Finite Group Databasebeta

← #10951977

Subgroups — #10951977

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
1C3×#3415C211yesmaximal, characteristic, complemented
2#84010C212yesmaximal, complemented
3#82031C211yesmaximal, characteristic, complemented
4#72122C212yesmaximal, complemented
5#71517C212yesmaximal, complemented
6#71514C212yesmaximal, complemented
7#71511C211yesmaximal, characteristic, complemented
8#71472C211yesmaximal, characteristic, complemented
9#71464C211yesmaximal, characteristic, complemented
10#71454C211yesmaximal, characteristic, complemented
11#71452C211yesmaximal, characteristic, complemented
12#33858—31nomaximal, Sylow
13C3×#576C2212yescomplemented
14#5835—22no—
15#5769C2212yescomplemented
16C3×#509—22no—
17C3×#509C2212yescomplemented
18C3×#508—21no—
19C3×#508C2211yescharacteristic
20C2×C3×D16—21no—
21C2×C3×D16C2211yescharacteristic, complemented
22C3×#504C2211yescharacteristic, complemented
23#5241—22no—
24#5199—22no—
25#5198—22no—
26#5196C2212yescomplemented
27#5193C2212yescomplemented
28#5160C2212yescomplemented
29#5151C2212yescomplemented
30C2×#826C2212yescomplemented
31#5144C2211yescharacteristic
32#5142C2212yescomplemented
33#4919—21no—
34#4914—22no—
35D3×D16—21no—
36C2×D48C2211yescharacteristic, complemented
37#4901C2211yescharacteristic, complemented
38#4567C2212yescomplemented
39#4563C2212yescomplemented
40#4561C2211yescharacteristic, complemented
41#4516C2212yescomplemented
42#4515C2211yescharacteristic, complemented
43#4511C2211yescharacteristic, complemented
44#4506C2211yescharacteristic, complemented
45Dic3×C16C2211yescharacteristic, complemented
46#3415—31no—
47#3415D311yescharacteristic, complemented
48#3292—32no—
49#2246—31no—
50#2221—31no—
51#2221—31no—
52#2213—32no—
53#2192—32no—
54#2192—32no—
55#2178—31no—
56#2170—31no—
57C3×H(5,𝔽2)—22no—
58#1009—22no—
59C3×AGL(1,ℤ/8)—22no—
60C3×AGL(1,ℤ/8)—42no—
61C3×#135—22no—
62C3×#135C2312yes—
63C2×C3×D8—22no—
64C2×C3×D8C2312yes—
65C3×#131C2311yescharacteristic
66#950—42no—
67#950D411yescharacteristic
68#949—22no—
69#949—22no—
70#949—42no—
71#948—22no—
72#940—22no—
73#932—22no—
74#932—42no—
75C2×#213—22no—
76C2×#213—42no—
77#919—22no—
78#914—42no—
79D3×D8—22no—
80C2×D24C2312yes—
81#901C2311yescharacteristic
82C3×Q32D411yescharacteristic, complemented
83C3×SD32—22no—
84C3×SD32D412yescomplemented
85C3×D16—22no—
86C3×D16—21no—
87C3×D16—21no—
88C3×D16D411yescharacteristic, complemented
89C3×M32—21no—
90C3×M32C2311yescharacteristic
91C2×C3×C16—21no—
92C2×C3×C16C2311yescharacteristic
93#837—22no—
94#837D411yescharacteristic
95#833—22no—
96#828—22no—
97#826—22no—
98#826—42no—
99#825D411yescharacteristic
100#823—22no—
101#822D411yescharacteristic
102#819C2312yes—
103C8×Dic3C2311yescharacteristic
104#812C2311yescharacteristic
105C2×C3⋊C16C2311yescharacteristic
106D48—21no—
107D48—21no—
108#798—21no—
109D3×C16—21no—
110#576—32no—
111#576—62no—
112#576C2×D312yescomplemented
113#509—22no—
114#509—32no—
115#509—62no—
116#509C2×D312yescomplemented
117#508—21no—
118#508—61no—
119#508C2×D311yescharacteristic
120C2×D16—21no—
121C2×D16—32no—
122C2×D16—31no—
123C2×D16—61no—
124C2×D16C2×D311yescharacteristic, complemented
125#504—31no—
126#504C2×D311yescharacteristic, complemented
127#493—32no—
128#472—32no—
129#472—32no—
130#472—62no—
131#453—62no—
132#453—62no—
133#443—31no—
134#443—32no—
135#368—31no—
136#364—31no—
137#361—32no—
138#361—32no—
139#359—32no—
140#350—31no—
141#350—31no—
142C4×C16—31no—
143C3×#41—22no—
144C3×#41—42no—
145C3×#41—42no—
146C3×#41C2×D411yescharacteristic
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—82no—
151C2×#67—82no—
152#239—42no—
153D3×D4—42no—
154D3×D4—82no—
155#235—42no—
156#235C2×D411yescharacteristic
157C2×D12—22no—
158C2×D12—42no—
159C3×Q16C2×D412yes—
160C3×SD16—22no—
161C3×SD16—42no—
162C3×D8—22no—
163C3×D8—22no—
164C3×D8—42no—
165C3×D8C2×D412yes—
166C3×M16—21no—
167C3×M16C2×D411yescharacteristic
168C2×C3×C8—21no—
169C2×C3×C8C2411yescharacteristic
170#216D811yescharacteristic
171#215—42no—
172#215—42no—
173#215D811yescharacteristic
174#214—42no—
175#214D811yescharacteristic
176#213—42no—
177#213—42no—
178#213—42no—
179#213—42no—
180#213—42no—
181#213D811yescharacteristic
182C4×Dic3C2×D411yescharacteristic
183#208—22no—
184#208—42no—
185#208C2×D411yescharacteristic
186C2×C3⋊C8—42no—
187C2×C3⋊C8C2×D411yescharacteristic
188D24—22no—
189D24—22no—
190ΓL(1,𝔽25)—21no—
191D3×C8—21no—
192C3×C16—21no—
193C3×C16C2×D411yescharacteristic
194C3×C16C2×D411yescharacteristic
195C3⋊C16—21no—
196C3⋊C16—21no—
197H(5,𝔽2)—22no—
198H(5,𝔽2)—62no—
199AGL(1,ℤ/8)—22no—
200AGL(1,ℤ/8)—42no—
201AGL(1,ℤ/8)—62no—
202AGL(1,ℤ/8)—62no—
203AGL(1,ℤ/8)—62no—
204AGL(1,ℤ/8)—122no—
205AGL(1,ℤ/8)—122no—
206AGL(1,ℤ/8)—122no—
207#135—22no—
208#135—31no—
209#135—62no—
210#135—122no—
211#135C22×D312yes—
212C2×D8—22no—
213C2×D8—32no—
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#6711yescharacteristic, complemented
223SD32—22no—
224SD32—62no—
225SD32#6712yescomplemented
226D16—22no—
227D16—21no—
228D16—21no—
229D16—62no—
230D16—61no—
231D16—61no—
232D16—122no—
233D16#6711yescharacteristic, complemented
234M32—21no—
235M32—31no—
236M32—61no—
237M32C22×D311yescharacteristic
238C2×C16—21no—
239C2×C16—31no—
240C2×C16—61no—
241C2×C16C22×D311yescharacteristic
242#108—31no—
243#108—32no—
244#104—31no—
245#104—31no—
246#104—62no—
247#100—62no—
248#100—62no—
249C4×C8—31no—
250C23×C3—42no—
251C23×C3—82no—
252C22×D3—42no—
253C22×D3—82no—
254C3×Q8—22no—
255C3×Q8C2×D811yescharacteristic
256C3×D4—21no—
257C3×D4—22no—
258C3×D4—42no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—82no—
262C3×D4—82no—
263C3×D4C2×D811yescharacteristic
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—82no—
267C2×C3×C4C22×D411yescharacteristic
268#67—21no—
269#67—82no—
270#67—164no—
271C2×Dic3—21no—
272D12—42no—
273D12—42no—
274D12—82no—
275D12C2×D811yescharacteristic
276C4×D3—21no—
277C4×D3—82no—
278Dic6C2×D811yescharacteristic
279C3×C8—21no—
280C3×C8C22×D411yescharacteristic
281C3×C8H(5,𝔽2)11yescharacteristic
282C3⋊C8—21no—
283C3⋊C8—42no—
284C3⋊C8C2×D811yescharacteristic
285C3⋊C8C2×D811yescharacteristic
286#41—22no—
287#41—31no—
288#41—42no—
289#41—42no—
290#41—122no—
291#41—122no—
292#41D3×D411yescharacteristic
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—62no—
297C2×D4—82no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—242no—
301C2×D4—242no—
302Q16—31no—
303Q16C2×#6712yes—
304SD16—22no—
305SD16—31no—
306SD16—31no—
307SD16—42no—
308SD16—122no—
309SD16—122no—
310SD16—122no—
311D8—22no—
312D8—22no—
313D8—31no—
314D8—42no—
315D8—62no—
316D8—62no—
317D8—122no—
318D8—122no—
319D8—122no—
320D8—122no—
321D8—122no—
322D8C2×#6712yes—
323M16—21no—
324M16—31no—
325M16—61no—
326M16—62no—
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—
338C16C2×#6711yescharacteristic
339C16C2×#6711yescharacteristic
340C22×C3—21no—
341C22×C3—42no—
342C22×C3—82no—
343C22×C3—162no—
344C22×C3—322no—
345C22×C3#57611yescharacteristic
346C2×D3—21no—
347C2×D3—82no—
348C2×D3—162no—
349C2×D3—322no—
350C3×C4—21no—
351C3×C4—42no—
352C3×C4D4211yescharacteristic
353C3×C4C22×D811yescharacteristic
354Dic3—22no—
355Dic3—21no—
356C23—42no—
357C23—82no—
358C23—122no—
359C23—242no—
360Q8—22no—
361Q8—31no—
362Q8D3×D811yescharacteristic
363D4—21no—
364D4—22no—
365D4—31no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—61no—
370D4—82no—
371D4—82no—
372D4—122no—
373D4—122no—
374D4—242no—
375D4—242no—
376D4—484no—
377D4D3×D811yescharacteristic
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—82no—
383C2×C4—242no—
384C2×C4C2×D3×D411yescharacteristic
385C8—21no—
386C8—31no—
387C8—31no—
388C8—61no—
389C8—122no—
390C8#100411yescharacteristic
391C8C22×#6711yescharacteristic
392C2×C3—21no—
393C2×C3—41no—
394C2×C3—82no—
395C2×C3—162no—
396C2×C3D4×D811yescharacteristic
397D3—41no—
398D3—162no—
399C22—21no—
400C22—42no—
401C22—61no—
402C22—82no—
403C22—162no—
404C22—242no—
405C22—322no—
406C22—482no—
407C22—962no—
408C22#575711yescharacteristic
409C4—21no—
410C4—42no—
411C4—62no—
412C4—61no—
413C4C2×D3×D811yescharacteristic
414C4D4×#6711yescharacteristic
415C3#3385811yescharacteristic, complemented, Sylow
416C2—21no—
417C2—41no—
418C2—82no—
419C2—121no—
420C2—162no—
421C2—482no—
422C2D8×#6711yescharacteristic
subgroupG/H
centerC2D8×#67
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3415C2
socleC2×C3D4×D8
hypercenter#3415D3
hypocenterC3#33858
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #3415

Complements

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

NH ≅ G/Ntype
1. C3×#3415C2semidirect
2. #84010C2semidirect
3. #82031C2semidirect
4. #72122C2semidirect
5. #71517C2semidirect
6. #71514C2semidirect
7. #71511C2semidirect
8. #71472C2semidirect
9. #71464C2semidirect
10. #71454C2semidirect
11. #71452C2semidirect
13. C3×#576C22semidirect
15. #5769C22semidirect
17. C3×#509C22semidirect
21. C2×C3×D16C22semidirect
22. C3×#504C22semidirect
26. #5196C22semidirect
27. #5193C22semidirect
28. #5160C22semidirect
29. #5151C22semidirect
30. C2×#826C22semidirect
32. #5142C22semidirect
36. C2×D48C22semidirect
37. #4901C22semidirect
38. #4567C22semidirect
39. #4563C22semidirect
40. #4561C22semidirect
41. #4516C22semidirect
42. #4515C22semidirect
43. #4511C22semidirect
44. #4506C22semidirect
45. Dic3×C16C22semidirect
47. #3415D3semidirect
82. C3×Q32D4semidirect
84. C3×SD32D4semidirect
88. C3×D16D4semidirect
112. #576C2×D3semidirect
116. #509C2×D3semidirect
124. C2×D16C2×D3semidirect
126. #504C2×D3semidirect
222. Q32#67semidirect
225. SD32#67semidirect
233. D16#67semidirect
415. C3#33858semidirect

Maximal subgroups

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

subgroupindex
C3×#34152
#840102
#820312
#721222
#715172
#715142
#715112
#714722
#714642
#714542
#714522
#338583