Finite Group Databasebeta

← #14112

Subgroups — #14112

Order 256.

923 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×C4×C4⋊C4C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊C4C211yesmaximal, characteristic
8#1444C211yesmaximal, characteristic, complemented
9#1441C211yesmaximal, characteristic, complemented
10#1439C211yesmaximal, characteristic
11#1439C211yesmaximal, characteristic
12#1436C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14#1436C211yesmaximal, characteristic, complemented
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×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C22⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C42C2211yescharacteristic
32C22×C42C2211yescharacteristic
33C22×C42C2211yescharacteristic
34C22×C42C2211yescharacteristic
35#389—24no—
36#388—24no—
37#388C418yescomplemented
38#385—24no—
39#385—24no—
40C4×C4⋊C4—24no—
41C4×C22⋊C4—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—22no—
63C23×C4—22no—
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—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4C2×C414yescomplemented
103C2×C4⋊C4D414yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4C2×C414yes—
118C2×C22⋊C4C2×C414yes—
119C2×C22⋊C4C2×C414yes—
120C2×C22⋊C4D414yes—
121C2×C22⋊C4Q814yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—41no—
131C2×C42—41no—
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—22no—
145#95—22no—
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—28no—
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—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4C22×C412yes—
250C22×C4C22×C412yes—
251C22×C4C22×C412yes—
252C22×C4C22×C412yes—
253C22×C4C22×C412yes—
254C22×C4C22×C412yes—
255C22×C4C2×D412yes—
256C22×C4C2×D412yes—
257C22×C4C2×D412yes—
258C22×C4C2×D412yes—
259C22×C4C2×D412yes—
260C22×C4C2×D412yes—
261C22×C4C2×D412yes—
262C22×C4C2×D412yes—
263C22×C4C2×Q812yes—
264C22×C4C2×Q812yes—
265C22×C4#4112yes—
266C22×C4#4112yes—
267C22×C4#4112yes—
268C22×C4#4112yes—
269C22×C4#4112yes—
270C22×C4#4112yes—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C22⋊C4—28no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4C4⋊C4116yescomplemented
285C42—28no—
286C42—44no—
287C42—44no—
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
331C23C2×#4111yescharacteristic
332C23C2×#4111yescharacteristic
333C23C2×#4111yescharacteristic
334C23C2×#4111yescharacteristic
335C23H(5,𝔽2)11yescharacteristic
336C23H(5,𝔽2)11yescharacteristic
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—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—416no—
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×C4C2×C4⋊C414yes—
434C2×C4C2×C4⋊C414yes—
435C2×C4C4×D414yes—
436C2×C4C4×D414yes—
437C2×C4#12114yes—
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
456C22C4×#4111yescharacteristic
457C22C4×#4111yescharacteristic
458C22#52011yescharacteristic
459C22#52011yescharacteristic
460C22C2×#12111yescharacteristic
461C22#52511yescharacteristic
462C22#52511yescharacteristic
463C22#52511yescharacteristic
464C22#52911yescharacteristic
465C22#53211yescharacteristic
466C22#53311yescharacteristic
467C22#53311yescharacteristic
468C22#53811yescharacteristic
469C22#53911yescharacteristic
470C22#54111yescharacteristic
471C22#54211yescharacteristic
472C22D4211yescharacteristic
473C22D4211yescharacteristic
474C22#54611yescharacteristic
475C22#54611yescharacteristic
476C22#54711yescharacteristic
477C22#54711yescharacteristic
478C22#54711yescharacteristic
479C22#54811yescharacteristic
480C22D4×Q811yescharacteristic
481C22D4×Q811yescharacteristic
482C22#55111yescharacteristic
483C22#55211yescharacteristic
484C22#55411yescharacteristic
485C22#55411yescharacteristic
486C22#55611yescharacteristic
487C22#55911yescharacteristic
488C4—24no—
489C4—24no—
490C4—24no—
491C4—24no—
492C4—24no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—81no—
501C4—81no—
502C2—44no—
503C2#229611yescharacteristic
504C2C4×#12111yescharacteristic
505C2#233411yescharacteristic
506C2D4×C4⋊C411yescharacteristic
507C2D4×C4⋊C411yescharacteristic
508C2#235911yescharacteristic
509C2#236711yescharacteristic
510C2#242411yescharacteristic
511C2#242911yescharacteristic
512C2#245511yescharacteristic
513C2#246111yescharacteristic
514C2#246511yescharacteristic
515C2#247811yescharacteristic
516C2#255611yescharacteristic
517C2#258911yescharacteristic
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. C2×C4×C4⋊C4C2semidirect
8. #1444C2semidirect
9. #1441C2semidirect
12. #1436C2semidirect
13. #1436C2semidirect
14. #1436C2semidirect
15. C4⋊#95C2semidirect
37. #388C4semidirect
102. C2×C4⋊C4C2×C4semidirect
133. C2×C42C2×C4semidirect
147. #95C2×C4semidirect
148. #95C2×C4semidirect
284. 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×C4⋊C42
C2×C4×C22⋊C42
#14442
#14412
#14392
#14362
C4⋊#952