Finite Group Databasebeta

← #30452

Subgroups — #30452

Order 256.

625 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#495C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic
5#2435C211yesmaximal, characteristic, complemented
6#2427C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#1934C211yesmaximal, characteristic, complemented
14#1923C211yesmaximal, characteristic, complemented
15#1879C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×Q16C2211yescharacteristic, complemented
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic, complemented
21C22×D8C2211yescharacteristic
22C22×D8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic, complemented
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic, complemented
30#495—24no—
31#463—24no—
32#450—24no—
33#447—24no—
34C2×C4⋊C8C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic, complemented
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic, complemented
46#397—21no—
47#394—21no—
48#394—21no—
49#394—21no—
50#391C2211yescharacteristic, complemented
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic, complemented
54#386—21no—
55#380—21no—
56#379—21no—
57#379—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4—41no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4—41no—
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C2×Q16—22no—
80C2×Q16—41no—
81C2×Q16—41no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—41no—
88C2×SD16—41no—
89C2×SD16—41no—
90C2×SD16—41no—
91C2×D8—22no—
92C2×D8—22no—
93C2×D8—41no—
94C2×D8—41no—
95C2×D8—41no—
96C2×D8D414yes—
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101#124—22no—
102#124—22no—
103#124—22no—
104#124—41no—
105#124—41no—
106#124—41no—
107C22⋊Q8—22no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111#121—22no—
112#121—41no—
113C4×D4—22no—
114C4×D4—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D411yescharacteristic
136C2×C22⋊C4D411yescharacteristic
137C2×C42C2311yescharacteristic
138C2×C42C2311yescharacteristic
139C4⋊C8—22no—
140#103—22no—
141#103—41no—
142#103—41no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—41no—
147#102D414yescomplemented
148C22⋊C8—22no—
149C22⋊C8—22no—
150C4×C8—22no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95D411yescharacteristic
155#95D411yescharacteristic
156C24—21no—
157C24—21no—
158C24—21no—
159C24—42no—
160C24—42no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—81no—
165C24—81no—
166C24C2×D411yescharacteristic
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—81no—
204C2×D4C2×D412yes—
205C2×D4C2×D412yes—
206C2×D4C2×D412yes—
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—42no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2411yescharacteristic
227Q16—44no—
228SD16—44no—
229SD16—44no—
230SD16—44no—
231D8—44no—
232D8—44no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8C2×D412yes—
244C2×C8C2×D412yes—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—81no—
253C4⋊C4C2×D412yes—
254C22⋊C4—22no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23#13512yes—
309C23C22×D411yescharacteristic
310Q8—41no—
311Q8—41no—
312Q8—41no—
313Q8—41no—
314Q8—41no—
315Q8—41no—
316Q8—41no—
317Q8—41no—
318D4—22no—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—48no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—41no—
331D4—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—84no—
367C2×C4C22×D411yescharacteristic
368C2×C4C22×D411yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C2×C4H(5,𝔽2)11yescharacteristic
371C2×C4H(5,𝔽2)11yescharacteristic
372C2×C4H(5,𝔽2)11yescharacteristic
373C8—22no—
374C8—22no—
375C8—41no—
376C8—41no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—816no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#53511yescharacteristic
407C22D4211yescharacteristic
408C22D4211yescharacteristic
409C22C2×#13511yescharacteristic
410C22#57511yescharacteristic
411C22#57611yescharacteristic
412C22#57711yescharacteristic
413C4—41no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—41no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4#53411yescharacteristic
424C4#53511yescharacteristic
425C4D4211yescharacteristic
426C4#56011yescharacteristic
427C2—42no—
428C2—82no—
429C2—82no—
430C2—82no—
431C2#266911yescharacteristic
432C2#321111yescharacteristic
433C2#322411yescharacteristic
434C2#326711yescharacteristic
435C2#327311yescharacteristic
436C2#328011yescharacteristic
437C2#328211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#495C2semidirect
2. C2×#463C2semidirect
3. C2×#450C2semidirect
5. #2435C2semidirect
6. #2427C2semidirect
7. #2055C2semidirect
8. #2042C2semidirect
9. #2015C2semidirect
10. #2012C2semidirect
11. #2001C2semidirect
12. #2000C2semidirect
13. #1934C2semidirect
14. #1923C2semidirect
15. #1879C2semidirect
17. C22×Q16C22semidirect
20. C22×SD16C22semidirect
25. C2×#124C22semidirect
29. C2×C4×D4C22semidirect
34. C2×C4⋊C8C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
42. C2×#102C22semidirect
45. C2×C4×C8C22semidirect
50. #391C22semidirect
53. #389C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect
147. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4952
C2×#4632
C2×#4502
C2×#4472
#24352
#24272
#20552
#20422
#20152
#20122
#20012
#20002
#19342
#19232
#18792