Finite Group Databasebeta

← #14405

Subgroups — #14405

Order 256.

875 conjugacy classes of subgroups.

The subgroup lattice has 519 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×#385C211yesmaximal, characteristic
6C2×#385C211yesmaximal, characteristic
7C2×#384C211yesmaximal, characteristic, complemented
8#1444C211yesmaximal, characteristic, complemented
9#1444C211yesmaximal, characteristic, complemented
10#1442C211yesmaximal, characteristic, complemented
11#1441C211yesmaximal, characteristic, complemented
12#1439C211yesmaximal, characteristic
13#1439C211yesmaximal, characteristic
14#1436C211yesmaximal, characteristic, complemented
15#1434C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊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×C22⋊C4C2211yescharacteristic
32C22×C22⋊C4C2211yescharacteristic
33C22×C42C2211yescharacteristic
34C22×C42C2211yescharacteristic
35#389—24no—
36#388—24no—
37#388C418yescomplemented
38#385—24no—
39#385—24no—
40#385—24no—
41#384—24no—
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—41no—
63C23×C4—41no—
64C23×C4—41no—
65C23×C4—41no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
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—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C2×C414yescomplemented
109C2×C4⋊C4D414yes—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4C2×C414yes—
124C2×C22⋊C4C2×C414yes—
125C2×C22⋊C4C2×C414yes—
126C2×C22⋊C4D414yes—
127C2×C22⋊C4Q814yes—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—41no—
133C2×C42C2×C414yescomplemented
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#95C2×C414yescomplemented
148#95C2×C414yescomplemented
149#95D414yes—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—21no—
161C24—21no—
162C24C22×C412yes—
163C24C2×D412yes—
164C24C2×Q812yes—
165C24C2411yescharacteristic
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—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—21no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4C22×C412yes—
254C22×C4C22×C412yes—
255C22×C4C22×C412yes—
256C22×C4C22×C412yes—
257C22×C4C22×C412yes—
258C22×C4C22×C412yes—
259C22×C4C2×D412yes—
260C22×C4C2×D412yes—
261C22×C4C2×D412yes—
262C22×C4C2×D412yes—
263C22×C4C2×D412yes—
264C22×C4C2×D412yes—
265C22×C4C2×D412yes—
266C22×C4C2×D412yes—
267C22×C4C2×Q812yes—
268C22×C4C2×Q812yes—
269C22×C4#4112yes—
270C22×C4#4112yes—
271C4⋊C4—28no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4C4⋊C4116yescomplemented
287C42—28no—
288C42—44no—
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—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C2×C4⋊C414yes—
324C23C23×C411yescharacteristic
325C23C22×D411yescharacteristic
326C23C22×D411yescharacteristic
327C23C22×D411yescharacteristic
328C23C22×Q811yescharacteristic
329C23C2×#4111yescharacteristic
330C23C2×#4111yescharacteristic
331C23H(5,𝔽2)11yescharacteristic
332C23H(5,𝔽2)11yescharacteristic
333C23H(5,𝔽2)11yescharacteristic
334C23H(5,𝔽2)11yescharacteristic
335C232−1+411yescharacteristic
336C232−1+411yescharacteristic
337C232−1+411yescharacteristic
338C232−1+411yescharacteristic
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—22no—
380C2×C4—22no—
381C2×C4—416no—
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×C4—41no—
433C2×C4—41no—
434C2×C4C2×C4⋊C414yes—
435C2×C4C2×C4⋊C414yes—
436C2×C4C4×D414yes—
437C2×C4C4×D414yes—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22C22×C4⋊C411yescharacteristic
454C22C2×C4×D411yescharacteristic
455C22C2×C4×D411yescharacteristic
456C22#52011yescharacteristic
457C22#52011yescharacteristic
458C22#52011yescharacteristic
459C22#52011yescharacteristic
460C22#53411yescharacteristic
461C22#53511yescharacteristic
462C22#53511yescharacteristic
463C22#53611yescharacteristic
464C22#53611yescharacteristic
465C22#53611yescharacteristic
466C22#53911yescharacteristic
467C22#53911yescharacteristic
468C22#54211yescharacteristic
469C22#54211yescharacteristic
470C22#54411yescharacteristic
471C22#54411yescharacteristic
472C22D4211yescharacteristic
473C22D4211yescharacteristic
474C22D4211yescharacteristic
475C22#54711yescharacteristic
476C22#54711yescharacteristic
477C22#54711yescharacteristic
478C22#54811yescharacteristic
479C22D4×Q811yescharacteristic
480C22D4×Q811yescharacteristic
481C22#55311yescharacteristic
482C22#55411yescharacteristic
483C22#55411yescharacteristic
484C22#56011yescharacteristic
485C22#56311yescharacteristic
486C22#56311yescharacteristic
487C22#56311yescharacteristic
488C4—24no—
489C4—24no—
490C4—24no—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C2—44no—
503C2#231711yescharacteristic
504C2#232211yescharacteristic
505C2#232611yescharacteristic
506C2D4×C4⋊C411yescharacteristic
507C2D4×C4⋊C411yescharacteristic
508C2#236111yescharacteristic
509C2#238311yescharacteristic
510C2#266911yescharacteristic
511C2#267411yescharacteristic
512C2#267711yescharacteristic
513C2#268011yescharacteristic
514C2#271811yescharacteristic
515C2#272011yescharacteristic
516C2#272911yescharacteristic
517C2#279011yescharacteristic
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
7. C2×#384C2semidirect
8. #1444C2semidirect
9. #1444C2semidirect
10. #1442C2semidirect
11. #1441C2semidirect
14. #1436C2semidirect
15. #1434C2semidirect
37. #388C4semidirect
108. C2×C4⋊C4C2×C4semidirect
133. C2×C42C2×C4semidirect
147. #95C2×C4semidirect
148. #95C2×C4semidirect
286. 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×#3842
#14442
#14422
#14412
#14392
#14362
#14342