Finite Group Databasebeta

← #24428

Subgroups — #24428

Order 256.

816 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic, complemented
2C2×#466C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5C2×#416C211yesmaximal, characteristic, complemented
6#2414C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic, complemented
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C23×C8C2211yescharacteristic, complemented
22#546—24no—
23#546—24no—
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121—21no—
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×#121C2211yescharacteristic, complemented
31C2×#120—21no—
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×#117C2211yescharacteristic, complemented
35C22×C22⋊C4—21no—
36#466—24no—
37#449—24no—
38#447—24no—
39C2×#107C2211yescharacteristic, complemented
40#416—24no—
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic, complemented
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic, complemented
54#394—21no—
55#392—21no—
56#390—21no—
57#390C2211yescharacteristic, complemented
58#389C2211yescharacteristic, complemented
59#388—21no—
60#380C2211yescharacteristic
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—41no—
70C2×#41—41no—
71C2×#41D414yescomplemented
72C22×Q8C2311yescharacteristic
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C22×D4C2311yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C2×SD16—22no—
93C2×SD16—41no—
94C2×SD16—41no—
95C2×D8—22no—
96C2×D8—22no—
97C2×D8—22no—
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—41no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—41no—
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108#124—22no—
109#124—22no—
110#123—28no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113#121—28no—
114#121—22no—
115#121—22no—
116#121—22no—
117#121—22no—
118#121—41no—
119#121D414yescomplemented
120#120—28no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125#117—22no—
126#117—41no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4D411yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C22⋊C4—28no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4D411yescharacteristic
147C2×C22⋊C4D411yescharacteristic
148C2×C42—21no—
149C2×C42C2311yescharacteristic
150#107—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—41no—
160#102D414yescomplemented
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—41no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24—24no—
168C24—21no—
169C24—21no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—42no—
174C24—81no—
175C24—81no—
176C24C2×D411yescharacteristic
177C24C2×D411yescharacteristic
178#41—28no—
179#41—44no—
180#41—44no—
181#41—44no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8C2×D412yes—
188C2×D4—24no—
189C2×D4—22no—
190C2×D4—24no—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—81no—
220C2×D4—81no—
221C2×D4—81no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4C2×D412yes—
225C2×D4C2×D412yes—
226C2×D4C2×D412yes—
227C2×D4#4112yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D412yes—
254C22×C4C2×D412yes—
255C22×C4C2411yescharacteristic
256SD16—44no—
257D8—44no—
258D8—44no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—44no—
271C2×C8C2×D412yes—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—81no—
281C4⋊C4C2×D412yes—
282C4⋊C4C2×D412yes—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C42—21no—
297C42—21no—
298C42—42no—
299C23—21no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—48no—
316C23—48no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—84no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—84no—
339C23#12012yes—
340C23#13512yes—
341C23C22×D411yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—41no—
345Q8—41no—
346D4—22no—
347D4—22no—
348D4—22no—
349D4—22no—
350D4—22no—
351D4—44no—
352D4—48no—
353D4—48no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—84no—
361D4—84no—
362D4—84no—
363D4#12114yes—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—48no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—84no—
400C2×C4#12012yes—
401C2×C4C22×D411yescharacteristic
402C2×C4C22×D411yescharacteristic
403C2×C4C22×D411yescharacteristic
404C2×C4C2×#4111yescharacteristic
405C2×C4H(5,𝔽2)11yescharacteristic
406C2×C4H(5,𝔽2)11yescharacteristic
407C8—22no—
408C8—41no—
409C8—41no—
410C8—41no—
411C22—22no—
412C22—24no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—41no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—42no—
425C22—41no—
426C22—42no—
427C22—41no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—816no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22C2×#12011yescharacteristic
448C22D4211yescharacteristic
449C22#54611yescharacteristic
450C22C2×#13511yescharacteristic
451C22#57511yescharacteristic
452C22#57611yescharacteristic
453C22#57611yescharacteristic
454C4—22no—
455C4—41no—
456C4—42no—
457C4—41no—
458C4—41no—
459C4—42no—
460C4—42no—
461C4—81no—
462C4—81no—
463C4C2×#12111yescharacteristic
464C4#53411yescharacteristic
465C4D4211yescharacteristic
466C4#54611yescharacteristic
467C2—42no—
468C2—42no—
469C2—44no—
470C2—82no—
471C2—161no—
472C2#241011yescharacteristic
473C2#300211yescharacteristic
474C2#300811yescharacteristic
475C2#326811yescharacteristic
476C2#328011yescharacteristic
477C2#329711yescharacteristic
478C2#331911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#546C2semidirect
2. C2×#466C2semidirect
3. C2×#449C2semidirect
4. C2×#447C2semidirect
5. C2×#416C2semidirect
6. #2414C2semidirect
7. #2042C2semidirect
8. #2036C2semidirect
9. #2012C2semidirect
10. #2000C2semidirect
11. #1879C2semidirect
12. #1875C2semidirect
13. #1869C2semidirect
14. #1862C2semidirect
15. #1786C2semidirect
18. C22×SD16C22semidirect
21. C23×C8C22semidirect
30. C2×#121C22semidirect
34. C2×#117C22semidirect
39. C2×#107C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
50. C2×#102C22semidirect
53. C2×C22⋊C8C22semidirect
57. #390C22semidirect
58. #389C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
71. C2×#41D4semidirect
119. #121D4semidirect
160. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4662
C2×#4492
C2×#4472
C2×#4162
#24142
#20422
#20362
#20122
#20002
#18792
#18752
#18692
#18622
#17862