Finite Group Databasebeta

← #80280

Subgroups — #80280

Order 384.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5836C211yesmaximal, characteristic, complemented
2#5835C211yesmaximal, characteristic, complemented
3#5775C211yesmaximal, characteristic, complemented
4#5774C211yesmaximal, characteristic, complemented
5C3×#453C211yesmaximal, characteristic, complemented
6#5200C211yesmaximal, characteristic, complemented
7#5199C211yesmaximal, characteristic, complemented
8#5077C211yesmaximal, characteristic, complemented
9#5074C211yesmaximal, characteristic, complemented
10#4827C211yesmaximal, characteristic, complemented
11#4825C211yesmaximal, characteristic, complemented
12#4823C211yesmaximal, characteristic, complemented
13#4823C211yesmaximal, characteristic, complemented
14#4749C211yesmaximal, characteristic, complemented
15#4748C211yesmaximal, characteristic, complemented
16#3288—31nomaximal, Sylow
17C3×H(5,𝔽2)C2211yescharacteristic, complemented
18#1010C2211yescharacteristic, complemented
19#1009C2211yescharacteristic
20#1004C2211yescharacteristic, complemented
21C3×AGL(1,ℤ/8)C2211yescharacteristic
22C3×AGL(1,ℤ/8)C2211yescharacteristic, complemented
23C3×#127C2211yescharacteristic, complemented
24#951—21no—
25#951C2211yescharacteristic, complemented
26#950—21no—
27#950—21no—
28#950C2211yescharacteristic
29#950C2211yescharacteristic
30#949—21no—
31#949C2211yescharacteristic, complemented
32#948C2211yescharacteristic, complemented
33#948C2211yescharacteristic, complemented
34C2×#214—21no—
35C2×#214C2211yescharacteristic, complemented
36C2×#214C2211yescharacteristic, complemented
37#932—21no—
38#932—21no—
39#932C2211yescharacteristic
40#932C2211yescharacteristic, complemented
41C2×#213—21no—
42C2×#213C2211yescharacteristic, complemented
43C2×#213C2211yescharacteristic, complemented
44#916—21no—
45#915—21no—
46#914—21no—
47D3×SD16—21no—
48#912—21no—
49#911—21no—
50#911—21no—
51D3×D8—21no—
52#909C2211yescharacteristic, complemented
53#908C2211yescharacteristic, complemented
54#907C2211yescharacteristic, complemented
55#907C2211yescharacteristic, complemented
56#875C2211yescharacteristic, complemented
57C3×#104C2211yescharacteristic, complemented
58C3×#104C2211yescharacteristic, complemented
59C3×#100C2211yescharacteristic, complemented
60#833C2211yescharacteristic, complemented
61#824C2211yescharacteristic, complemented
62#823C2211yescharacteristic, complemented
63#822C2211yescharacteristic, complemented
64#822C2211yescharacteristic, complemented
65#805C2211yescharacteristic, complemented
66#805C2211yescharacteristic, complemented
67#803C2211yescharacteristic, complemented
68#577—31no—
69#577—31no—
70#576—31no—
71#576—31no—
72#496—31no—
73#472—31no—
74#472—31no—
75#471—31no—
76#471—31no—
77#455—31no—
78#454—31no—
79#453—31no—
80#453D311yescharacteristic, complemented
81#444—31no—
82#444—31no—
83C3×#41—21no—
84C3×#41—21no—
85C3×#41C2311yescharacteristic
86C3×#41C2311yescharacteristic
87C2×C3×D4—21no—
88C2×C3×D4—21no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4C2311yescharacteristic
92C2×C3×D4C2311yescharacteristic
93C2×#67—41no—
94#239—21no—
95D3×Q8—21no—
96#237—21no—
97#237—21no—
98#237—21no—
99#237—41no—
100D3×D4—21no—
101D3×D4—21no—
102D3×D4—21no—
103D3×D4—41no—
104#235—21no—
105#235—21no—
106#235C2311yescharacteristic
107#235C2311yescharacteristic
108C2×D12—21no—
109C2×D12C2311yescharacteristic
110C2×Dic6—21no—
111C2×Dic6C2311yescharacteristic
112C3×SD16—21no—
113C3×SD16—21no—
114C3×D8—21no—
115C3×D8—21no—
116C3×M16C2311yescharacteristic
117C3×M16C2311yescharacteristic
118C3×C42C2311yescharacteristic
119#216—21no—
120#216D411yescharacteristic, complemented
121#216D411yescharacteristic, complemented
122#215—21no—
123#215D411yescharacteristic, complemented
124#215D411yescharacteristic, complemented
125#214—21no—
126#214—21no—
127#214—21no—
128#214—21no—
129#214—21no—
130#214—21no—
131#214—21no—
132#214D411yescharacteristic, complemented
133#214D411yescharacteristic, complemented
134#213—21no—
135#213—21no—
136#213—21no—
137#213—21no—
138#213—21no—
139#213—21no—
140#213—21no—
141#213D411yescharacteristic, complemented
142#213D411yescharacteristic, complemented
143(C2×D3)⋊C4—41no—
144#208—21no—
145#208—21no—
146#208C2311yescharacteristic
147#208C2311yescharacteristic
148C2×C3⋊C8—21no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8C2311yescharacteristic
151C2×C3⋊C8C2311yescharacteristic
152Dic12—21no—
153D24—21no—
154#204—21no—
155#204—21no—
156ΓL(1,𝔽25)—21no—
157ΓL(1,𝔽25)—21no—
158D3×C8—21no—
159D3×C8—21no—
1602−1+4—31no—
161H(5,𝔽2)—31no—
162H(5,𝔽2)—31no—
163H(5,𝔽2)C2×D311yescharacteristic, complemented
164#137—31no—
165#137—31no—
166#137—61no—
167#137—61no—
168AGL(1,ℤ/8)—31no—
169AGL(1,ℤ/8)—31no—
170AGL(1,ℤ/8)—31no—
171AGL(1,ℤ/8)—31no—
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
179AGL(1,ℤ/8)C2×D311yescharacteristic, complemented
180#135—31no—
181#135—31no—
182#135—61no—
183#135—61no—
184#135—61no—
185#135—61no—
186C2×SD16—31no—
187C2×SD16—31no—
188C2×SD16—61no—
189C2×SD16—61no—
190C2×D8—31no—
191C2×D8—31no—
192C2×D8—61no—
193C2×D8—61no—
194#131—31no—
195#131—31no—
196#131—31no—
197#131—31no—
198#127C2×D311yescharacteristic, complemented
199#124—31no—
200#108—31no—
201#108—31no—
202#104—31no—
203#104—31no—
204#104C2×D311yescharacteristic, complemented
205#104C2×D311yescharacteristic, complemented
206#101—31no—
207#100—31no—
208#100—31no—
209#100C2×D311yescharacteristic, complemented
210#97—31no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C22×D3—21no—
215C22×D3—41no—
216C22×D3—41no—
217C3×Q8C2×D411yescharacteristic
218C3×Q8C2×D411yescharacteristic
219C3×D4—21no—
220C3×D4—21no—
221C3×D4—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—21no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—82no—
228C3×D4C2×D411yescharacteristic
229C3×D4C2×D411yescharacteristic
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4C2411yescharacteristic
235#67—21no—
236#67—21no—
237#67—41no—
238#67—41no—
239#67—41no—
240#67—41no—
241C2×Dic3—21no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244D12—21no—
245D12—21no—
246D12—41no—
247D12C2×D411yescharacteristic
248D12C2×D411yescharacteristic
249C4×D3—21no—
250C4×D3—21no—
251C4×D3—41no—
252C4×D3—41no—
253Dic6—21no—
254Dic6—21no—
255Dic6—41no—
256Dic6C2×D411yescharacteristic
257Dic6C2×D411yescharacteristic
258C3×C8—21no—
259C3×C8—21no—
260C3⋊C8—21no—
261C3⋊C8—21no—
262C3⋊C8C2×D411yescharacteristic
263C3⋊C8C2×D411yescharacteristic
264C3⋊C8C2×D411yescharacteristic
265C3⋊C8C2×D411yescharacteristic
266#41—21no—
267#41—21no—
268#41—31no—
269#41—31no—
270#41—61no—
271#41—61no—
272#41—61no—
273#41—61no—
274#41—61no—
275#41—61no—
276#41—121no—
277#41C22×D311yescharacteristic
278#41C22×D311yescharacteristic
279C2×Q8—31no—
280C2×Q8—61no—
281C2×Q8—61no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—31no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—121no—
292C2×D4—121no—
293C2×D4C22×D311yescharacteristic
294C2×D4C22×D311yescharacteristic
295Q16—31no—
296Q16—31no—
297Q16—61no—
298Q16—61no—
299SD16—21no—
300SD16—21no—
301SD16—31no—
302SD16—31no—
303SD16—31no—
304SD16—31no—
305SD16—61no—
306SD16—61no—
307SD16—61no—
308SD16—61no—
309SD16—61no—
310SD16—61no—
311SD16—61no—
312SD16—61no—
313SD16—61no—
314SD16—61no—
315D8—21no—
316D8—21no—
317D8—31no—
318D8—31no—
319D8—61no—
320D8—61no—
321D8—61no—
322D8—61no—
323D8—61no—
324D8—61no—
325D8—61no—
326D8—61no—
327M16—31no—
328M16—31no—
329M16—61no—
330M16—61no—
331M16—61no—
332M16—61no—
333M16C22×D311yescharacteristic
334M16C22×D311yescharacteristic
335C2×C8—31no—
336C2×C8—31no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C22⋊C4—121no—
342C42C22×D311yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—161no—
350C22×C3H(5,𝔽2)11yescharacteristic
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—41no—
354C2×D3—81no—
355C2×D3—161no—
356C2×D3—161no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—41no—
360C3×C4C22×D411yescharacteristic
361C3×C4C22×D411yescharacteristic
362Dic3—21no—
363Dic3—21no—
364Dic3—41no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—61no—
369C23—121no—
370C23—121no—
371Q8—31no—
372Q8—31no—
373Q8—61no—
374Q8—61no—
375Q8—121no—
376Q8D3×D411yescharacteristic
377Q8D3×D411yescharacteristic
378D4—21no—
379D4—21no—
380D4—21no—
381D4—21no—
382D4—21no—
383D4—21no—
384D4—31no—
385D4—31no—
386D4—41no—
387D4—41no—
388D4—61no—
389D4—61no—
390D4—61no—
391D4—61no—
392D4—82no—
393D4—121no—
394D4—121no—
395D4—121no—
396D4—121no—
397D4—121no—
398D4D3×D411yescharacteristic
399D4D3×D411yescharacteristic
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4C23×D311yescharacteristic
412C8—21no—
413C8—21no—
414C8—31no—
415C8—31no—
416C8—31no—
417C8—31no—
418C8—61no—
419C8—61no—
420C2×C3—21no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—81no—
424C2×C3—81no—
425C2×C3D4211yescharacteristic
426D3—41no—
427D3—41no—
428D3—81no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—61no—
434C22—61no—
435C22—81no—
436C22—81no—
437C22—121no—
438C22—161no—
439C22—241no—
440C22—481no—
441C22—481no—
442C22#100411yescharacteristic
443C4—21no—
444C4—21no—
445C4—41no—
446C4—61no—
447C4—61no—
448C4—121no—
449C4C2×D3×D411yescharacteristic
450C4C2×D3×D411yescharacteristic
451C3#328811yescharacteristic, complemented, Sylow
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—121no—
458C2—121no—
459C2—241no—
460C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#453C2
socleC2×C3D42
hypercenter#453D3
hypocenterC3#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #5836C2semidirect
2. #5835C2semidirect
3. #5775C2semidirect
4. #5774C2semidirect
5. C3×#453C2semidirect
6. #5200C2semidirect
7. #5199C2semidirect
8. #5077C2semidirect
9. #5074C2semidirect
10. #4827C2semidirect
11. #4825C2semidirect
12. #4823C2semidirect
13. #4823C2semidirect
14. #4749C2semidirect
15. #4748C2semidirect
17. C3×H(5,𝔽2)C22semidirect
18. #1010C22semidirect
20. #1004C22semidirect
22. C3×AGL(1,ℤ/8)C22semidirect
23. C3×#127C22semidirect
25. #951C22semidirect
31. #949C22semidirect
32. #948C22semidirect
33. #948C22semidirect
35. C2×#214C22semidirect
36. C2×#214C22semidirect
40. #932C22semidirect
42. C2×#213C22semidirect
43. C2×#213C22semidirect
52. #909C22semidirect
53. #908C22semidirect
54. #907C22semidirect
55. #907C22semidirect
56. #875C22semidirect
57. C3×#104C22semidirect
58. C3×#104C22semidirect
59. C3×#100C22semidirect
60. #833C22semidirect
61. #824C22semidirect
62. #823C22semidirect
63. #822C22semidirect
64. #822C22semidirect
65. #805C22semidirect
66. #805C22semidirect
67. #803C22semidirect
80. #453D3semidirect
120. #216D4semidirect
121. #216D4semidirect
123. #215D4semidirect
124. #215D4semidirect
132. #214D4semidirect
133. #214D4semidirect
141. #213D4semidirect
142. #213D4semidirect
163. H(5,𝔽2)C2×D3semidirect
179. AGL(1,ℤ/8)C2×D3semidirect
198. #127C2×D3semidirect
204. #104C2×D3semidirect
205. #104C2×D3semidirect
209. #100C2×D3semidirect
451. C3#3288semidirect

Maximal subgroups

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

subgroupindex
#58362
#58352
#57752
#57742
C3×#4532
#52002
#51992
#50772
#50742
#48272
#48252
#48232
#47492
#47482
#32883