Finite Group Databasebeta

← #14767

Subgroups — #14767

Order 256.

897 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#400C211yesmaximal, characteristic, complemented
2C2×#385C211yesmaximal, characteristic
3C2×#380C211yesmaximal, characteristic
4C22×#95C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic, complemented
6#1443C211yesmaximal, characteristic, complemented
7#1443C211yesmaximal, characteristic, complemented
8#1442C211yesmaximal, characteristic, complemented
9#1442C211yesmaximal, characteristic, complemented
10#1442C211yesmaximal, characteristic, complemented
11#1441C211yesmaximal, characteristic, complemented
12C22⋊#95C211yesmaximal, characteristic
13C22⋊#95C211yesmaximal, characteristic
14C22⋊#95C211yesmaximal, characteristic
15C22⋊#95C211yesmaximal, characteristic
16C24×C4C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27#400—24no—
28#385—24no—
29#380—24no—
30C2×#95—24no—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2211yescharacteristic
39C2×#95C2211yescharacteristic
40C2×#95C2211yescharacteristic
41C2×#95C2211yescharacteristic
42C2×#95C2211yescharacteristic
43C2×#95C2211yescharacteristic
44C2×#95C2211yescharacteristic
45C2×#95C2211yescharacteristic
46C2×#95C2211yescharacteristic
47C2×#95C2211yescharacteristic
48C2×#95C2211yescharacteristic
49C2×#95C2211yescharacteristic
50C2×#95C2211yescharacteristic
51C2×#95C2211yescharacteristic
52C2×#95C2211yescharacteristic
53C2×#95C2211yescharacteristic
54C2×#95C2211yescharacteristic
55C25C2311yescharacteristic
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—22no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4Q814yes—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4D414yes—
115#95—22no—
116#95—22no—
117#95—22no—
118#95—22no—
119#95—22no—
120#95—22no—
121#95—22no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—22no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95D414yes—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24C2×D412yes—
148C24C2×Q812yes—
149C24#4112yes—
150C24#4112yes—
151C24#4112yes—
152C24#4112yes—
153C24#4112yes—
154C24C2411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—28no—
160C22×C4—28no—
161C22×C4—28no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
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—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
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—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
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—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4C2×D412yes—
265C22×C4C2×D412yes—
266C22×C4C2×D412yes—
267C22×C4C2×D412yes—
268C22×C4C2×D412yes—
269C22×C4C2×Q812yes—
270C22×C4C2×Q812yes—
271C22×C4#4112yes—
272C22×C4#4112yes—
273C22×C4#4112yes—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C22⋊C4—28no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23C22⋊Q814yes—
311C23#12314yes—
312C23#12414yes—
313C23#12514yes—
314C23#12514yes—
315C23#12614yes—
316C23#12614yes—
317C23C22×D411yescharacteristic
318C23C22×D411yescharacteristic
319C23C22×Q811yescharacteristic
320C23C2×#4111yescharacteristic
321C23C2×#4111yescharacteristic
322C23C2×#4111yescharacteristic
323C23C2×#4111yescharacteristic
324C23C2×#4111yescharacteristic
325C23C2×#4111yescharacteristic
326C23C2×#4111yescharacteristic
327C23C2×#4111yescharacteristic
328C23H(5,𝔽2)11yescharacteristic
329C23H(5,𝔽2)11yescharacteristic
330C23H(5,𝔽2)11yescharacteristic
331C232−1+411yescharacteristic
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—416no—
360C2×C4—416no—
361C2×C4—416no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
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—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—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4C22⋊Q814yes—
433C22—24no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—24no—
447C22#40018yescomplemented
448C22C2×C22⋊Q811yescharacteristic
449C22C2×C22⋊Q811yescharacteristic
450C22C2×#12311yescharacteristic
451C22#52511yescharacteristic
452C22#52511yescharacteristic
453C22C2×#12411yescharacteristic
454C22C2×#12511yescharacteristic
455C22C2×#12511yescharacteristic
456C22C2×#12611yescharacteristic
457C22C2×#12611yescharacteristic
458C22#52911yescharacteristic
459C22#52911yescharacteristic
460C22#53311yescharacteristic
461C22#53811yescharacteristic
462C22#53811yescharacteristic
463C22#53911yescharacteristic
464C22#53911yescharacteristic
465C22#54311yescharacteristic
466C22#54411yescharacteristic
467C22D4211yescharacteristic
468C22#54611yescharacteristic
469C22#54611yescharacteristic
470C22#54611yescharacteristic
471C22#54611yescharacteristic
472C22#54711yescharacteristic
473C22#54811yescharacteristic
474C22#55111yescharacteristic
475C22#55111yescharacteristic
476C22#55111yescharacteristic
477C22#55111yescharacteristic
478C22#55211yescharacteristic
479C22#55211yescharacteristic
480C22#55311yescharacteristic
481C22#55311yescharacteristic
482C22#55311yescharacteristic
483C4—24no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C2—28no—
498C2C2×#40011yescharacteristic
499C2#243711yescharacteristic
500C2#243811yescharacteristic
501C2#244211yescharacteristic
502C2#244311yescharacteristic
503C2#244411yescharacteristic
504C2#247211yescharacteristic
505C2#248011yescharacteristic
506C2#248111yescharacteristic
507C2#248811yescharacteristic
508C2#249311yescharacteristic
509C2#251011yescharacteristic
510C2#256911yescharacteristic
511C2#264611yescharacteristic
512C2#285111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#400C2semidirect
5. #1445C2semidirect
6. #1443C2semidirect
7. #1443C2semidirect
8. #1442C2semidirect
9. #1442C2semidirect
10. #1442C2semidirect
11. #1441C2semidirect
447. C22#400semidirect

Maximal subgroups

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

subgroupindex
C2×#4002
C2×#3852
C2×#3802
C22×#952
#14452
#14432
#14422
#14412
C22⋊#952