Finite Group Databasebeta

← #13789

Subgroups — #13789

Order 256.

913 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#388C211yesmaximal, characteristic
2C2×#388C211yesmaximal, characteristic
3C2×#385C211yesmaximal, characteristic
4C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C4C211yesmaximal, characteristic
6#1437C211yesmaximal, characteristic
7#1437C211yesmaximal, characteristic
8#1436C211yesmaximal, characteristic, complemented
9#1436C211yesmaximal, characteristic, complemented
10#1436C211yesmaximal, characteristic, complemented
11#1436C211yesmaximal, characteristic, complemented
12C4⋊#95C211yesmaximal, characteristic, complemented
13#1434C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic
15C4×#95C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×C4⋊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
31C22×C42C2211yescharacteristic
32C22×C42C2211yescharacteristic
33#388C418yescomplemented
34#388C418yescomplemented
35#385C418yescomplemented
36C4×C4⋊C4—24no—
37C4×C22⋊C4—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
56C25C2311yescharacteristic
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—41no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C2×C414yes—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4C2×C414yes—
87C2×C4⋊C4C2×C414yes—
88C2×C4⋊C4C2×C414yescomplemented
89C2×C4⋊C4C2×C414yescomplemented
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4C2×C414yes—
101C2×C22⋊C4C2×C414yes—
102C2×C22⋊C4C2×C414yes—
103C2×C22⋊C4C2×C414yes—
104C2×C22⋊C4C2×C414yes—
105C2×C22⋊C4C2×C414yes—
106C2×C22⋊C4C2×C414yes—
107C2×C22⋊C4C2×C414yes—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42—41no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C2×C414yescomplemented
124C2×C42C2×C414yescomplemented
125C2×C42D414yes—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—22no—
131#95—22no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95C2×C414yes—
139#95C2×C414yes—
140#95C2×C414yescomplemented
141#95C2×C414yescomplemented
142#95C2×C414yescomplemented
143#95C2×C414yescomplemented
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—
157C24C22×C412yes—
158C24C22×C412yes—
159C24C2411yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
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—28no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4C22×C412yes—
242C22×C4C22×C412yes—
243C22×C4C22×C412yes—
244C22×C4C22×C412yes—
245C22×C4C22×C412yes—
246C22×C4C22×C412yes—
247C22×C4C22×C412yes—
248C22×C4C22×C412yes—
249C22×C4C22×C412yes—
250C22×C4C22×C412yes—
251C22×C4C22×C412yes—
252C22×C4C22×C412yes—
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×C4#4112yes—
263C22×C4#4112yes—
264C22×C4#4112yes—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C22⋊C4—28no—
270C22⋊C4—28no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4C42116yescomplemented
275C42—28no—
276C42—44no—
277C42—44no—
278C42—44no—
279C42—44no—
280C42—44no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
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—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C2×C4214yes—
316C23C23×C411yescharacteristic
317C23C23×C411yescharacteristic
318C23C23×C411yescharacteristic
319C23C22×D411yescharacteristic
320C23C2×#4111yescharacteristic
321C23C2×#4111yescharacteristic
322C23C2×#4111yescharacteristic
323C23H(5,𝔽2)11yescharacteristic
324C23H(5,𝔽2)11yescharacteristic
325C23H(5,𝔽2)11yescharacteristic
326C23H(5,𝔽2)11yescharacteristic
327C23H(5,𝔽2)11yescharacteristic
328C232−1+411yescharacteristic
329C232−1+411yescharacteristic
330C232−1+411yescharacteristic
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
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—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—416no—
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×C4C2×C4214yes—
426C2×C4C2×C4214yes—
427C2×C4C4×D414yes—
428C2×C4C4×D414yes—
429C2×C4C4×D414yes—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22C22×C4211yescharacteristic
446C22C2×C4×D411yescharacteristic
447C22C2×C4×D411yescharacteristic
448C22C2×C4×D411yescharacteristic
449C22C4×#4111yescharacteristic
450C22C4×#4111yescharacteristic
451C22C4×#4111yescharacteristic
452C22#51811yescharacteristic
453C22#51811yescharacteristic
454C22#51811yescharacteristic
455C22#51811yescharacteristic
456C22#51911yescharacteristic
457C22#52011yescharacteristic
458C22#52011yescharacteristic
459C22#52011yescharacteristic
460C22#52011yescharacteristic
461C22#52011yescharacteristic
462C22#52011yescharacteristic
463C22#52011yescharacteristic
464C22#53411yescharacteristic
465C22#53511yescharacteristic
466C22#53611yescharacteristic
467C22#53711yescharacteristic
468C22#53811yescharacteristic
469C22#53811yescharacteristic
470C22#53811yescharacteristic
471C22#53911yescharacteristic
472C22#53911yescharacteristic
473C22#53911yescharacteristic
474C22#54011yescharacteristic
475C22#54111yescharacteristic
476C22#54211yescharacteristic
477C22#54211yescharacteristic
478C22#54211yescharacteristic
479C22#54211yescharacteristic
480C4—24no—
481C4—24no—
482C4—24no—
483C4—24no—
484C4—24no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C2—44no—
495C2C42×D411yescharacteristic
496C2#227411yescharacteristic
497C2#227511yescharacteristic
498C2#232111yescharacteristic
499C2#232211yescharacteristic
500C2#232311yescharacteristic
501C2#232411yescharacteristic
502C2#232511yescharacteristic
503C2#232611yescharacteristic
504C2#233211yescharacteristic
505C2#233411yescharacteristic
506C2#233411yescharacteristic
507C2#233511yescharacteristic
508C2#233511yescharacteristic
509C2#233711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC22C22×C42
prime radicalC25C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×C4×C4⋊C4C2semidirect
8. #1436C2semidirect
9. #1436C2semidirect
10. #1436C2semidirect
11. #1436C2semidirect
12. C4⋊#95C2semidirect
13. #1434C2semidirect
15. C4×#95C2semidirect
33. #388C4semidirect
34. #388C4semidirect
35. #385C4semidirect
88. C2×C4⋊C4C2×C4semidirect
89. C2×C4⋊C4C2×C4semidirect
123. C2×C42C2×C4semidirect
124. C2×C42C2×C4semidirect
140. #95C2×C4semidirect
141. #95C2×C4semidirect
142. #95C2×C4semidirect
143. #95C2×C4semidirect
274. C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#3882
C2×#3852
C2×C4×C4⋊C42
C2×C4×C22⋊C42
#14372
#14362
C4⋊#952
#14342
C22⋊#952
C4×#952