Finite Group Databasebeta

← #14220

Subgroups — #14220

Order 256.

893 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#393C211yesmaximal, characteristic
2C2×#391C211yesmaximal, characteristic, complemented
3C2×#391C211yesmaximal, characteristic, complemented
4C2×#385C211yesmaximal, characteristic
5C2×C4×C22⋊C4C211yesmaximal, characteristic
6#1443C211yesmaximal, characteristic, complemented
7#1442C211yesmaximal, characteristic, complemented
8#1439C211yesmaximal, characteristic
9#1437C211yesmaximal, characteristic
10#1437C211yesmaximal, characteristic
11#1436C211yesmaximal, characteristic, complemented
12#1436C211yesmaximal, characteristic, complemented
13#1436C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic
15C4×#95C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C24×C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4C2211yescharacteristic
22C22×C22⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C42C2211yescharacteristic
28C22×C42C2211yescharacteristic
29C22×C42C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#393C418yescomplemented
32#391—24no—
33#391—24no—
34#385—24no—
35C4×C22⋊C4—24no—
36C2×#95C2211yescharacteristic
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
56C25C2311yescharacteristic
57C22×Q8—22no—
58C22×Q8—22no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—41no—
62C22×Q8—41no—
63C22×Q8C2×C414yescomplemented
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
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—41no—
93C2×C4⋊C4—41no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4C2×C414yes—
109C2×C22⋊C4C2×C414yes—
110C2×C22⋊C4C2×C414yes—
111C2×C22⋊C4Q814yescomplemented
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124#95—22no—
125#95—22no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—22no—
136#95—41no—
137#95—41no—
138#95—41no—
139#95C2×C414yescomplemented
140#95C2×C414yescomplemented
141#95C2×C414yescomplemented
142#95D414yes—
143#95D414yes—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24—21no—
151C24—21no—
152C24—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24C22×C412yes—
157C24C2×Q812yes—
158C24#4112yes—
159C24C2411yescharacteristic
160C2×Q8—28no—
161C2×Q8—28no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
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—28no—
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×C4C22×C412yes—
253C22×C4C22×C412yes—
254C22×C4C22×C412yes—
255C22×C4C22×C412yes—
256C22×C4C22×C412yes—
257C22×C4C22×C412yes—
258C22×C4C2×D412yes—
259C22×C4C2×D412yes—
260C22×C4C2×D412yes—
261C22×C4C2×D412yes—
262C22×C4C2×D412yes—
263C22×C4C2×D412yes—
264C22×C4C2×Q812yes—
265C22×C4C2×Q812yes—
266C22×C4#4112yes—
267C22×C4#4112yes—
268C22×C4#4112yes—
269C22×C4#4112yes—
270C22×C4#4112yes—
271C22×C4#4112yes—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C22⋊C4—28no—
276C22⋊C4—28no—
277C22⋊C4—28no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C42—44no—
282C42—44no—
283C42—44no—
284C42—44no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
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—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23C4×Q814yes—
320C23C23×C411yescharacteristic
321C23C22×D411yescharacteristic
322C23C22×D411yescharacteristic
323C23C22×Q811yescharacteristic
324C23C2×#4111yescharacteristic
325C23C2×#4111yescharacteristic
326C23C2×#4111yescharacteristic
327C23C2×#4111yescharacteristic
328C23C2×#4111yescharacteristic
329C23C2×#4111yescharacteristic
330C23C2×#4111yescharacteristic
331C23H(5,𝔽2)11yescharacteristic
332C23H(5,𝔽2)11yescharacteristic
333C23H(5,𝔽2)11yescharacteristic
334C232−1+411yescharacteristic
335Q8—416no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
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—416no—
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×C4C4×D414yes—
432C2×C4C4×D414yes—
433C2×C4C22⋊Q814yes—
434C2×C4#12414yes—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22C2×C4×D411yescharacteristic
451C22C2×C4×D411yescharacteristic
452C22C2×C4×Q811yescharacteristic
453C22C4×#4111yescharacteristic
454C22C4×#4111yescharacteristic
455C22#51811yescharacteristic
456C22#52011yescharacteristic
457C22C2×C22⋊Q811yescharacteristic
458C22#52511yescharacteristic
459C22C2×#12411yescharacteristic
460C22#52911yescharacteristic
461C22#52911yescharacteristic
462C22#52911yescharacteristic
463C22#53211yescharacteristic
464C22#53311yescharacteristic
465C22#53811yescharacteristic
466C22#53811yescharacteristic
467C22#54211yescharacteristic
468C22#54211yescharacteristic
469C22D4211yescharacteristic
470C22#54611yescharacteristic
471C22#54611yescharacteristic
472C22#54611yescharacteristic
473C22#54611yescharacteristic
474C22#54811yescharacteristic
475C22D4×Q811yescharacteristic
476C22#55111yescharacteristic
477C22#55111yescharacteristic
478C22#55111yescharacteristic
479C22#55211yescharacteristic
480C22#55411yescharacteristic
481C22#55411yescharacteristic
482C22#55411yescharacteristic
483C22#55611yescharacteristic
484C22#55911yescharacteristic
485C4—24no—
486C4—24no—
487C4—24no—
488C4—24no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—81no—
498C4—81no—
499C2—44no—
500C2C4×C22⋊Q811yescharacteristic
501C2C4×#12411yescharacteristic
502C2#233711yescharacteristic
503C2#235811yescharacteristic
504C2#236011yescharacteristic
505C2#236711yescharacteristic
506C2#236811yescharacteristic
507C2#245011yescharacteristic
508C2#245211yescharacteristic
509C2#247211yescharacteristic
510C2#247711yescharacteristic
511C2#249211yescharacteristic
512C2#252711yescharacteristic
513C2#255711yescharacteristic
514C2#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
2. C2×#391C2semidirect
3. C2×#391C2semidirect
6. #1443C2semidirect
7. #1442C2semidirect
11. #1436C2semidirect
12. #1436C2semidirect
13. #1436C2semidirect
15. C4×#95C2semidirect
31. #393C4semidirect
63. C22×Q8C2×C4semidirect
111. C2×C22⋊C4Q8semidirect
139. #95C2×C4semidirect
140. #95C2×C4semidirect
141. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3932
C2×#3912
C2×#3852
C2×C4×C22⋊C42
#14432
#14422
#14392
#14372
#14362
C22⋊#952
C4×#952