Finite Group Databasebeta

← #14052

Subgroups — #14052

Order 256.

1133 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C2×#391C211yesmaximal, characteristic, complemented
3C2×#385C211yesmaximal, characteristic
4C2×C4×C22⋊C4C211yesmaximal, characteristic
5C22×#95C211yesmaximal, characteristic
6#1446C211yesmaximal, characteristic, complemented
7#1445C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1436C211yesmaximal, characteristic, complemented
11#1436C211yesmaximal, characteristic, complemented
12C4⋊#95C211yesmaximal, characteristic, complemented
13C22⋊#95C211yesmaximal, characteristic
14C22⋊#95C211yesmaximal, characteristic
15C22⋊#95C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C2×C22⋊Q8—24no—
21C2×C22⋊Q8—24no—
22C2×C22⋊Q8C418yescomplemented
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C42C2211yescharacteristic
33C22×C42C2211yescharacteristic
34#391—24no—
35#385—24no—
36C4×C22⋊C4—24no—
37C2×#95—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
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C25C2311yescharacteristic
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8C2×C414yescomplemented
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—41no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4C2×C414yes—
86C23×C4Q814yescomplemented
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C23×C4C2311yescharacteristic
101C22⋊Q8—232no—
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—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C2×C414yescomplemented
115C2×C4⋊C4C2×C414yescomplemented
116C2×C4⋊C4C2×C414yescomplemented
117C2×C22⋊C4—22no—
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⋊C4C2×C414yes—
125C2×C22⋊C4C2×C414yes—
126C2×C22⋊C4D414yes—
127C2×C22⋊C4D414yes—
128C2×C22⋊C4Q814yescomplemented
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—41no—
134C2×C42—41no—
135#95—22no—
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—41no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95D414yes—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24—21no—
163C24C22×C412yes—
164C24C2×D412yes—
165C24C2×D412yes—
166C24C2×Q812yes—
167C24C2×Q812yes—
168C24#4112yes—
169C24#4112yes—
170C24C2411yescharacteristic
171C2×Q8—28no—
172C2×Q8—28no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
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—28no—
195C22×C4—28no—
196C22×C4—28no—
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—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
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×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4C22×C412yes—
278C22×C4C22×C412yes—
279C22×C4C22×C412yes—
280C22×C4C22×C412yes—
281C22×C4C22×C412yes—
282C22×C4C22×C412yes—
283C22×C4C2×D412yes—
284C22×C4C2×D412yes—
285C22×C4C2×D412yes—
286C22×C4C2×D412yes—
287C22×C4C2×D412yes—
288C22×C4C2×D412yes—
289C22×C4C2×D412yes—
290C22×C4C2×Q812yes—
291C22×C4C2×Q812yes—
292C22×C4C2×Q812yes—
293C22×C4C2×Q812yes—
294C22×C4#4112yes—
295C22×C4#4112yes—
296C22×C4#4112yes—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—44no—
305C22⋊C4C22⋊C4116yes—
306C42—44no—
307C42—44no—
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—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23C2×C22⋊C414yes—
337C23C4×Q814yes—
338C23C4×Q814yes—
339C23C22⋊Q814yes—
340C23C22⋊Q814yes—
341C23#12414yes—
342C23C4⋊Q814yes—
343C23C23×C411yescharacteristic
344C23C22×D411yescharacteristic
345C23C22×D411yescharacteristic
346C23C22×D411yescharacteristic
347C23C22×Q811yescharacteristic
348C23C22×Q811yescharacteristic
349C23C2×#4111yescharacteristic
350C23C2×#4111yescharacteristic
351C23C2×#4111yescharacteristic
352C23C2×#4111yescharacteristic
353C23C2×#4111yescharacteristic
354C23H(5,𝔽2)11yescharacteristic
355C23H(5,𝔽2)11yescharacteristic
356C23H(5,𝔽2)11yescharacteristic
357C232−1+411yescharacteristic
358Q8—416no—
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—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—416no—
408C2×C4—416no—
409C2×C4—416no—
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×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4C2×C22⋊C414yes—
457C2×C4C2×C22⋊C414yes—
458C2×C4C4×D414yes—
459C2×C4C22⋊Q814yescomplemented
460C22—24no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22#39118yescomplemented
475C22C22×C22⋊C411yescharacteristic
476C22C2×C4×D411yescharacteristic
477C22C2×C4×Q811yescharacteristic
478C22C2×C4×Q811yescharacteristic
479C22C4×#4111yescharacteristic
480C22#51811yescharacteristic
481C22#52011yescharacteristic
482C22C2×C22⋊Q811yescharacteristic
483C22C2×C22⋊Q811yescharacteristic
484C22C2×C22⋊Q811yescharacteristic
485C22#52511yescharacteristic
486C22C2×#12411yescharacteristic
487C22#52911yescharacteristic
488C22C2×C4⋊Q811yescharacteristic
489C22#53311yescharacteristic
490C22#53811yescharacteristic
491C22#53911yescharacteristic
492C22#54311yescharacteristic
493C22#54411yescharacteristic
494C22D4211yescharacteristic
495C22D4211yescharacteristic
496C22#54611yescharacteristic
497C22#54611yescharacteristic
498C22#54611yescharacteristic
499C22#54611yescharacteristic
500C22#54611yescharacteristic
501C22#54711yescharacteristic
502C22#54711yescharacteristic
503C22D4×Q811yescharacteristic
504C22#55111yescharacteristic
505C22#55111yescharacteristic
506C22#55211yescharacteristic
507C22#55411yescharacteristic
508C22#55411yescharacteristic
509C22#55411yescharacteristic
510C4—24no—
511C4—24no—
512C4—24no—
513C4—24no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C2—28no—
525C2C2×#39111yescharacteristic
526C2C4×C22⋊Q811yescharacteristic
527C2#232911yescharacteristic
528C2D4×C22⋊C411yescharacteristic
529C2#233911yescharacteristic
530C2#236011yescharacteristic
531C2#236811yescharacteristic
532C2#243711yescharacteristic
533C2#243811yescharacteristic
534C2#244011yescharacteristic
535C2#244911yescharacteristic
536C2#245011yescharacteristic
537C2#247211yescharacteristic
538C2#248811yescharacteristic
539C2#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
2. C2×#391C2semidirect
6. #1446C2semidirect
7. #1445C2semidirect
8. #1441C2semidirect
9. #1441C2semidirect
10. #1436C2semidirect
11. #1436C2semidirect
12. C4⋊#95C2semidirect
22. C2×C22⋊Q8C4semidirect
65. C22×Q8C2×C4semidirect
86. C23×C4Q8semidirect
114. C2×C4⋊C4C2×C4semidirect
115. C2×C4⋊C4C2×C4semidirect
116. C2×C4⋊C4C2×C4semidirect
128. C2×C22⋊C4Q8semidirect
459. C2×C4C22⋊Q8semidirect
474. C22#391semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C2×#3912
C2×#3852
C2×C4×C22⋊C42
C22×#952
#14462
#14452
#14412
#14362
C4⋊#952
C22⋊#952