Finite Group Databasebeta

← #14488

Subgroups — #14488

Order 256.

775 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×#385C211yesmaximal, characteristic
4#1444C211yesmaximal, characteristic, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1442C211yesmaximal, characteristic, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1439C211yesmaximal, characteristic
9#1439C211yesmaximal, characteristic
10#1436C211yesmaximal, characteristic, complemented
11C4⋊#95C211yesmaximal, characteristic, complemented
12C4⋊#95C211yesmaximal, characteristic, complemented
13#1434C211yesmaximal, characteristic, complemented
14C22⋊#95C211yesmaximal, characteristic
15C22⋊#95C211yesmaximal, characteristic
16C24×C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2211yescharacteristic
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27C22×C22⋊C4C2211yescharacteristic
28C22×C22⋊C4C2211yescharacteristic
29C22×C42C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#388—24no—
32#388C418yescomplemented
33#385—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—41no—
59C23×C4—41no—
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—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4C2×C414yescomplemented
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4C2×C414yes—
108C2×C22⋊C4C2×C414yes—
109C2×C22⋊C4C2×C414yes—
110C2×C22⋊C4D414yes—
111C2×C22⋊C4Q814yes—
112C2×C42—22no—
113C2×C42—22no—
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—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
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—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—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4C22×C412yes—
246C22×C4C22×C412yes—
247C22×C4C22×C412yes—
248C22×C4C22×C412yes—
249C22×C4C22×C412yes—
250C22×C4C22×C412yes—
251C22×C4C2×D412yes—
252C22×C4C2×D412yes—
253C22×C4C2×Q812yes—
254C22×C4C2×Q812yes—
255C22×C4#4112yes—
256C22×C4#4112yes—
257C22×C4#4112yes—
258C22×C4#4112yes—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4C4⋊C4116yescomplemented
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
315C23H(5,𝔽2)11yescharacteristic
316C23H(5,𝔽2)11yescharacteristic
317C23H(5,𝔽2)11yescharacteristic
318C23H(5,𝔽2)11yescharacteristic
319C23H(5,𝔽2)11yescharacteristic
320C23H(5,𝔽2)11yescharacteristic
321C232−1+411yescharacteristic
322C232−1+411yescharacteristic
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—416no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
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×C4—41no—
420C2×C4C2×C4⋊C414yes—
421C2×C4C2×C4⋊C414yes—
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
440C22#51811yescharacteristic
441C22#51811yescharacteristic
442C22#52011yescharacteristic
443C22#52011yescharacteristic
444C22#53511yescharacteristic
445C22#53611yescharacteristic
446C22#53811yescharacteristic
447C22#53811yescharacteristic
448C22#53811yescharacteristic
449C22#53911yescharacteristic
450C22#54011yescharacteristic
451C22#54211yescharacteristic
452C22#54211yescharacteristic
453C22#54211yescharacteristic
454C22#54411yescharacteristic
455C22#54411yescharacteristic
456C22#54611yescharacteristic
457C22#54611yescharacteristic
458C22#54611yescharacteristic
459C22#54611yescharacteristic
460C22#55111yescharacteristic
461C22#55211yescharacteristic
462C22#55311yescharacteristic
463C22#55411yescharacteristic
464C22#55411yescharacteristic
465C22#55411yescharacteristic
466C22#55411yescharacteristic
467C22#55611yescharacteristic
468C22#56011yescharacteristic
469C22#56011yescharacteristic
470C22#56211yescharacteristic
471C22#56211yescharacteristic
472C4—24no—
473C4—24no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C2—44no—
487C2#231711yescharacteristic
488C2#233511yescharacteristic
489C2#233711yescharacteristic
490C2#234911yescharacteristic
491C2#234911yescharacteristic
492C2#237311yescharacteristic
493C2#237711yescharacteristic
494C2#268511yescharacteristic
495C2#268711yescharacteristic
496C2#270011yescharacteristic
497C2#270311yescharacteristic
498C2#274211yescharacteristic
499C2#276811yescharacteristic
500C2#278811yescharacteristic
501C2#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
4. #1444C2semidirect
5. #1442C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
10. #1436C2semidirect
11. C4⋊#95C2semidirect
12. C4⋊#95C2semidirect
13. #1434C2semidirect
32. #388C4semidirect
93. C2×C4⋊C4C2×C4semidirect
117. C2×C42C2×C4semidirect
136. #95C2×C4semidirect
137. #95C2×C4semidirect
270. 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×#3852
#14442
#14422
#14412
#14392
#14362
C4⋊#952
#14342
C22⋊#952