Finite Group Databasebeta

← #24008

Subgroups — #24008

Order 256.

724 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#482C211yesmaximal, characteristic, complemented
4#2280C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1875C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic, complemented
12#1854C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1779C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C24×C4C2211yescharacteristic
18#525—24no—
19#525—24no—
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#120—21no—
26C2×C4×D4—21no—
27C2×C4×D4—21no—
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31#485—24no—
32#482—24no—
33C2×#107C2211yescharacteristic, complemented
34C2×#107C2211yescharacteristic, complemented
35C2×#106C2211yescharacteristic, complemented
36C2×#106C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#103C2211yescharacteristic, complemented
39C2×#103C2211yescharacteristic, complemented
40C2×#103C2211yescharacteristic, complemented
41C2×#102C2211yescharacteristic, complemented
42C2×#102C2211yescharacteristic, complemented
43C2×#102C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic, complemented
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51#381—21no—
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#380C2211yescharacteristic
56#379—21no—
57C4×C22⋊C4—21no—
58C4×C22⋊C4—21no—
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63C25D411yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4D411yescharacteristic
83C23×C4D411yescharacteristic
84C23×C4D411yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#123—28no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8D414yes—
97#121—22no—
98#121—22no—
99#121—22no—
100#121D414yes—
101#120—28no—
102C4×D4—28no—
103C4×D4—28no—
104#117—22no—
105#117—22no—
106#117—22no—
107#117—22no—
108#117—41no—
109#117—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132#107—22no—
133#107—22no—
134#106—22no—
135#106—22no—
136#103—22no—
137#103—22no—
138#103—41no—
139#103—41no—
140#102—22no—
141#102—22no—
142#102—41no—
143#102—41no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—22no—
155C24—22no—
156C24—22no—
157C24—21no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24—41no—
162C24C2×D411yescharacteristic
163C24C2×D411yescharacteristic
164C24C2×D411yescharacteristic
165#41—44no—
166#41—44no—
167#41—44no—
168#41—44no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8C2×D412yes—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4C2×D412yes—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D412yes—
230C22×C4C2×D412yes—
231C22×C4C2×D411yescharacteristic
232C22×C4#4112yes—
233C22×C4#4112yes—
234C22×C4C2411yescharacteristic
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4C2×D412yes—
260C4⋊C4C2×D412yes—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C42—24no—
273C42—42no—
274C42—42no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23—44no—
300C23—44no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23#12012yes—
308C23#12312yes—
309C23#13512yes—
310C23C22×D411yescharacteristic
311Q8—22no—
312Q8—22no—
313Q8—41no—
314Q8—41no—
315D4—22no—
316D4—22no—
317D4—48no—
318D4—48no—
319D4—48no—
320D4—41no—
321D4—41no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4#12012yes—
367C2×C4#12312yes—
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C2×#4111yescharacteristic
371C2×C4C2×#4111yescharacteristic
372C2×C4H(5,𝔽2)11yescharacteristic
373C2×C4H(5,𝔽2)11yescharacteristic
374C8—41no—
375C8—41no—
376C8—41no—
377C8—41no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—416no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22C2×#12011yescharacteristic
408C22C2×#12311yescharacteristic
409C22#53411yescharacteristic
410C22C2×#13511yescharacteristic
411C22#57511yescharacteristic
412C22#57511yescharacteristic
413C22#57511yescharacteristic
414C4—22no—
415C4—22no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4#54611yescharacteristic
425C4#54611yescharacteristic
426C4#54611yescharacteristic
427C4#54611yescharacteristic
428C2—42no—
429C2—42no—
430C2—42no—
431C2—82no—
432C2#240511yescharacteristic
433C2#300211yescharacteristic
434C2#300511yescharacteristic
435C2#309111yescharacteristic
436C2#309311yescharacteristic
437C2#319111yescharacteristic
438C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#525C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#485C2semidirect
3. C2×#482C2semidirect
4. #2280C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1875C2semidirect
10. #1873C2semidirect
11. #1854C2semidirect
12. #1854C2semidirect
13. #1788C2semidirect
14. #1788C2semidirect
15. #1779C2semidirect
33. C2×#107C22semidirect
34. C2×#107C22semidirect
35. C2×#106C22semidirect
36. C2×#106C22semidirect
37. C2×#103C22semidirect
38. C2×#103C22semidirect
39. C2×#103C22semidirect
40. C2×#103C22semidirect
41. C2×#102C22semidirect
42. C2×#102C22semidirect
43. C2×#102C22semidirect
44. C2×#102C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #340C22semidirect
62. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
C2×#4852
C2×#4822
#22802
#20362
#20352
#20342
#20332
#18752
#18732
#18542
#17882
#17792