Finite Group Databasebeta

← #18822

Subgroups — #18822

Order 256.

483 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic, complemented
2#3251C211yesmaximal, characteristic, complemented
3#3228C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1654C211yesmaximal, characteristic, complemented
6#1651C211yesmaximal, characteristic, complemented
7#1650C211yesmaximal, characteristic, complemented
8#1647C211yesmaximal, characteristic, complemented
9#1628C211yesmaximal, characteristic, complemented
10#1627C211yesmaximal, characteristic, complemented
11#1625C211yesmaximal, characteristic, complemented
12#1625C211yesmaximal, characteristic, complemented
13#1561C211yesmaximal, characteristic, complemented
14#1495C211yesmaximal, characteristic, complemented
15#1486C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)—21no—
17#556C2211yescharacteristic
18#554—21no—
19#552—21no—
20#551—22no—
21D4×Q8C2211yescharacteristic
22#548—21no—
23#547C2211yescharacteristic
24#547C2211yescharacteristic
25#546—21no—
26#542—21no—
27#539—22no—
28#533C2211yescharacteristic
29#532C2211yescharacteristic
30#529C2211yescharacteristic
31#525—21no—
32C2×C22⋊Q8—21no—
33#520—21no—
34C2×C4×D4C2211yescharacteristic
35#500C2211yescharacteristic, complemented
36#498C2211yescharacteristic, complemented
37#486C2211yescharacteristic, complemented
38#486C2211yescharacteristic, complemented
39#482—22no—
40#478—22no—
41#478—21no—
42#475C2211yescharacteristic, complemented
43#474C2211yescharacteristic, complemented
44#469—21no—
45#468—21no—
46#463—21no—
47#463—21no—
48#463—21no—
49#461C2211yescharacteristic
50#461C2211yescharacteristic
51#460C2211yescharacteristic, complemented
52#459C2211yescharacteristic
53#459C2211yescharacteristic
54#459C2211yescharacteristic, complemented
55#431C2211yescharacteristic
56#423C2211yescharacteristic
57#423C2211yescharacteristic
58#417—21no—
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, complemented
61#332C2211yescharacteristic, complemented
62#332C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41—21no—
74C22×Q8—21no—
75C22×D4D411yescharacteristic
76C23×C4—21no—
77AGL(1,ℤ/8)—44no—
78C2×SD16—21no—
79C2×SD16—21no—
80C2×SD16—21no—
81C2×D8—21no—
82C2×D8—21no—
83C2×D8—21no—
84C2×M16—21no—
85C2×M16—21no—
86C4⋊Q8—21no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89C4⋊Q8C2311yescharacteristic
90#127C2311yescharacteristic
91#126—22no—
92#126—22no—
93#126—21no—
94#125—22no—
95#125D411yescharacteristic
96#125D412yes—
97#124—22no—
98#124—21no—
99#124—21no—
100#124D411yescharacteristic
101#123—22no—
102#123—22no—
103#123—21no—
104#123—22no—
105#123—22no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8—22no—
109C22⋊Q8—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8—21no—
116C22⋊Q8D411yescharacteristic
117#121—22no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121—21no—
122#121D411yescharacteristic
123#120—22no—
124C4×Q8—21no—
125C4×Q8—21no—
126C4×Q8—21no—
127C4×Q8C2311yescharacteristic
128C4×D4—22no—
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4D411yescharacteristic
138C4×D4D411yescharacteristic
139C4×D4C2311yescharacteristic
140C4×D4C2311yescharacteristic
141C4×D4C2311yescharacteristic
142#117—22no—
143#117—21no—
144#117—21no—
145#117D412yes—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4D411yescharacteristic
150C2×C22⋊C4—22no—
151C2×C22⋊C4—21no—
152C2×C42C2311yescharacteristic
153#107—21no—
154#107—22no—
155#106—21no—
156#106—22no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163C4⋊C8C2311yescharacteristic
164#103—21no—
165#103—21no—
166#103—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—21no—
172#102—22no—
173#102—22no—
174C22⋊C8—21no—
175C4×C8C2311yescharacteristic
176C4×C8C2311yescharacteristic
177C24—21no—
178#41—42no—
179#41—42no—
180#41—42no—
181#41—42no—
182#41—81no—
183#41—81no—
184C2×Q8—22no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8C2×D411yescharacteristic
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4C2×D411yescharacteristic
202C2×D4C2×D411yescharacteristic
203C2×D4C2×D411yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219SD16—42no—
220SD16—42no—
221SD16—81no—
222D8—42no—
223D8—42no—
224D8—81no—
225M16—42no—
226M16—81no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—21no—
238C4⋊C4—22no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4C2×D411yescharacteristic
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D411yescharacteristic
251C4⋊C4C2×D412yes—
252C4⋊C4C2×D411yescharacteristic
253C4⋊C4C2×D412yes—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—21no—
260C22⋊C4—22no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C42—22no—
272C42—22no—
273C42—21no—
274C42C2×D411yescharacteristic
275C42C2×D411yescharacteristic
276C42C2×D411yescharacteristic
277C42C2411yescharacteristic
278C23—21no—
279C23—21no—
280C23—21no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—81no—
285C23#12011yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294Q8—82no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—42no—
301D4—42no—
302D4—42no—
303D4—44no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—82no—
309D4—82no—
310D4AGL(1,ℤ/8)12yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#12011yescharacteristic
338C2×C4#12011yescharacteristic
339C2×C4#12011yescharacteristic
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C22×D411yescharacteristic
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C8—41no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—84no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×#12011yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4C2×AGL(1,ℤ/8)11yescharacteristic
379C4#57511yescharacteristic
380C4#57611yescharacteristic
381C4#57711yescharacteristic
382C2—41no—
383C2—42no—
384C2—81no—
385C2—81no—
386C2—161no—
387C2#300611yescharacteristic
388C2#301211yescharacteristic
389C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3520C2semidirect
2. #3251C2semidirect
3. #3228C2semidirect
4. #3036C2semidirect
5. #1654C2semidirect
6. #1651C2semidirect
7. #1650C2semidirect
8. #1647C2semidirect
9. #1628C2semidirect
10. #1627C2semidirect
11. #1625C2semidirect
12. #1625C2semidirect
13. #1561C2semidirect
14. #1495C2semidirect
15. #1486C2semidirect
35. #500C22semidirect
36. #498C22semidirect
37. #486C22semidirect
38. #486C22semidirect
42. #475C22semidirect
43. #474C22semidirect
51. #460C22semidirect
54. #459C22semidirect
59. #332C22semidirect
60. #332C22semidirect
61. #332C22semidirect
62. #332C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
69. D4⋊C8C22semidirect
70. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35202
#32512
#32282
#30362
#16542
#16512
#16502
#16472
#16282
#16272
#16252
#15612
#14952
#14862