Finite Group Databasebeta

← #14736

Subgroups — #14736

Order 256.

969 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398C211yesmaximal, characteristic, complemented
2C2×#385C211yesmaximal, characteristic
3C2×#385C211yesmaximal, characteristic
4C2×#385C211yesmaximal, characteristic
5C2×#380C211yesmaximal, characteristic
6C22×#95C211yesmaximal, characteristic
7#1445C211yesmaximal, characteristic, complemented
8#1444C211yesmaximal, characteristic, complemented
9#1442C211yesmaximal, characteristic, complemented
10#1442C211yesmaximal, characteristic, complemented
11#1441C211yesmaximal, characteristic, complemented
12#1441C211yesmaximal, characteristic, complemented
13#1441C211yesmaximal, characteristic, complemented
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×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31#398—24no—
32#385—24no—
33#385—24no—
34#385—24no—
35#380—24no—
36C2×#95—24no—
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
55C2×#95C2211yescharacteristic
56C2×#95C2211yescharacteristic
57C25C2311yescharacteristic
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—22no—
71C23×C4—22no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—41no—
78C23×C4Q814yes—
79C23×C4C2311yescharacteristic
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
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—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—22no—
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⋊C4—41no—
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⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4D414yes—
127C2×C22⋊C4D414yes—
128C2×C22⋊C4Q814yes—
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—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95D414yes—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24C2×D412yes—
158C24C2×D412yes—
159C24C2×Q812yes—
160C24C2×Q812yes—
161C24#4112yes—
162C24#4112yes—
163C24#4112yes—
164C24C2411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—28no—
170C22×C4—28no—
171C22×C4—28no—
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—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—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×C4—44no—
270C22×C4C2×D412yes—
271C22×C4C2×D412yes—
272C22×C4C2×D412yes—
273C22×C4C2×D412yes—
274C22×C4C2×D412yes—
275C22×C4C2×D412yes—
276C22×C4C2×D412yes—
277C22×C4C2×Q812yes—
278C22×C4C2×Q812yes—
279C22×C4C2×Q812yes—
280C22×C4C2×Q812yes—
281C22×C4#4112yes—
282C22×C4#4112yes—
283C22×C4#4112yes—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C22⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—28no—
295C22⋊C4—44no—
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—
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#12114yes—
325C23C22⋊Q814yes—
326C23C22⋊Q814yes—
327C23#12314yes—
328C23#12514yes—
329C23#12514yes—
330C23C4⋊Q814yes—
331C23C22×D411yescharacteristic
332C23C22×D411yescharacteristic
333C23C22×D411yescharacteristic
334C23C22×Q811yescharacteristic
335C23C22×Q811yescharacteristic
336C23C2×#4111yescharacteristic
337C23C2×#4111yescharacteristic
338C23C2×#4111yescharacteristic
339C23C2×#4111yescharacteristic
340C23C2×#4111yescharacteristic
341C23C2×#4111yescharacteristic
342C23H(5,𝔽2)11yescharacteristic
343C23H(5,𝔽2)11yescharacteristic
344C23H(5,𝔽2)11yescharacteristic
345C232−1+411yescharacteristic
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—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—416no—
386C2×C4—416no—
387C2×C4—416no—
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×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×C4C22⋊Q814yes—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22#39818yescomplemented
462C22C2×#12111yescharacteristic
463C22C2×C22⋊Q811yescharacteristic
464C22C2×C22⋊Q811yescharacteristic
465C22C2×C22⋊Q811yescharacteristic
466C22C2×#12311yescharacteristic
467C22#52511yescharacteristic
468C22#52511yescharacteristic
469C22C2×#12511yescharacteristic
470C22C2×#12511yescharacteristic
471C22#52911yescharacteristic
472C22#52911yescharacteristic
473C22C2×C4⋊Q811yescharacteristic
474C22#53311yescharacteristic
475C22#53811yescharacteristic
476C22#53811yescharacteristic
477C22#53911yescharacteristic
478C22#53911yescharacteristic
479C22#54311yescharacteristic
480C22#54411yescharacteristic
481C22D4211yescharacteristic
482C22D4211yescharacteristic
483C22#54611yescharacteristic
484C22#54611yescharacteristic
485C22#54611yescharacteristic
486C22#54611yescharacteristic
487C22#54611yescharacteristic
488C22#54711yescharacteristic
489C22#54711yescharacteristic
490C22D4×Q811yescharacteristic
491C22#55111yescharacteristic
492C22#55111yescharacteristic
493C22#55211yescharacteristic
494C22#55411yescharacteristic
495C22#55411yescharacteristic
496C22#55411yescharacteristic
497C4—24no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—81no—
511C2—28no—
512C2C2×#39811yescharacteristic
513C2#241811yescharacteristic
514C2#241911yescharacteristic
515C2#242411yescharacteristic
516C2#242511yescharacteristic
517C2#243711yescharacteristic
518C2#243811yescharacteristic
519C2#244011yescharacteristic
520C2#244911yescharacteristic
521C2#245011yescharacteristic
522C2#247311yescharacteristic
523C2#250511yescharacteristic
524C2#254911yescharacteristic
525C2#264611yescharacteristic
526C2#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×#398C2semidirect
7. #1445C2semidirect
8. #1444C2semidirect
9. #1442C2semidirect
10. #1442C2semidirect
11. #1441C2semidirect
12. #1441C2semidirect
13. #1441C2semidirect
461. C22#398semidirect

Maximal subgroups

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

subgroupindex
C2×#3982
C2×#3852
C2×#3802
C22×#952
#14452
#14442
#14422
#14412
C22⋊#952