Finite Group Databasebeta

← #14004

Subgroups — #14004

Order 256.

1053 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#389C211yesmaximal, characteristic, complemented
2C2×#385C211yesmaximal, characteristic
3C2×#385C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic
5C2×#380C211yesmaximal, characteristic
6C2×C4×C22⋊C4C211yesmaximal, characteristic
7C22×#95C211yesmaximal, characteristic
8#1444C211yesmaximal, characteristic, complemented
9#1442C211yesmaximal, characteristic, complemented
10#1441C211yesmaximal, characteristic, complemented
11#1441C211yesmaximal, characteristic, complemented
12#1436C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14C4⋊#95C211yesmaximal, characteristic, complemented
15C22⋊#95C211yesmaximal, characteristic
16C24×C4C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C42C2211yescharacteristic
32C22×C42C2211yescharacteristic
33#389—24no—
34#385—24no—
35#385—24no—
36#385C418yescomplemented
37#380—24no—
38C4×C22⋊C4—24no—
39C2×#95—24no—
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
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C25C2311yescharacteristic
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—22no—
71C23×C4—22no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4C2×C414yes—
79C23×C4Q814yes—
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×C414yescomplemented
117C2×C4⋊C4C2×C414yescomplemented
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4C2×C414yes—
126C2×C22⋊C4C2×C414yes—
127C2×C22⋊C4D414yes—
128C2×C22⋊C4D414yes—
129C2×C22⋊C4Q814yes—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—41no—
135C2×C42—41no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95—41no—
150#95—41no—
151#95C2×C414yescomplemented
152#95C2×C414yescomplemented
153#95D414yes—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24C22×C412yes—
163C24C2×D412yes—
164C24C2×D412yes—
165C24C2×Q812yes—
166C24C2×Q812yes—
167C24#4112yes—
168C24#4112yes—
169C24C2411yescharacteristic
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—28no—
187C22×C4—28no—
188C22×C4—28no—
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—28no—
238C22×C4—28no—
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—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
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×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4C22×C412yes—
270C22×C4C22×C412yes—
271C22×C4C22×C412yes—
272C22×C4C22×C412yes—
273C22×C4C22×C412yes—
274C22×C4C22×C412yes—
275C22×C4C2×D412yes—
276C22×C4C2×D412yes—
277C22×C4C2×D412yes—
278C22×C4C2×D412yes—
279C22×C4C2×D412yes—
280C22×C4C2×D412yes—
281C22×C4C2×D412yes—
282C22×C4C2×Q812yes—
283C22×C4C2×Q812yes—
284C22×C4C2×Q812yes—
285C22×C4C2×Q812yes—
286C22×C4#4112yes—
287C22×C4#4112yes—
288C22×C4#4112yes—
289C4⋊C4—28no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—44no—
300C22⋊C4C4⋊C4116yescomplemented
301C42—44no—
302C42—44no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23C2×C4⋊C414yes—
332C23C4×D414yes—
333C23C4×Q814yes—
334C23#12114yes—
335C23C22⋊Q814yes—
336C23#12514yes—
337C23C4⋊Q814yes—
338C23C23×C411yescharacteristic
339C23C22×D411yescharacteristic
340C23C22×D411yescharacteristic
341C23C22×D411yescharacteristic
342C23C22×Q811yescharacteristic
343C23C22×Q811yescharacteristic
344C23C2×#4111yescharacteristic
345C23C2×#4111yescharacteristic
346C23C2×#4111yescharacteristic
347C23C2×#4111yescharacteristic
348C23C2×#4111yescharacteristic
349C23H(5,𝔽2)11yescharacteristic
350C23H(5,𝔽2)11yescharacteristic
351C23H(5,𝔽2)11yescharacteristic
352C232−1+411yescharacteristic
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—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—416no—
402C2×C4—416no—
403C2×C4—416no—
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×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4C2×C4⋊C414yes—
451C2×C4C2×C4⋊C414yes—
452C2×C4C4×D414yes—
453C2×C4C22⋊Q814yes—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22#38918yescomplemented
469C22C22×C4⋊C411yescharacteristic
470C22C2×C4×D411yescharacteristic
471C22C2×C4×D411yescharacteristic
472C22C2×C4×Q811yescharacteristic
473C22C4×#4111yescharacteristic
474C22#51811yescharacteristic
475C22#52011yescharacteristic
476C22C2×#12111yescharacteristic
477C22C2×C22⋊Q811yescharacteristic
478C22C2×C22⋊Q811yescharacteristic
479C22#52511yescharacteristic
480C22C2×#12511yescharacteristic
481C22#52911yescharacteristic
482C22C2×C4⋊Q811yescharacteristic
483C22#53311yescharacteristic
484C22#53811yescharacteristic
485C22#53911yescharacteristic
486C22#54311yescharacteristic
487C22#54411yescharacteristic
488C22D4211yescharacteristic
489C22D4211yescharacteristic
490C22#54611yescharacteristic
491C22#54611yescharacteristic
492C22#54611yescharacteristic
493C22#54611yescharacteristic
494C22#54611yescharacteristic
495C22#54711yescharacteristic
496C22#54711yescharacteristic
497C22D4×Q811yescharacteristic
498C22#55111yescharacteristic
499C22#55111yescharacteristic
500C22#55211yescharacteristic
501C22#55411yescharacteristic
502C22#55411yescharacteristic
503C22#55411yescharacteristic
504C4—24no—
505C4—24no—
506C4—24no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—81no—
517C4—81no—
518C2—28no—
519C2C2×#38911yescharacteristic
520C2C4×C22⋊Q811yescharacteristic
521C2#232911yescharacteristic
522C2D4×C4⋊C411yescharacteristic
523C2#234911yescharacteristic
524C2#235811yescharacteristic
525C2#235911yescharacteristic
526C2#242411yescharacteristic
527C2#242511yescharacteristic
528C2#243711yescharacteristic
529C2#243811yescharacteristic
530C2#244011yescharacteristic
531C2#247311yescharacteristic
532C2#250511yescharacteristic
533C2#254911yescharacteristic
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
8. #1444C2semidirect
9. #1442C2semidirect
10. #1441C2semidirect
11. #1441C2semidirect
12. #1436C2semidirect
13. #1436C2semidirect
14. C4⋊#95C2semidirect
36. #385C4semidirect
116. C2×C4⋊C4C2×C4semidirect
117. C2×C4⋊C4C2×C4semidirect
151. #95C2×C4semidirect
152. #95C2×C4semidirect
300. C22⋊C4C4⋊C4semidirect
468. C22#389semidirect

Maximal subgroups

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

subgroupindex
C2×#3892
C2×#3852
C2×#3802
C2×C4×C22⋊C42
C22×#952
#14442
#14422
#14412
#14362
C4⋊#952
C22⋊#952