Finite Group Databasebeta

← #24452

Subgroups — #24452

Order 256.

796 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#546C211yesmaximal, characteristic
2C2×#468C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#450C211yesmaximal, characteristic, complemented
5C2×#417C211yesmaximal, characteristic, complemented
6#2417C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1870C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×M16C2211yescharacteristic, complemented
22#546—24no—
23#546—24no—
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic, complemented
29C2×#121—21no—
30C2×#121C2211yescharacteristic
31C2×#120—21no—
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic, complemented
35C22×C22⋊C4—21no—
36#468—24no—
37#452—24no—
38#450—24no—
39C2×#107C2211yescharacteristic, complemented
40#417—24no—
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#103C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic, complemented
54#398—21no—
55#396—21no—
56#391C2211yescharacteristic, complemented
57#389C2211yescharacteristic, complemented
58#386—21no—
59#380—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×#41D414yes—
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4—21no—
76C22×D4—22no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—21no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4D411yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C2×Q16—22no—
92C2×Q16—41no—
93C2×Q16—41no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—41no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—41no—
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107#124—22no—
108#124—22no—
109#123—28no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—41no—
116#121—28no—
117#121—22no—
118#121D414yes—
119#120—28no—
120C4×D4—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4D411yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C22⋊C4—28no—
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⋊C4D411yescharacteristic
146C2×C22⋊C4D411yescharacteristic
147C2×C42C2311yescharacteristic
148#107—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—41no—
158#102D414yescomplemented
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—41no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24—24no—
168C24—21no—
169C24—42no—
170C24—42no—
171C24—42no—
172C24—42no—
173C24—81no—
174C24—81no—
175C24C2×D411yescharacteristic
176C24C2×D411yescharacteristic
177#41—28no—
178#41—44no—
179#41—44no—
180#41—44no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8C2×D412yes—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—24no—
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—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—81no—
216C2×D4—81no—
217C2×D4C2×D412yes—
218C2×D4C2×D412yes—
219C2×D4C2×D412yes—
220C2×D4#4112yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D412yes—
251C22×C4C2×D412yes—
252C22×C4C2411yescharacteristic
253Q16—44no—
254SD16—44no—
255SD16—44no—
256M16—44no—
257C2×C8—21no—
258C2×C8—21no—
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×C8C2×D412yes—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
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—81no—
293C42—21no—
294C42—21no—
295C23—21no—
296C23—22no—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—48no—
312C23—48no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23#12012yes—
329C23#13712yes—
330C23C22×D411yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338Q8—41no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—22no—
344D4—44no—
345D4—48no—
346D4—48no—
347D4—41no—
348D4—41no—
349D4—84no—
350D4#12114yes—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—48no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4#12012yes—
392C2×C4C22×D411yescharacteristic
393C2×C4C22×D411yescharacteristic
394C2×C4C22×D411yescharacteristic
395C2×C4C2×#4111yescharacteristic
396C2×C4H(5,𝔽2)11yescharacteristic
397C2×C4H(5,𝔽2)11yescharacteristic
398C8—22no—
399C8—41no—
400C8—41no—
401C8—41no—
402C22—22no—
403C22—24no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—41no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—816no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22C2×#12011yescharacteristic
432C22D4211yescharacteristic
433C22#54611yescharacteristic
434C22C2×#13711yescharacteristic
435C22#57511yescharacteristic
436C22#57611yescharacteristic
437C22#57711yescharacteristic
438C4—22no—
439C4—41no—
440C4—42no—
441C4—41no—
442C4—41no—
443C4—42no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4C2×#12111yescharacteristic
449C4#53411yescharacteristic
450C4D4211yescharacteristic
451C4#54611yescharacteristic
452C2—42no—
453C2—42no—
454C2—44no—
455C2—82no—
456C2#241011yescharacteristic
457C2#300711yescharacteristic
458C2#301211yescharacteristic
459C2#326611yescharacteristic
460C2#327311yescharacteristic
461C2#331411yescharacteristic
462C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#546C2
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
2. C2×#468C2semidirect
3. C2×#452C2semidirect
4. C2×#450C2semidirect
5. C2×#417C2semidirect
6. #2417C2semidirect
7. #2042C2semidirect
8. #2036C2semidirect
9. #2015C2semidirect
10. #2001C2semidirect
11. #1877C2semidirect
12. #1874C2semidirect
13. #1870C2semidirect
14. #1862C2semidirect
15. #1786C2semidirect
18. C22×Q16C22semidirect
21. C22×M16C22semidirect
28. C2×C22⋊Q8C22semidirect
34. C22×C4⋊C4C22semidirect
39. C2×#107C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
44. C2×#103C22semidirect
53. C2×C22⋊C8C22semidirect
56. #391C22semidirect
57. #389C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
158. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C2×#4682
C2×#4522
C2×#4502
C2×#4172
#24172
#20422
#20362
#20152
#20012
#18772
#18742
#18702
#18622
#17862