Finite Group Databasebeta

← #14114

Subgroups — #14114

Order 256.

815 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#388C211yesmaximal, characteristic
2C2×#388C211yesmaximal, characteristic
3C2×C4×C22⋊C4C211yesmaximal, characteristic
4#1442C211yesmaximal, characteristic, complemented
5#1441C211yesmaximal, characteristic, complemented
6#1439C211yesmaximal, characteristic
7#1439C211yesmaximal, characteristic
8#1436C211yesmaximal, characteristic, complemented
9#1436C211yesmaximal, characteristic, complemented
10#1436C211yesmaximal, characteristic, complemented
11C4⋊#95C211yesmaximal, characteristic, complemented
12#1434C211yesmaximal, characteristic, complemented
13C22⋊#95C211yesmaximal, characteristic
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
31#388—24no—
32#388C418yescomplemented
33C4×C22⋊C4—24no—
34C2×#95C2211yescharacteristic
35C2×#95C2211yescharacteristic
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
54C25C2311yescharacteristic
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4C2311yescharacteristic
63C23×C4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4C2×C414yescomplemented
88C2×C22⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4C2×C414yes—
102C2×C22⋊C4C2×C414yes—
103C2×C22⋊C4C2×C414yes—
104C2×C22⋊C4D414yes—
105C2×C22⋊C4Q814yes—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—41no—
116C2×C42—41no—
117C2×C42C2×C414yescomplemented
118#95—22no—
119#95—22no—
120#95—22no—
121#95—22no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—22no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95—22no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—41no—
136#95C2×C414yescomplemented
137#95C2×C414yescomplemented
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—21no—
145C24—21no—
146C24—21no—
147C24—21no—
148C24—21no—
149C24—21no—
150C24C22×C412yes—
151C24C2×D412yes—
152C24C2×Q812yes—
153C24C2411yescharacteristic
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—28no—
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—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×C4C2×D412yes—
248C22×C4C2×D412yes—
249C22×C4C2×Q812yes—
250C22×C4C2×Q812yes—
251C22×C4#4112yes—
252C22×C4#4112yes—
253C22×C4#4112yes—
254C22×C4#4112yes—
255C22×C4#4112yes—
256C22×C4#4112yes—
257C22×C4#4112yes—
258C22×C4#4112yes—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C22⋊C4—28no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4C4⋊C4116yescomplemented
269C42—44no—
270C42—44no—
271C42—44no—
272C42—44no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23C2×C4⋊C414yes—
308C23C23×C411yescharacteristic
309C23C22×D411yescharacteristic
310C23C22×Q811yescharacteristic
311C23C2×#4111yescharacteristic
312C23C2×#4111yescharacteristic
313C23C2×#4111yescharacteristic
314C23C2×#4111yescharacteristic
315C23C2×#4111yescharacteristic
316C23C2×#4111yescharacteristic
317C23C2×#4111yescharacteristic
318C23C2×#4111yescharacteristic
319C23H(5,𝔽2)11yescharacteristic
320C23H(5,𝔽2)11yescharacteristic
321C23H(5,𝔽2)11yescharacteristic
322C23H(5,𝔽2)11yescharacteristic
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
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—416no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
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×C4C2×C4⋊C414yes—
420C2×C4C2×C4⋊C414yes—
421C2×C4#12614yes—
422C22—24no—
423C22—24no—
424C22—24no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22C22×C4⋊C411yescharacteristic
438C22C4×#4111yescharacteristic
439C22C4×#4111yescharacteristic
440C22C4×#4111yescharacteristic
441C22C4×#4111yescharacteristic
442C22#51811yescharacteristic
443C22#51811yescharacteristic
444C22#52511yescharacteristic
445C22#52511yescharacteristic
446C22C2×#12611yescharacteristic
447C22#52911yescharacteristic
448C22#52911yescharacteristic
449C22#52911yescharacteristic
450C22#53311yescharacteristic
451C22#53311yescharacteristic
452C22#53811yescharacteristic
453C22#53811yescharacteristic
454C22#54011yescharacteristic
455C22#54011yescharacteristic
456C22#54611yescharacteristic
457C22#54611yescharacteristic
458C22#54611yescharacteristic
459C22#54611yescharacteristic
460C22#55111yescharacteristic
461C22#55111yescharacteristic
462C22#55111yescharacteristic
463C22#55311yescharacteristic
464C22#55311yescharacteristic
465C22#55411yescharacteristic
466C22#55411yescharacteristic
467C22#55411yescharacteristic
468C22#55411yescharacteristic
469C22#55511yescharacteristic
470C22#55511yescharacteristic
471C22#55911yescharacteristic
472C4—24no—
473C4—24no—
474C4—24no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—81no—
486C2—44no—
487C2#229611yescharacteristic
488C2C4×#12611yescharacteristic
489C2#233311yescharacteristic
490C2#234911yescharacteristic
491C2#234911yescharacteristic
492C2#237211yescharacteristic
493C2#237411yescharacteristic
494C2#245711yescharacteristic
495C2#245911yescharacteristic
496C2#247611yescharacteristic
497C2#248011yescharacteristic
498C2#249711yescharacteristic
499C2#252211yescharacteristic
500C2#257411yescharacteristic
501C2#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
4. #1442C2semidirect
5. #1441C2semidirect
8. #1436C2semidirect
9. #1436C2semidirect
10. #1436C2semidirect
11. C4⋊#95C2semidirect
12. #1434C2semidirect
15. C4×#95C2semidirect
32. #388C4semidirect
87. C2×C4⋊C4C2×C4semidirect
117. C2×C42C2×C4semidirect
136. #95C2×C4semidirect
137. #95C2×C4semidirect
268. C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3882
C2×C4×C22⋊C42
#14422
#14412
#14392
#14362
C4⋊#952
#14342
C22⋊#952
C4×#952