Finite Group Databasebeta

← #24441

Subgroups — #24441

Order 256.

761 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic, complemented
5C2×#417C211yesmaximal, characteristic, complemented
6#2416C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1870C211yesmaximal, characteristic, complemented
14#1863C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C22×M16C2211yescharacteristic
22#548—24no—
23#548—24no—
24C2×#124—21no—
25C2×#124C2211yescharacteristic
26C2×C22⋊Q8—21no—
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121—21no—
29C2×#121—21no—
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×#121C2211yescharacteristic
33C2×C4×Q8C2211yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2211yescharacteristic
36C22×C4⋊C4C2211yescharacteristic
37#468—24no—
38#449—24no—
39#448—24no—
40C2×#107C2211yescharacteristic
41#417—24no—
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55#396—21no—
56#394—21no—
57#390C2211yescharacteristic
58#389C2211yescharacteristic
59#386—21no—
60#380C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#336C2211yescharacteristic
64#336C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—41no—
69C2×#41—41no—
70C2×#41D414yescomplemented
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C2×SD16—22no—
93C2×SD16—22no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—41no—
98C2×D8—22no—
99C2×D8—41no—
100C2×D8—41no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—41no—
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108#124—28no—
109#124—22no—
110#124—22no—
111C22⋊Q8—28no—
112C22⋊Q8—22no—
113C22⋊Q8D414yescomplemented
114#121—28no—
115#121—22no—
116#121—22no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—41no—
121#121—44no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×D4—28no—
125C4×D4—22no—
126C4×D4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4D411yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4D411yescharacteristic
148C2×C22⋊C4D411yescharacteristic
149C2×C42—21no—
150C2×C42C2311yescharacteristic
151#107—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—22no—
156#103—41no—
157#103D414yescomplemented
158#102—22no—
159#102—22no—
160#102—22no—
161#102—22no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—41no—
165#95—21no—
166#95—21no—
167C24—21no—
168C24—21no—
169C24C2×D411yescharacteristic
170C24C2×D411yescharacteristic
171#41—28no—
172#41—44no—
173#41—44no—
174#41—44no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8C2×D412yes—
191C2×Q8C2×D412yes—
192C2×Q8C2×D412yes—
193C2×Q8#4112yes—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—81no—
220C2×D4C2×D412yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
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—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—81no—
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D412yes—
251C22×C4C2×D412yes—
252C22×C4C2411yescharacteristic
253SD16—44no—
254SD16—44no—
255D8—44no—
256M16—44no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8C2×D412yes—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4C2×D412yes—
281C4⋊C4C2×D412yes—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—81no—
293C42—24no—
294C42—21no—
295C42—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—22no—
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#12012yes—
323C23AGL(1,ℤ/8)12yes—
324C23C22×D411yescharacteristic
325Q8—22no—
326Q8—22no—
327Q8—22no—
328Q8—22no—
329Q8—22no—
330Q8—48no—
331Q8—41no—
332Q8—41no—
333Q8#12114yescomplemented
334D4—22no—
335D4—22no—
336D4—48no—
337D4—44no—
338D4—48no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4—84no—
346D4—84no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—48no—
369C2×C4—48no—
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—84no—
387C2×C4#12012yes—
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C22×D411yescharacteristic
391C2×C4C2×#4111yescharacteristic
392C2×C4H(5,𝔽2)11yescharacteristic
393C2×C4H(5,𝔽2)11yescharacteristic
394C8—22no—
395C8—41no—
396C8—41no—
397C8—41no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C2×#12011yescharacteristic
427C22D4211yescharacteristic
428C22#54611yescharacteristic
429C22C2×AGL(1,ℤ/8)11yescharacteristic
430C22#57511yescharacteristic
431C22#57711yescharacteristic
432C22#57811yescharacteristic
433C4—22no—
434C4—24no—
435C4—41no—
436C4—42no—
437C4—41no—
438C4—42no—
439C4—42no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4C2×#12111yescharacteristic
444C4#53411yescharacteristic
445C4D4211yescharacteristic
446C4#54611yescharacteristic
447C2—42no—
448C2—42no—
449C2—82no—
450C2—161no—
451C2#241011yescharacteristic
452C2#300611yescharacteristic
453C2#301311yescharacteristic
454C2#326511yescharacteristic
455C2#327711yescharacteristic
456C2#330811yescharacteristic
457C2#332611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#548C2semidirect
2. C2×#468C2semidirect
3. C2×#449C2semidirect
4. C2×#448C2semidirect
5. C2×#417C2semidirect
6. #2416C2semidirect
7. #2042C2semidirect
8. #2033C2semidirect
9. #2012C2semidirect
10. #1999C2semidirect
11. #1880C2semidirect
12. #1872C2semidirect
13. #1870C2semidirect
14. #1863C2semidirect
15. #1787C2semidirect
70. C2×#41D4semidirect
113. C22⋊Q8D4semidirect
157. #103D4semidirect
333. Q8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4682
C2×#4492
C2×#4482
C2×#4172
#24162
#20422
#20332
#20122
#19992
#18802
#18722
#18702
#18632
#17872