Finite Group Databasebeta

← #24220

Subgroups — #24220

Order 256.

778 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#535C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#460C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5C2×#450C211yesmaximal, characteristic, complemented
6#2323C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2014C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1982C211yesmaximal, characteristic, complemented
13#1798C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×D4C2211yescharacteristic
18C22×Q16C2211yescharacteristic, complemented
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic, complemented
22#535—24no—
23#535—24no—
24C2×#127C2211yescharacteristic
25C2×#127C2211yescharacteristic
26C2×#124C2211yescharacteristic
27C2×#124C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×#121—21no—
30C2×#121—21no—
31C2×#120—21no—
32C2×#120—21no—
33C2×C4×Q8C2211yescharacteristic, complemented
34C2×#117C2211yescharacteristic
35#464—24no—
36#460—24no—
37#452—24no—
38#450—24no—
39C2×C4⋊C8C2211yescharacteristic, complemented
40C2×C4⋊C8C2211yescharacteristic, complemented
41C2×#103C2211yescharacteristic, complemented
42C2×#103C2211yescharacteristic, complemented
43C2×#103C2211yescharacteristic, complemented
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×#102C2211yescharacteristic, complemented
51C2×#102C2211yescharacteristic, complemented
52C2×#102C2211yescharacteristic, complemented
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55#391C2211yescharacteristic, complemented
56#389—21no—
57#389C2211yescharacteristic, complemented
58#388—21no—
59#388—21no—
60#387—21no—
61#380C2211yescharacteristic
62C4×C4⋊C4C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×D4—21no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—22no—
83C22×D4—22no—
84C22×D4—41no—
85C22×D4—41no—
86C22×D4C2311yescharacteristic
87C22×D4C2311yescharacteristic
88C22×D4C2311yescharacteristic
89C23×C4—21no—
90C23×C4C2311yescharacteristic
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—41no—
94C2×SD16—22no—
95C2×SD16—22no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—22no—
99C2×SD16—22no—
100C2×SD16—41no—
101C2×SD16—41no—
102C2×SD16—41no—
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107#127—22no—
108#127—22no—
109#127—22no—
110#127D414yes—
111#124—22no—
112#124—22no—
113#124—22no—
114#124D414yes—
115C22⋊Q8—22no—
116C22⋊Q8D414yescomplemented
117#121—28no—
118#121—28no—
119#120—28no—
120#120—28no—
121C4×Q8—22no—
122C4×Q8—22no—
123#117—22no—
124#117—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4D411yescharacteristic
129C2×C4⋊C4D411yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4D411yescharacteristic
140C2×C22⋊C4D411yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42C2311yescharacteristic
144C2×C42C2311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—41no—
150#102—22no—
151#102—22no—
152#102—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—41no—
157#102—41no—
158#102—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—24no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—42no—
171C24—42no—
172C24—81no—
173C24—81no—
174C24—81no—
175C24—81no—
176C24C2×D411yescharacteristic
177#41—44no—
178#41—44no—
179#41—44no—
180#41—44no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8C2×D412yes—
192C2×Q8C2×D412yes—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—24no—
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—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—81no—
223C2×D4—81no—
224C2×D4—81no—
225C2×D4—81no—
226C2×D4—81no—
227C2×D4—81no—
228C2×D4—81no—
229C2×D4—81no—
230C2×D4C2×D412yes—
231C2×D4C2×D412yes—
232C2×D4C2×D412yes—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D412yes—
251C22×C4#4112yes—
252C22×C4C2411yescharacteristic
253Q16—44no—
254SD16—44no—
255SD16—44no—
256SD16—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×C8—21no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4C2×D412yes—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C42—42no—
291C42—81no—
292C42C2×D412yes—
293C42C2×D412yes—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—48no—
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—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—84no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23#12112yes—
329C23C22×D411yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—41no—
338D4—22no—
339D4—22no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—22no—
344D4—48no—
345D4—48no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—41no—
352D4—84no—
353D4—84no—
354D4—84no—
355D4—84no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—48no—
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#12012yes—
389C2×C4#12012yes—
390C2×C4#12112yes—
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C22×D411yescharacteristic
394C2×C4C2×#4111yescharacteristic
395C2×C4H(5,𝔽2)11yescharacteristic
396C2×C4H(5,𝔽2)11yescharacteristic
397C8—41no—
398C8—41no—
399C8—41no—
400C8—41no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—816no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22C2×#12011yescharacteristic
431C22C2×#12111yescharacteristic
432C22#53511yescharacteristic
433C22#57611yescharacteristic
434C22#57711yescharacteristic
435C22#57711yescharacteristic
436C22#57711yescharacteristic
437C4—22no—
438C4—22no—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—41no—
443C4—41no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4D4211yescharacteristic
448C4D4211yescharacteristic
449C4#54611yescharacteristic
450C4#54611yescharacteristic
451C2—42no—
452C2—82no—
453C2—82no—
454C2—82no—
455C2#240811yescharacteristic
456C2#300911yescharacteristic
457C2#301211yescharacteristic
458C2#304311yescharacteristic
459C2#304511yescharacteristic
460C2#322211yescharacteristic
461C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#464C2semidirect
3. C2×#460C2semidirect
4. C2×#452C2semidirect
5. C2×#450C2semidirect
6. #2323C2semidirect
7. #2042C2semidirect
8. #2041C2semidirect
9. #2015C2semidirect
10. #2014C2semidirect
11. #1983C2semidirect
12. #1982C2semidirect
13. #1798C2semidirect
14. #1788C2semidirect
15. #1786C2semidirect
18. C22×Q16C22semidirect
21. C22×SD16C22semidirect
33. C2×C4×Q8C22semidirect
39. C2×C4⋊C8C22semidirect
40. C2×C4⋊C8C22semidirect
41. C2×#103C22semidirect
42. C2×#103C22semidirect
43. C2×#103C22semidirect
50. C2×#102C22semidirect
51. C2×#102C22semidirect
52. C2×#102C22semidirect
55. #391C22semidirect
57. #389C22semidirect
62. C4×C4⋊C4C22semidirect
63. #336C22semidirect
64. #336C22semidirect
116. C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5352
C2×#4642
C2×#4602
C2×#4522
C2×#4502
#23232
#20422
#20412
#20152
#20142
#19832
#19822
#17982
#17882
#17862