Finite Group Databasebeta

← #18786

Subgroups — #18786

Order 256.

661 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#41C211yesmaximal, characteristic, complemented
2#3205C211yesmaximal, characteristic, complemented
3#3038C211yesmaximal, characteristic, complemented
4C2×#459C211yesmaximal, characteristic, complemented
5#1627C212yesmaximal, complemented
6#1625C212yesmaximal, complemented
7#1621C212yesmaximal, complemented
8#1619C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1486C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C2×AGL(1,ℤ/8)—21no—
14C22×D8—21no—
15#550C2211yescharacteristic, complemented
16D4×Q8C2211yescharacteristic, complemented
17#548—21no—
18#547—21no—
19#547C2211yescharacteristic, complemented
20#546—22no—
21#546—21no—
22D42—21no—
23D42C2211yescharacteristic, complemented
24#532—21no—
25#532C2211yescharacteristic
26C2×#127C2211yescharacteristic, complemented
27#525—22no—
28#525—21no—
29C4×#41C2211yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2211yescharacteristic
32#493C2212yescomplemented
33#486C2212yescomplemented
34#460—21no—
35#460C2212yescomplemented
36#459—22no—
37#459—21no—
38#459—21no—
39#459—21no—
40#459C2212yes—
41#459C2212yescomplemented
42#459C2212yescomplemented
43#447—22no—
44#431C2211yescharacteristic, complemented
45#423C2211yescharacteristic, complemented
46C2×C4⋊C8C2211yescharacteristic, complemented
47#418—21no—
48C2×#102—21no—
49#332C2212yescomplemented
50#331C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53D4⋊C8C2212yescomplemented
54D4⋊C8C2212yescomplemented
55C2×#41—22no—
56C2×#41—22no—
57C2×#41—21no—
58C2×#41—21no—
59C2×#41—21no—
60C2×#41—44no—
61C2×#41—44no—
62C2×#41D411yescharacteristic
63C22×Q8—21no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—21no—
70C22×D4D411yescharacteristic
71C23×C4—22no—
72C23×C4—21no—
73AGL(1,ℤ/8)—44no—
74C2×SD16—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—24no—
79C2×D8—44no—
80C2×M16—21no—
81C22×C8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84#127—21no—
85#127—21no—
86#127—21no—
87#127—21no—
88#127C2312yes—
89#127C2311yescharacteristic
90#124—22no—
91#124—21no—
92#123—22no—
93#123—22no—
94#123—22no—
95C22⋊Q8—22no—
96C22⋊Q8—21no—
97C22⋊Q8—22no—
98C22⋊Q8—21no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—21no—
103#121—21no—
104#121—21no—
105#121—21no—
106#121—21no—
107#121D412yes—
108#120—22no—
109#120—22no—
110#120—22no—
111C4×Q8D411yescharacteristic
112C4×Q8C2311yescharacteristic
113C4×D4—22no—
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4—21no—
122C4×D4—21no—
123C4×D4D411yescharacteristic
124C4×D4D411yescharacteristic
125C4×D4D411yescharacteristic
126C4×D4D412yes—
127C4×D4C2311yescharacteristic
128C4×D4C2311yescharacteristic
129C4×D4C2311yescharacteristic
130#117—21no—
131#117D411yescharacteristic
132C2×C4⋊C4—21no—
133C2×C4⋊C4D411yescharacteristic
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C42—21no—
137C2×C42C2311yescharacteristic
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8C2312yes—
142C4⋊C8C2312yes—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—22no—
147#102—24no—
148C22⋊C8—21no—
149C4×C8C2312yes—
150C24—22no—
151C24—21no—
152C24—41no—
153#41—22no—
154#41—48no—
155#41—48no—
156#41—42no—
157#41—42no—
158#41—42no—
159#41—42no—
160#41—42no—
161#41—42no—
162#41D814yescomplemented
163C2×Q8—22no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C2×D411yescharacteristic
168C2×D4—21no—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—41no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—48no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—82no—
196C2×D4C2×D411yescharacteristic
197C2×D4C2×D411yescharacteristic
198C2×D4C2×D411yescharacteristic
199C2×D4C2×D412yes—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—48no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218SD16—44no—
219D8—44no—
220D8—44no—
221D8—44no—
222D8—84no—
223M16—42no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—42no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4C2×D411yescharacteristic
235C4⋊C4C2×D411yescharacteristic
236C4⋊C4C2×D411yescharacteristic
237C4⋊C4C2×D411yescharacteristic
238C4⋊C4C2×D412yes—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—21no—
244C22⋊C4—21no—
245C22⋊C4—22no—
246C22⋊C4—21no—
247C22⋊C4—21no—
248C22⋊C4—22no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42C2×D411yescharacteristic
256C42C2×D411yescharacteristic
257C42C2×D411yescharacteristic
258C42C2411yescharacteristic
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—42no—
264C23—44no—
265C23—44no—
266C23—42no—
267C23—42no—
268C23—81no—
269C23—82no—
270C23—82no—
271C23—82no—
272C23—82no—
273C23#12011yescharacteristic
274Q8—21no—
275Q8—44no—
276Q8—41no—
277Q8—41no—
278Q8C2×D812yescomplemented
279D4—24no—
280D4—21no—
281D4—21no—
282D4—21no—
283D4—21no—
284D4—42no—
285D4—42no—
286D4—44no—
287D4—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291D4—44no—
292D4—41no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—82no—
301D4—82no—
302D4—82no—
303D4—82no—
304D4C2×D812yescomplemented
305D4AGL(1,ℤ/8)12yescomplemented
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4#12011yescharacteristic
331C2×C4#12011yescharacteristic
332C2×C4#12011yescharacteristic
333C2×C4C2×D812yes—
334C2×C4AGL(1,ℤ/8)12yes—
335C2×C4C22×D411yescharacteristic
336C2×C4C22×D411yescharacteristic
337C2×C4C22×D411yescharacteristic
338C8—42no—
339C8—42no—
340C8—42no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22—42no—
353C22—42no—
354C22—41no—
355C22—84no—
356C22—88no—
357C22—82no—
358C22—82no—
359C22—82no—
360C22—164no—
361C22#44712yes—
362C22C2×#12011yescharacteristic
363C4—22no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4#44712yes—
373C4C22×D811yescharacteristic
374C4C2×AGL(1,ℤ/8)11yescharacteristic
375C4C2×AGL(1,ℤ/8)11yescharacteristic
376C4#57511yescharacteristic
377C2—22no—
378C2—42no—
379C2—42no—
380C2—81no—
381C2—82no—
382C2—162no—
383C2C2×#44711yescharacteristic
384C2#300611yescharacteristic
385C2#302111yescharacteristic
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. D4×#41C2semidirect
2. #3205C2semidirect
3. #3038C2semidirect
4. C2×#459C2semidirect
5. #1627C2semidirect
6. #1625C2semidirect
7. #1621C2semidirect
8. #1619C2semidirect
9. #1547C2semidirect
10. #1486C2semidirect
11. #1477C2semidirect
15. #550C22semidirect
16. D4×Q8C22semidirect
19. #547C22semidirect
23. D42C22semidirect
26. C2×#127C22semidirect
32. #493C22semidirect
33. #486C22semidirect
35. #460C22semidirect
41. #459C22semidirect
42. #459C22semidirect
44. #431C22semidirect
45. #423C22semidirect
46. C2×C4⋊C8C22semidirect
49. #332C22semidirect
50. #331C22semidirect
51. Q8⋊C8C22semidirect
52. D4⋊C8C22semidirect
53. D4⋊C8C22semidirect
54. D4⋊C8C22semidirect
162. #41D8semidirect
278. Q8C2×D8semidirect
304. D4C2×D8semidirect
305. D4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
D4×#412
#32052
#30382
C2×#4592
#16272
#16252
#16212
#16192
#15472
#14862
#14772