Finite Group Databasebeta

← #14116

Subgroups — #14116

Order 256.

865 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389C211yesmaximal, characteristic, complemented
2C2×#388C211yesmaximal, characteristic
3C2×#388C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic
5C2×C4×C22⋊C4C211yesmaximal, characteristic
6#1442C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1439C211yesmaximal, characteristic
9#1439C211yesmaximal, characteristic
10#1436C211yesmaximal, characteristic, complemented
11#1436C211yesmaximal, characteristic, complemented
12#1436C211yesmaximal, characteristic, complemented
13C4⋊#95C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic
15C4×#95C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊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
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C42C2211yescharacteristic
30C22×C42C2211yescharacteristic
31C22×C42C2211yescharacteristic
32C22×C42C2211yescharacteristic
33#389—24no—
34#388—24no—
35#388C418yescomplemented
36#385—24no—
37C4×C22⋊C4—24no—
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
55C2×#95C2211yescharacteristic
56C25C2311yescharacteristic
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—22no—
62C23×C4—41no—
63C23×C4—41no—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4C2×C414yescomplemented
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4C2×C414yes—
110C2×C22⋊C4C2×C414yes—
111C2×C22⋊C4C2×C414yes—
112C2×C22⋊C4D414yes—
113C2×C22⋊C4Q814yes—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42C2×C414yescomplemented
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95—22no—
137#95—22no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95C2×C414yescomplemented
142#95C2×C414yescomplemented
143#95D414yes—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24C22×C412yes—
157C24C2×D412yes—
158C24C2×Q812yes—
159C24C2411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
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—28no—
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—21no—
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—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4C22×C412yes—
246C22×C4C22×C412yes—
247C22×C4C22×C412yes—
248C22×C4C22×C412yes—
249C22×C4C22×C412yes—
250C22×C4C22×C412yes—
251C22×C4C2×D412yes—
252C22×C4C2×D412yes—
253C22×C4C2×D412yes—
254C22×C4C2×D412yes—
255C22×C4C2×D412yes—
256C22×C4C2×Q812yes—
257C22×C4C2×Q812yes—
258C22×C4#4112yes—
259C22×C4#4112yes—
260C22×C4#4112yes—
261C22×C4#4112yes—
262C22×C4#4112yes—
263C22×C4#4112yes—
264C22×C4#4112yes—
265C4⋊C4—28no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C22⋊C4—28no—
272C22⋊C4—28no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4C4⋊C4116yescomplemented
277C42—44no—
278C42—44no—
279C42—44no—
280C42—44no—
281C23—22no—
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—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C2×C4⋊C414yes—
316C23C23×C411yescharacteristic
317C23C22×D411yescharacteristic
318C23C22×D411yescharacteristic
319C23C22×Q811yescharacteristic
320C23C2×#4111yescharacteristic
321C23C2×#4111yescharacteristic
322C23C2×#4111yescharacteristic
323C23C2×#4111yescharacteristic
324C23C2×#4111yescharacteristic
325C23C2×#4111yescharacteristic
326C23C2×#4111yescharacteristic
327C23H(5,𝔽2)11yescharacteristic
328C23H(5,𝔽2)11yescharacteristic
329C23H(5,𝔽2)11yescharacteristic
330C232−1+411yescharacteristic
331C2×C4—22no—
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—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—416no—
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×C4C2×C4⋊C414yes—
427C2×C4C2×C4⋊C414yes—
428C2×C4C4×D414yes—
429C2×C4#12414yes—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22C22×C4⋊C411yescharacteristic
446C22C2×C4×D411yescharacteristic
447C22C4×#4111yescharacteristic
448C22C4×#4111yescharacteristic
449C22C4×#4111yescharacteristic
450C22#51811yescharacteristic
451C22#52011yescharacteristic
452C22#52511yescharacteristic
453C22#52511yescharacteristic
454C22C2×#12411yescharacteristic
455C22#52911yescharacteristic
456C22#52911yescharacteristic
457C22#53211yescharacteristic
458C22#53311yescharacteristic
459C22#53311yescharacteristic
460C22#53811yescharacteristic
461C22#53911yescharacteristic
462C22#54011yescharacteristic
463C22#54211yescharacteristic
464C22D4211yescharacteristic
465C22#54611yescharacteristic
466C22#54611yescharacteristic
467C22#54611yescharacteristic
468C22#54611yescharacteristic
469C22#54711yescharacteristic
470C22D4×Q811yescharacteristic
471C22#55111yescharacteristic
472C22#55211yescharacteristic
473C22#55311yescharacteristic
474C22#55311yescharacteristic
475C22#55311yescharacteristic
476C22#55411yescharacteristic
477C22#55411yescharacteristic
478C22#55411yescharacteristic
479C22#55611yescharacteristic
480C4—24no—
481C4—24no—
482C4—24no—
483C4—24no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—81no—
494C2—44no—
495C2#229611yescharacteristic
496C2C4×#12411yescharacteristic
497C2#233511yescharacteristic
498C2D4×C4⋊C411yescharacteristic
499C2#234911yescharacteristic
500C2#236411yescharacteristic
501C2#237311yescharacteristic
502C2#246411yescharacteristic
503C2#246611yescharacteristic
504C2#247011yescharacteristic
505C2#247211yescharacteristic
506C2#249211yescharacteristic
507C2#252211yescharacteristic
508C2#255611yescharacteristic
509C2#258711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#389C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
10. #1436C2semidirect
11. #1436C2semidirect
12. #1436C2semidirect
13. C4⋊#95C2semidirect
15. C4×#95C2semidirect
35. #388C4semidirect
95. C2×C4⋊C4C2×C4semidirect
125. C2×C42C2×C4semidirect
141. #95C2×C4semidirect
142. #95C2×C4semidirect
276. C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
C2×#3882
C2×#3852
C2×C4×C22⋊C42
#14422
#14412
#14392
#14362
C4⋊#952
C22⋊#952
C4×#952