495—32no—
128#473—32no—
129#472—32no—
130#471—62no—
131#455—62no—
132#454—62no—
133#443—31no—
134#443C2×D312yescomplemented
135#368C2×D311yescharacteristic, complemented
136#364—31no—
137#362—32no—
138#361—32no—
139#359C2×D312yescomplemented
140#350—31no—
141#350—31no—
142C4×C16C2×D311yescharacteristic, complemented
143C3×#41—22no—
144#239—42no—
145D3×Q8—42no—
146#237—42no—
147#237—82no—
148D3×D4—42no—
149D3×D4—82no—
150#235—42no—
151#235—42no—
152#235C2×D411yescharacteristic
153#235C2×D411yescharacteristic
154C2×D12—22no—
155C2×D12—42no—
156C2×Dic6—22no—
157C2×Dic6—42no—
158C3×Q16C2×D412yes—
159C3×SD16—22no—
160C3×D8C2×D412yes—
161C3×M16—22no—
162C3×M16—42no—
163C2×C3×C8—42no—
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2411yescharacteristic
166C3×C42C2×D411yescharacteristic
167#216—42no—
168#215—42no—
169#214—42no—
170#213—42no—
171(C2×D3)⋊C4—82no—
172#208C2×D411yescharacteristic
173#208C2×D411yescharacteristic
174Dic12—22no—
175Dic12—22no—
176Dic12—42no—
177Dic12D811yescharacteristic
178D24—22no—
179D24—22no—
180D24—42no—
181D24D811yescharacteristic
182#204—42no—
183#204—42no—
184#204—42no—
185#204—42no—
186#204D811yescharacteristic
187#204D811yescharacteristic
188ΓL(1,𝔽25)—21no—
189ΓL(1,𝔽25)—21no—
190D3×C8—21no—
191D3×C8—21no—
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—62no—
209#135—62no—
210#135—122no—
211#135C22×D312yes—
212C2×Q16—32no—
213C2×Q16—62no—
214C2×SD16—62no—
215C2×SD16—122no—
216C2×D8—32no—
217C2×D8—62no—
218#131—22no—
219#131—31no—
220#131—31no—
221#124—62no—
222Q32—62no—
223Q32—61no—
224Q32—61no—
225Q32D1211yescharacteristic
226SD32—62no—
227SD32—62no—
228SD32—122no—
229SD32D1212yes—
230D16—62no—
231D16—61no—
232D16—61no—
233D16D1211yescharacteristic
234M32—31no—
235M32—31no—
236M32—61no—
237M32—61no—
238C2×C16—61no—
239C2×C16—61no—
240C2×C16C22×D311yescharacteristic
241C2×C16C22×D311yescharacteristic
242#108—31no—
243#108C22×D312yes—
244#104—22no—
245#104—31no—
246#104—31no—
247#101—62no—
248#100—62no—
249C4×C8C22×D311yescharacteristic
250C22×D3—42no—
251C22×D3—82no—
252C3×Q8—22no—
253C3×D4—22no—
254C3×D4—42no—
255C2×C3×C4—21no—
256C2×C3×C4—42no—
257C2×C3×C4C22×D411yescharacteristic
258#67—21no—
259#67—21no—
260#67—82no—
261#67—82no—
262C2×Dic3—42no—
263C2×Dic3—82no—
264D12—42no—
265D12—42no—
266D12—82no—
267D12C2×D811yescharacteristic
268D12C2×D811yescharacteristic
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—82no—
272C4×D3—82no—
273Dic6—42no—
274Dic6—42no—
275Dic6—82no—
276Dic6C2×D811yescharacteristic
277Dic6C2×D811yescharacteristic
278C3×C8—42no—
279C3×C8C2×D811yescharacteristic
280C3×C8C2×D811yescharacteristic
281C3×C8C22×D411yescharacteristic
282C3×C8H(5,𝔽2)11yescharacteristic
283C3⋊C8—21no—
284C3⋊C8—21no—
285#41—22no—
286#41—31no—
287#41—31no—
288#41—122no—
289#41—122no—
290#41—122no—
291#41—122no—
292#41—242no—
293C2×Q8—62no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×D4—62no—
297C2×D4—122no—
298C2×D4—122no—
299C2×D4—242no—
300Q16—31no—
301Q16—62no—
302Q16—62no—
303Q16—122no—
304Q16—122no—
305Q16C2×D1212yes—
306SD16—22no—
307SD16—31no—
308SD16—31no—
309SD16—122no—
310SD16—122no—
311SD16—122no—
312SD16—122no—
313SD16—122no—
314SD16—122no—
315D8—31no—
316D8—62no—
317D8—62no—
318D8—122no—
319D8—122no—
320D8C2×D1212yes—
321M16—22no—
322M16—31no—
323M16—31no—
324M16—42no—
325M16—61no—
326M16—61no—
327C2×C8—42no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8D3×D411yescharacteristic
331C2×C8C23×D311yescharacteristic
332C22⋊C4—242no—
333C42D3×D411yescharacteristic
334C16—21no—
335C16—61no—
336C16—61no—
337C16C2×D1211yescharacteristic
338C16C2×D1211yescharacteristic
339C22×C3—42no—
340C22×C3#57611yescharacteristic
341C2×D3—21no—
342C2×D3—21no—
343C2×D3—82no—
344C2×D3—162no—
345C2×D3—322no—
346C3×C4—41no—
347C3×C4—42no—
348C3×C4D4211yescharacteristic
349C3×C4C22×D811yescharacteristic
350Dic3—21no—
351Dic3—21no—
352Dic3—82no—
353C23—122no—
354C23—242no—
355Q8—22no—
356Q8—31no—
357Q8—31no—
358Q8—122no—
359Q8—122no—
360Q8—242no—
361D4—22no—
362D4—31no—
363D4—31no—
364D4—42no—
365D4—61no—
366D4—61no—
367D4—122no—
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—42no—
382C8—61no—
383C8—61no—
384C8D3×D811yescharacteristic
385C8D3×D811yescharacteristic
386C8C22×D1211yescharacteristic
387C8#100911yescharacteristic
388C2×C3—21no—
389C2×C3—82no—
390C2×C3D4×D811yescharacteristic
391D3—41no—
392D3—41no—
393D3—162no—
394C22—42no—
395C22—61no—
396C22—61no—
397C22—242no—
398C22—482no—
399C22—962no—
400C22#576911yescharacteristic
401C4—41no—
402C4—42no—
403C4—61no—
404C4—61no—
405C4—242no—
406C4D4×D1211yescharacteristic
407C4C2×D3×D811yescharacteristic
408C3#3385911yescharacteristic, complemented, Sylow
409C2—21no—
410C2—82no—
411C2—121no—
412C2—121no—
413C2—482no—
414C2D8×D1211yescharacteristic
subgroupG/H
centerC2D8×D12
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2178C2
socleC2×C3D4×D8
hypercenter#2178D3
hypocenterC3#33859
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2178

Complements

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

NH ≅ G/Ntype
1. #82033C2semidirect
2. #82031C2semidirect
3. #79892C2semidirect
4. C3×#2178C2semidirect
5. #70925C2semidirect
6. #69362C2semidirect
7. #69361C2semidirect
8. #69244C2semidirect
9. #69243C2semidirect
10. #69062C2semidirect
11. #69056C2semidirect
13. #5771C22semidirect
17. C3×#443C22semidirect
18. #5195C22semidirect
27. C2×Dic24C22semidirect
28. C2×#800C22semidirect
31. #4901C22semidirect
32. #4897C22semidirect
37. #4696C22semidirect
38. C3×#368C22semidirect
39. C3×#359C22semidirect
40. C3×C4×C16C22semidirect
42. #4508C22semidirect
43. #4496C22semidirect
44. #4495C22semidirect
45. #4463C22semidirect
55. #2178D3semidirect
134. #443C2×D3semidirect
135. #368C2×D3semidirect
139. #359C2×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
#820332
#820312
#798922
C3×#21782
#709252
#693622
#693612
#692442
#692432
#690622
#690562
#338593