Finite Group Databasebeta

← #24031

Subgroups — #24031

Order 256.

667 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3488C211yesmaximal, characteristic
2#3093C211yesmaximal, characteristic, complemented
3#2896C211yesmaximal, characteristic, complemented
4#2500C211yesmaximal, characteristic, complemented
5#2040C211yesmaximal, characteristic, complemented
6#2039C211yesmaximal, characteristic, complemented
7#2037C211yesmaximal, characteristic, complemented
8#2037C211yesmaximal, characteristic, complemented
9#1900C211yesmaximal, characteristic, complemented
10#1871C211yesmaximal, characteristic, complemented
11#1860C211yesmaximal, characteristic, complemented
12#1859C211yesmaximal, characteristic, complemented
13#1792C211yesmaximal, characteristic, complemented
14#1792C211yesmaximal, characteristic, complemented
15#1789C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C22×M16C2211yescharacteristic, complemented
19#550—21no—
20#548—21no—
21#546—21no—
22#546—22no—
23D42—22no—
24D42—21no—
25#537—21no—
26#537C2211yescharacteristic
27#535—21no—
28#535C2211yescharacteristic
29#525—21no—
30#525C2211yescharacteristic
31C2×#121—21no—
32C2×#121C2211yescharacteristic
33#520C2211yescharacteristic
34C2×#117C2211yescharacteristic, complemented
35#485—21no—
36#485—21no—
37#482—21no—
38#482—21no—
39#428C2211yescharacteristic, complemented
40#428C2211yescharacteristic, complemented
41C2×#104C2211yescharacteristic
42C2×#104C2211yescharacteristic
43C2×#104C2211yescharacteristic, complemented
44C2×#104C2211yescharacteristic, complemented
45C2×#104C2211yescharacteristic, complemented
46C2×#104C2211yescharacteristic, complemented
47#419—21no—
48#418—21no—
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic, complemented
52#417C2211yescharacteristic, complemented
53#417C2211yescharacteristic, complemented
54#417C2211yescharacteristic, complemented
55#416—21no—
56#416—21no—
57C2×#100C2211yescharacteristic, complemented
58C2×#100C2211yescharacteristic, complemented
59#407C2211yescharacteristic, complemented
60#394—22no—
61#390C2211yescharacteristic, complemented
62#390C2211yescharacteristic, complemented
63#388—22no—
64#384C2211yescharacteristic, complemented
65#382C2211yescharacteristic, complemented
66C4×C22⋊C4C2211yescharacteristic, complemented
67#343C2211yescharacteristic, complemented
68#343C2211yescharacteristic, complemented
69#339C2211yescharacteristic, complemented
70#339C2211yescharacteristic, complemented
71H(5,𝔽2)—22no—
72H(5,𝔽2)—22no—
73H(5,𝔽2)—41no—
74H(5,𝔽2)—41no—
75C2×#41—21no—
76C2×#41—21no—
77C2×#41—21no—
78C2×#41—21no—
79C2×#41—22no—
80C2×#41—22no—
81C2×#41—41no—
82C2×#41—41no—
83C2×#41D411yescharacteristic
84C2×#41D411yescharacteristic
85C2×#41C2311yescharacteristic
86C2×#41C2311yescharacteristic
87C2×#41C2311yescharacteristic
88C22×Q8D411yescharacteristic
89C22×D4—21no—
90C22×D4—22no—
91C22×D4—22no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4D411yescharacteristic
95C22×D4D412yes—
96C22×D4C2311yescharacteristic
97C23×C4—21no—
98C23×C4C2311yescharacteristic
99C2×M16—21no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16C2311yescharacteristic
103C2×M16C2311yescharacteristic
104C2×M16C2311yescharacteristic
105C2×M16C2311yescharacteristic
106C22×C8—21no—
107#127—22no—
108#127—21no—
109#124—22no—
110#124—21no—
111#123—22no—
112#123—21no—
113C22⋊Q8—22no—
114C22⋊Q8—21no—
115C22⋊Q8—21no—
116C22⋊Q8D411yescharacteristic
117C22⋊Q8D411yescharacteristic
118#121—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—21no—
127#121—21no—
128#121—21no—
129#121—22no—
130#121—21no—
131#121—21no—
132#121D411yescharacteristic
133#121D411yescharacteristic
134#120—22no—
135#120—22no—
136#120—22no—
137#120—21no—
138#120—22no—
139C4×Q8—21no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—21no—
143C4×D4—22no—
144C4×D4—21no—
145C4×D4—21no—
146#117—21no—
147#117—21no—
148#117—21no—
149#117—21no—
150#117—21no—
151#117C2311yescharacteristic
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4C2311yescharacteristic
155C2×C4⋊C4C2311yescharacteristic
156C2×C4⋊C4C2311yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4D412yes—
163C2×C42—21no—
164C2×C42C2311yescharacteristic
165C2×C42C2311yescharacteristic
166#107—21no—
167#107—21no—
168#106—21no—
169#106—21no—
170#104—22no—
171#104—22no—
172#104—41no—
173#104—41no—
174#104—41no—
175#104—41no—
176#103—21no—
177#103—21no—
178#103—21no—
179#103—21no—
180#103—21no—
181#103—21no—
182#102—21no—
183#102—21no—
184#102—21no—
185#102—21no—
186#102—21no—
187#102—21no—
188#100—41no—
189#100—41no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192#95—21no—
193#95—22no—
194C24—22no—
195C24—21no—
196C24C2×D412yes—
197C24C2×D411yescharacteristic
198#41—21no—
199#41—21no—
200#41—22no—
201#41—22no—
202#41—21no—
203#41—21no—
204#41—21no—
205#41—21no—
206#41—21no—
207#41—21no—
208#41—21no—
209#41—21no—
210#41—42no—
211#41—42no—
212#41—41no—
213#41—41no—
214#41—42no—
215#41—42no—
216#41—42no—
217#41—42no—
218#41—42no—
219#41—42no—
220#41#4112yes—
221#41#4112yes—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8C2×D411yescharacteristic
226C2×Q8C2×D411yescharacteristic
227C2×Q8C2×D411yescharacteristic
228C2×D4—21no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—22no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—82no—
260C2×D4—84no—
261C2×D4C2×D411yescharacteristic
262C2×D4C2×D411yescharacteristic
263C2×D4C2×D411yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4C2×D411yescharacteristic
283C22×C4C2×D411yescharacteristic
284C22×C4C2×D411yescharacteristic
285C22×C4C2×D411yescharacteristic
286C22×C4C2×D411yescharacteristic
287C22×C4C2×D412yes—
288C22×C4C2411yescharacteristic
289M16—21no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—21no—
296M16—21no—
297M16—42no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—82no—
314C4⋊C4C2×D411yescharacteristic
315C4⋊C4C2×D411yescharacteristic
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—21no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—82no—
333C42—22no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—41no—
340C42—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—82no—
358C23—82no—
359C23#12011yescharacteristic
360C23#12011yescharacteristic
361C23C22×D411yescharacteristic
362Q8—21no—
363Q8—21no—
364Q8—21no—
365Q8—21no—
366Q8—21no—
367Q8—42no—
368Q8#12312yes—
369D4—22no—
370D4—22no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—21no—
375D4—21no—
376D4—42no—
377D4—42no—
378D4—44no—
379D4—42no—
380D4—44no—
381D4—44no—
382D4—42no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4#12312yes—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4#12011yescharacteristic
436C2×C4#12011yescharacteristic
437C2×C4C22×D411yescharacteristic
438C2×C4C22×D411yescharacteristic
439C2×C4C2×#4111yescharacteristic
440C2×C4C2×#4111yescharacteristic
441C2×C4H(5,𝔽2)11yescharacteristic
442C2×C4H(5,𝔽2)11yescharacteristic
443C8—41no—
444C8—41no—
445C8—41no—
446C8—41no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—42no—
461C22—41no—
462C22—42no—
463C22—41no—
464C22—42no—
465C22—41no—
466C22—41no—
467C22—81no—
468C22—81no—
469C22—82no—
470C22—82no—
471C22—84no—
472C22—164no—
473C22C2×#12011yescharacteristic
474C22#54611yescharacteristic
475C22#54611yescharacteristic
476C4—21no—
477C4—21no—
478C4—22no—
479C4—42no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—42no—
484C4—41no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4C2×#12311yescharacteristic
489C4#53411yescharacteristic
490C4#54611yescharacteristic
491C4#54611yescharacteristic
492C2—21no—
493C2—21no—
494C2—41no—
495C2—41no—
496C2—42no—
497C2—81no—
498C2—82no—
499C2—81no—
500C2#240511yescharacteristic
501C2#301811yescharacteristic
502C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3488C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3093C2semidirect
3. #2896C2semidirect
4. #2500C2semidirect
5. #2040C2semidirect
6. #2039C2semidirect
7. #2037C2semidirect
8. #2037C2semidirect
9. #1900C2semidirect
10. #1871C2semidirect
11. #1860C2semidirect
12. #1859C2semidirect
13. #1792C2semidirect
14. #1792C2semidirect
15. #1789C2semidirect
18. C22×M16C22semidirect
34. C2×#117C22semidirect
39. #428C22semidirect
40. #428C22semidirect
43. C2×#104C22semidirect
44. C2×#104C22semidirect
45. C2×#104C22semidirect
46. C2×#104C22semidirect
51. #417C22semidirect
52. #417C22semidirect
53. #417C22semidirect
54. #417C22semidirect
57. C2×#100C22semidirect
58. C2×#100C22semidirect
59. #407C22semidirect
61. #390C22semidirect
62. #390C22semidirect
64. #384C22semidirect
65. #382C22semidirect
66. C4×C22⋊C4C22semidirect
67. #343C22semidirect
68. #343C22semidirect
69. #339C22semidirect
70. #339C22semidirect

Maximal subgroups

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

subgroupindex
#34882
#30932
#28962
#25002
#20402
#20392
#20372
#19002
#18712
#18602
#18592
#17922
#17892