Finite Group Databasebeta

← #80930

Subgroups — #80930

Order 384.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5777C211yesmaximal, characteristic, complemented
2D3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5750C211yesmaximal, characteristic, complemented
5C3×#471C211yesmaximal, characteristic, complemented
6#5203C211yesmaximal, characteristic, complemented
7#5199C211yesmaximal, characteristic, complemented
8#5177C211yesmaximal, characteristic, complemented
9#5142C211yesmaximal, characteristic, complemented
10#4897C211yesmaximal, characteristic, complemented
11#4893C211yesmaximal, characteristic, complemented
12#4754C211yesmaximal, characteristic, complemented
13#4751C211yesmaximal, characteristic, complemented
14#4749C211yesmaximal, characteristic, complemented
15D3×#100C211yesmaximal, characteristic, complemented
16#3068—31nomaximal, Sylow
17D3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C3×#137C2211yescharacteristic, complemented
21C3×AGL(1,ℤ/8)C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic, complemented
23C3×#131C2211yescharacteristic, complemented
24#949C2211yescharacteristic
25#949C2211yescharacteristic, complemented
26#948C2211yescharacteristic, complemented
27#942C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic, complemented
29#932C2211yescharacteristic
30C2×#213C2211yescharacteristic, complemented
31#919—21no—
32#918—21no—
33#916—21no—
34#914—21no—
35#914—21no—
36#914—21no—
37#914—21no—
38#914—21no—
39#914—21no—
40D3×SD16—21no—
41#911—21no—
42D3×D8—21no—
43#908C2211yescharacteristic
44#908C2211yescharacteristic, complemented
45#907—21no—
46#907C2211yescharacteristic, complemented
47D3×M16—21no—
48D3×M16C2211yescharacteristic
49D3×M16C2211yescharacteristic
50D3×M16C2211yescharacteristic
51C2×D24C2211yescharacteristic, complemented
52#901—21no—
53#901C2211yescharacteristic, complemented
54C2×ΓL(1,𝔽25)—21no—
55C2×ΓL(1,𝔽25)C2211yescharacteristic
56C3×#108C2211yescharacteristic, complemented
57C3×#101C2211yescharacteristic, complemented
58C3×#100C2211yescharacteristic, complemented
59#836C2211yescharacteristic, complemented
60#833C2211yescharacteristic, complemented
61#823C2211yescharacteristic, complemented
62#823C2211yescharacteristic, complemented
63#823C2211yescharacteristic, complemented
64#823C2211yescharacteristic, complemented
65#822C2211yescharacteristic, complemented
66#822C2211yescharacteristic, complemented
67#819C2211yescharacteristic, complemented
68#575—31no—
69C2×AGL(1,ℤ/8)—31no—
70C2×AGL(1,ℤ/8)—31no—
71#568—31no—
72#472—31no—
73#472—31no—
74#472—31no—
75#472—31no—
76#471—31no—
77#471—31no—
78#471—31no—
79#471D311yescharacteristic, complemented
80#430—31no—
81#413—31no—
82C2×#100—31no—
83C23×D3—21no—
84C3×#41C2311yescharacteristic
85C2×C3×Q8C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C2×#67—21no—
88#239—21no—
89#239—21no—
90#239—21no—
91#239—21no—
92#239D411yescharacteristic
93D3×Q8D411yescharacteristic
94#237—21no—
95#237D411yescharacteristic
96D3×D4—21no—
97D3×D4—21no—
98D3×D4—21no—
99D3×D4—21no—
100D3×D4—21no—
101D3×D4D411yescharacteristic
102#235—21no—
103#235C2311yescharacteristic
104C2×D12—21no—
105C2×D12C2311yescharacteristic
106C2×D12C2311yescharacteristic
107C2×C4×D3—21no—
108C2×C4×D3—21no—
109C2×C4×D3C2311yescharacteristic
110C3×Q16—21no—
111C3×SD16—21no—
112C3×SD16—21no—
113C3×SD16—21no—
114C3×SD16—21no—
115C3×D8—21no—
116C3×M16—21no—
117C3×M16C2311yescharacteristic
118C3×M16C2311yescharacteristic
119C3×M16C2311yescharacteristic
120C2×C3×C8—21no—
121C2×C3×C8C2311yescharacteristic
122#216—21no—
123#215—21no—
124#215—21no—
125#215—21no—
126#215—21no—
127#215—21no—
128#214—21no—
129#213—21no—
130#213—21no—
131#213—21no—
132#213—21no—
133#213—21no—
134#208—21no—
135#208C2311yescharacteristic
136#208C2311yescharacteristic
137#208C2311yescharacteristic
138C2×C3⋊C8—21no—
139C2×C3⋊C8C2311yescharacteristic
140D24—21no—
141D24—21no—
142D24—21no—
143D24—21no—
144#204—21no—
145#204—21no—
146ΓL(1,𝔽25)—21no—
147ΓL(1,𝔽25)—21no—
148ΓL(1,𝔽25)—21no—
149ΓL(1,𝔽25)—21no—
150ΓL(1,𝔽25)—21no—
151ΓL(1,𝔽25)D411yescharacteristic
152ΓL(1,𝔽25)D411yescharacteristic
153ΓL(1,𝔽25)D411yescharacteristic
154ΓL(1,𝔽25)D411yescharacteristic
155D3×C8—21no—
156D3×C8—21no—
157D3×C8—21no—
158D3×C8—21no—
159D3×C8—21no—
160C2×#41—31no—
161C2×#41—31no—
162C22×D4—31no—
163#137—31no—
164#137—61no—
165#137C2×D311yescharacteristic, complemented
166AGL(1,ℤ/8)—31no—
167AGL(1,ℤ/8)—31no—
168AGL(1,ℤ/8)—31no—
169AGL(1,ℤ/8)—31no—
170AGL(1,ℤ/8)—31no—
171AGL(1,ℤ/8)—61no—
172AGL(1,ℤ/8)—61no—
173AGL(1,ℤ/8)—61no—
174AGL(1,ℤ/8)—61no—
175AGL(1,ℤ/8)—61no—
176AGL(1,ℤ/8)—61no—
177AGL(1,ℤ/8)—61no—
178AGL(1,ℤ/8)C2×D311yescharacteristic
179#135—61no—
180#135—61no—
181C2×SD16—31no—
182C2×SD16—61no—
183C2×SD16C2×D311yescharacteristic, complemented
184C2×D8—31no—
185C2×D8—31no—
186C2×D8—61no—
187#131—31no—
188#131—31no—
189#131—31no—
190#131—61no—
191#131—61no—
192#131C2×D311yescharacteristic, complemented
193C2×M16—31no—
194C2×M16—31no—
195C2×M16—31no—
196C2×M16—31no—
197C2×M16—61no—
198C2×M16—61no—
199#108—31no—
200#108—31no—
201#108—31no—
202#108C2×D311yescharacteristic, complemented
203#101—31no—
204#101C2×D311yescharacteristic, complemented
205#100—31no—
206#100—31no—
207#100—31no—
208#100—31no—
209#100—31no—
210#100C2×D311yescharacteristic, complemented
211C23×C3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—82no—
218C22×D3#4111yescharacteristic
219C3×Q8—21no—
220C3×Q8—21no—
221C3×Q8C2×D411yescharacteristic
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4C2×D411yescharacteristic
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4C2411yescharacteristic
229#67—21no—
230#67—21no—
231#67—21no—
232#67—41no—
233#67—41no—
234C2×Dic3—21no—
235C2×Dic3#4111yescharacteristic
236D12—21no—
237D12—21no—
238D12—21no—
239D12—21no—
240D12—21no—
241D12—41no—
242D12—41no—
243D12C2×D411yescharacteristic
244C4×D3—21no—
245C4×D3—21no—
246C4×D3—21no—
247C4×D3—41no—
248C4×D3—41no—
249C4×D3C2×D411yescharacteristic
250C4×D3C2×D411yescharacteristic
251C4×D3C2×D411yescharacteristic
252C4×D3C2×D411yescharacteristic
253Dic6—21no—
254Dic6C2×D411yescharacteristic
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8C2×D411yescharacteristic
259C3×C8C2×D411yescharacteristic
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8—21no—
263C3⋊C8C2×D411yescharacteristic
264C3⋊C8C2×D411yescharacteristic
265C24—61no—
266#41—31no—
267#41—31no—
268#41—31no—
269#41—61no—
270#41—61no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41C22×D311yescharacteristic
276C2×Q8—31no—
277C2×Q8C22×D311yescharacteristic
278C2×D4—31no—
279C2×D4—31no—
280C2×D4—31no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4C22×D311yescharacteristic
289C22×C4—31no—
290C22×C4—61no—
291C22×C4—61no—
292Q16—21no—
293Q16—61no—
294SD16—21no—
295SD16—21no—
296SD16—21no—
297SD16—21no—
298SD16—61no—
299SD16—61no—
300SD16—61no—
301SD16—61no—
302SD16—61no—
303SD16—61no—
304SD16—61no—
305SD16—61no—
306D8—21no—
307D8—61no—
308D8—61no—
309D8—61no—
310D8—61no—
311D8—61no—
312D8—61no—
313D8—61no—
314D8—61no—
315D8—61no—
316M16—21no—
317M16—31no—
318M16—31no—
319M16—31no—
320M16—31no—
321M16—31no—
322M16—31no—
323M16—31no—
324M16—61no—
325M16—61no—
326M16—61no—
327M16—61no—
328M16—61no—
329M16—61no—
330M16C22×D311yescharacteristic
331M16C22×D311yescharacteristic
332M16C22×D311yescharacteristic
333C2×C8—21no—
334C2×C8—31no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8C22×D311yescharacteristic
342C22×C3—21no—
343C22×C3—41no—
344C22×C3—81no—
345C22×C3C2×#4111yescharacteristic
346C2×D3—21no—
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—81no—
351C2×D3—82no—
352C2×D3—82no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3#12111yescharacteristic
356C2×D3#12111yescharacteristic
357C3×C4—21no—
358C3×C4—41no—
359C3×C4C22×D411yescharacteristic
360C3×C4C22×D411yescharacteristic
361Dic3—21no—
362Dic3#12111yescharacteristic
363Dic3#12111yescharacteristic
364C23—21no—
365C23—31no—
366C23—61no—
367C23—61no—
368C23—61no—
369C23—121no—
370C23—121no—
371C23—242no—
372Q8—21no—
373Q8—21no—
374Q8—31no—
375Q8—61no—
376Q8D3×D411yescharacteristic
377D4—21no—
378D4—21no—
379D4—21no—
380D4—31no—
381D4—61no—
382D4—61no—
383D4—61no—
384D4—61no—
385D4—61no—
386D4—61no—
387D4—61no—
388D4—61no—
389D4—121no—
390D4—121no—
391D4—121no—
392D4—121no—
393D4D3×D411yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—31no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—121no—
406C2×C4—121no—
407C2×C4C23×D311yescharacteristic
408C8—21no—
409C8—21no—
410C8—21no—
411C8—31no—
412C8—31no—
413C8—61no—
414C8—61no—
415C8—61no—
416C8D3×D411yescharacteristic
417C8D3×D411yescharacteristic
418C2×C3—21no—
419C2×C3—41no—
420C2×C3—81no—
421C2×C3C2×#12111yescharacteristic
422D3—21no—
423D3—21no—
424D3—41no—
425D3—81no—
426D3—81no—
427C22—21no—
428C22—31no—
429C22—31no—
430C22—41no—
431C22—61no—
432C22—81no—
433C22—121no—
434C22—121no—
435C22—121no—
436C22—241no—
437C22—241no—
438C22—241no—
439C22—242no—
440C22—242no—
441C22D3×#4111yescharacteristic
442C4—21no—
443C4—31no—
444C4—31no—
445C4—41no—
446C4—61no—
447C4C2×D3×D411yescharacteristic
448C4C2×D3×D411yescharacteristic
449C3#306811yescharacteristic, complemented, Sylow
450C2—21no—
451C2—41no—
452C2—61no—
453C2—61no—
454C2—81no—
455C2—121no—
456C2—241no—
457C2—241no—
458C2D3×#12111yescharacteristic
subgroupG/H
centerC2D3×#121
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#471C2
socleC2×C3C2×#121
hypercenter#471D3
hypocenterC3#3068
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #5777C2semidirect
2. D3×AGL(1,ℤ/8)C2semidirect
3. C2×#914C2semidirect
4. #5750C2semidirect
5. C3×#471C2semidirect
6. #5203C2semidirect
7. #5199C2semidirect
8. #5177C2semidirect
9. #5142C2semidirect
10. #4897C2semidirect
11. #4893C2semidirect
12. #4754C2semidirect
13. #4751C2semidirect
14. #4749C2semidirect
15. D3×#100C2semidirect
20. C3×#137C22semidirect
22. C2×C3×SD16C22semidirect
23. C3×#131C22semidirect
25. #949C22semidirect
26. #948C22semidirect
27. #942C22semidirect
28. C2×#215C22semidirect
30. C2×#213C22semidirect
44. #908C22semidirect
46. #907C22semidirect
51. C2×D24C22semidirect
53. #901C22semidirect
56. C3×#108C22semidirect
57. C3×#101C22semidirect
58. C3×#100C22semidirect
59. #836C22semidirect
60. #833C22semidirect
61. #823C22semidirect
62. #823C22semidirect
63. #823C22semidirect
64. #823C22semidirect
65. #822C22semidirect
66. #822C22semidirect
67. #819C22semidirect
79. #471D3semidirect
165. #137C2×D3semidirect
183. C2×SD16C2×D3semidirect
192. #131C2×D3semidirect
202. #108C2×D3semidirect
204. #101C2×D3semidirect
210. #100C2×D3semidirect
449. C3#3068semidirect

Maximal subgroups

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

subgroupindex
#57772
D3×AGL(1,ℤ/8)2
C2×#9142
#57502
C3×#4712
#52032
#51992
#51772
#51422
#48972
#48932
#47542
#47512
#47492
D3×#1002
#30683