Finite Group Databasebeta

← #14726

Subgroups — #14726

Order 256.

1747 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C4C211yesmaximal, characteristic
2C2×#396C212yesmaximal, complemented
3C2×#385C211yesmaximal, characteristic, complemented
4C2×#385C211yesmaximal, characteristic, complemented
5C2×#380C211yesmaximal, characteristic, complemented
6C2×#379C211yesmaximal, characteristic
7C2×#379C211yesmaximal, characteristic
8#1439C212yesmaximal, complemented
9#1439C212yesmaximal, complemented
10#1439C212yesmaximal, complemented
11C22⋊#95C211yesmaximal, characteristic, complemented
12C26C2211yescharacteristic
13C24×C4C2211yescharacteristic
14C24×C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic, complemented
16C22×C4⋊C4C2212yescomplemented
17C22×C4⋊C4C2212yescomplemented
18C22×C4⋊C4C2211yescharacteristic, complemented
19C22×C22⋊C4—24no—
20C22×C22⋊C4—28no—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2212yes—
26C22×C22⋊C4C2212yes—
27C22×C22⋊C4C2212yes—
28C22×C22⋊C4C2212yes—
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31#396—28no—
32#385—24no—
33#385—24no—
34#380—24no—
35#379—24no—
36#379—24no—
37C2×#95C2212yescomplemented
38C2×#95C2212yescomplemented
39C2×#95C2211yescharacteristic, complemented
40C2×#95C2211yescharacteristic, complemented
41C2×#95C2212yescomplemented
42C2×#95C2212yescomplemented
43C25—24no—
44C25—22no—
45C25—24no—
46C25—22no—
47C25—22no—
48C25—22no—
49C25—22no—
50C25—22no—
51C25—42no—
52C25—42no—
53C25Q814yes—
54C25C2312yes—
55C25C2311yescharacteristic
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—24no—
60C23×C4—24no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—42no—
64C23×C4—42no—
65C23×C4D414yes—
66C23×C4C2311yescharacteristic
67C23×C4C2312yes—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2312yes—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2312yes—
74C23×C4C2311yescharacteristic
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—232no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4D418yes—
114C2×C22⋊C4D414yes—
115C2×C22⋊C4D414yes—
116#95—24no—
117#95—24no—
118#95—22no—
119#95—22no—
120#95—24no—
121#95—42no—
122C24—22no—
123C24—22no—
124C24—22no—
125C24—22no—
126C24—28no—
127C24—28no—
128C24—216no—
129C24—216no—
130C24—28no—
131C24—22no—
132C24—22no—
133C24—22no—
134C24—22no—
135C24—22no—
136C24—22no—
137C24—22no—
138C24—22no—
139C24—22no—
140C24—22no—
141C24—22no—
142C24—22no—
143C24—28no—
144C24—28no—
145C24—48no—
146C24—48no—
147C24—48no—
148C24—48no—
149C24—48no—
150C24—48no—
151C24—48no—
152C24—48no—
153C24—48no—
154C24—44no—
155C24—44no—
156C24—48no—
157C24—44no—
158C24—44no—
159C24—44no—
160C24—44no—
161C24C2×D414yes—
162C24C2×D412yes—
163C24C2×D412yes—
164C24C2×D412yes—
165C24C2×Q814yes—
166C24C2×Q812yes—
167C24#4114yes—
168C24#4112yes—
169C24#4112yes—
170C24#4112yes—
171C24C2411yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—28no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—44no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4C2×D412yes—
227C22×C4C2×D414yes—
228C22×C4C2×D412yes—
229C22×C4C2×D414yes—
230C22×C4C2×D412yes—
231C22×C4C2×D412yes—
232C22×C4C2×D414yes—
233C4⋊C4—48no—
234C4⋊C4—48no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C22⋊C4—216no—
238C22⋊C4—28no—
239C22⋊C4—28no—
240C22⋊C4—216no—
241C22⋊C4—48no—
242C22⋊C4—48no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C23—22no—
248C23—24no—
249C23—22no—
250C23—24no—
251C23—22no—
252C23—22no—
253C23—24no—
254C23—24no—
255C23—24no—
256C23—24no—
257C23—24no—
258C23—24no—
259C23—24no—
260C23—24no—
261C23—24no—
262C23—24no—
263C23—232no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23—24no—
271C23—24no—
272C23—24no—
273C23—24no—
274C23—24no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23—24no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—432no—
283C23—432no—
284C23—416no—
285C23—416no—
286C23—416no—
287C23—416no—
288C23—432no—
289C23—432no—
290C23—416no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—416no—
308C23#12114yes—
309C23#12114yes—
310C23#12114yes—
311C23C22⋊Q818yes—
312C23C22⋊Q814yes—
313C23C22⋊Q814yes—
314C23C22⋊Q814yes—
315C23#12614yes—
316C23C22×D411yescharacteristic
317C23C22×D412yes—
318C23C22×D411yescharacteristic
319C23C22×D411yescharacteristic
320C23C22×Q811yescharacteristic
321C23C2×#4111yescharacteristic
322C23C2×#4111yescharacteristic
323C23C2×#4111yescharacteristic
324C23C2×#4112yes—
325C23H(5,𝔽2)12yes—
326C23H(5,𝔽2)12yes—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—416no—
345C2×C4—416no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4#12014yes—
380C22—24no—
381C22—24no—
382C22—24no—
383C22—24no—
384C22—24no—
385C22—28no—
386C22—28no—
387C22—28no—
388C22—28no—
389C22—24no—
390C22—28no—
391C22—28no—
392C22—28no—
393C22—464no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—44no—
398C22—44no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22#39618yescomplemented
407C22C2×#12011yescharacteristic
408C22C2×#12111yescharacteristic
409C22C2×#12111yescharacteristic
410C22C2×#12111yescharacteristic
411C22C2×C22⋊Q811yescharacteristic
412C22C2×C22⋊Q812yes—
413C22C2×C22⋊Q811yescharacteristic
414C22C2×C22⋊Q811yescharacteristic
415C22#52511yescharacteristic
416C22#52512yes—
417C22C2×#12611yescharacteristic
418C22#53411yescharacteristic
419C22#53511yescharacteristic
420C22#53812yes—
421C22#54312yes—
422C22D4212yes—
423C22D4212yes—
424C22#54612yes—
425C22#54612yes—
426C22#54612yes—
427C22#54612yes—
428C22#55112yes—
429C22#55112yes—
430C4—24no—
431C4—42no—
432C4—44no—
433C4—42no—
434C4—44no—
435C4—42no—
436C4—81no—
437C4—81no—
438C4—82no—
439C2—28no—
440C2—48no—
441C2C2×#39611yescharacteristic
442C2#240811yescharacteristic
443C2#241011yescharacteristic
444C2#241111yescharacteristic
445C2#242412yes—
446C2#243712yes—
447C2#243712yes—
448C2#248012yes—
449C2#253411yescharacteristic
450C2#262611yescharacteristic
451C2#284811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC26C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#396C2semidirect
3. C2×#385C2semidirect
4. C2×#385C2semidirect
5. C2×#380C2semidirect
8. #1439C2semidirect
9. #1439C2semidirect
10. #1439C2semidirect
11. C22⋊#95C2semidirect
15. C22×C4⋊C4C22semidirect
16. C22×C4⋊C4C22semidirect
17. C22×C4⋊C4C22semidirect
18. C22×C4⋊C4C22semidirect
37. C2×#95C22semidirect
38. C2×#95C22semidirect
39. C2×#95C22semidirect
40. C2×#95C22semidirect
41. C2×#95C22semidirect
42. C2×#95C22semidirect
406. C22#396semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C42
C2×#3962
C2×#3852
C2×#3802
C2×#3792
#14392
C22⋊#952