Finite Group Databasebeta

← #18754

Subgroups — #18754

Order 256.

521 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3481C211yesmaximal, characteristic, complemented
2#3207C211yesmaximal, characteristic, complemented
3#3115C211yesmaximal, characteristic, complemented
4#3035C211yesmaximal, characteristic, complemented
5#1644C211yesmaximal, characteristic, complemented
6#1643C211yesmaximal, characteristic, complemented
7#1640C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1622C211yesmaximal, characteristic, complemented
10#1622C211yesmaximal, characteristic, complemented
11#1620C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1551C211yesmaximal, characteristic, complemented
14#1488C211yesmaximal, characteristic, complemented
15#1477C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#559C2211yescharacteristic
18#554C2211yescharacteristic
19#554C2211yescharacteristic
20#553—21no—
21#553—21no—
22#551—22no—
23#546—21no—
24#546—21no—
25D42C2211yescharacteristic
26#532C2211yescharacteristic
27#532C2211yescharacteristic, complemented
28#529—21no—
29#529—21no—
30#525—22no—
31C4×#41C2211yescharacteristic
32C4×#41C2211yescharacteristic
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35#493C2211yescharacteristic
36#492C2211yescharacteristic, complemented
37#487C2211yescharacteristic, complemented
38#486C2211yescharacteristic
39#482—21no—
40#475C2211yescharacteristic, complemented
41#474C2211yescharacteristic
42#464—21no—
43#463—22no—
44#463—21no—
45#462C2211yescharacteristic, complemented
46#461C2211yescharacteristic
47#460C2211yescharacteristic, complemented
48#460C2211yescharacteristic, complemented
49#459C2211yescharacteristic
50#459C2211yescharacteristic
51#449—21no—
52#449—22no—
53#442—21no—
54#440—21no—
55#432C2211yescharacteristic, complemented
56#431C2211yescharacteristic, complemented
57C2×C4⋊C8C2211yescharacteristic, complemented
58#416—21no—
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#331C2211yescharacteristic
62#331C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65Q8⋊C8C2211yescharacteristic, complemented
66Q8⋊C8C2211yescharacteristic, complemented
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71C2×#41—21no—
72C2×#41D411yescharacteristic
73C2×#41D411yescharacteristic
74C2×#41D412yes—
75C22×D4—21no—
76C22×D4—21no—
77C23×C4—22no—
78#135—44no—
79C2×Q16—21no—
80C2×SD16—21no—
81C2×SD16—21no—
82C2×SD16—21no—
83C2×SD16—22no—
84C2×D8—21no—
85C2×D8—21no—
86C2×D8—22no—
87C22×C8—21no—
88C4⋊Q8C2311yescharacteristic
89C4⋊Q8C2311yescharacteristic
90#127C2311yescharacteristic
91#127C2311yescharacteristic
92#126—22no—
93#126—22no—
94#125—21no—
95#125—21no—
96#124—21no—
97#124—21no—
98#124—21no—
99#124D412yes—
100#123—22no—
101#123—22no—
102#123—22no—
103#123—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#121—22no—
109#121—22no—
110#121—21no—
111#121—21no—
112#121—21no—
113#121—21no—
114#121—21no—
115#121—21no—
116#120—22no—
117#120—22no—
118C4×Q8D411yescharacteristic
119C4×Q8D411yescharacteristic
120C4×Q8C2311yescharacteristic
121C4×Q8C2311yescharacteristic
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—21no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4D411yescharacteristic
139C4×D4D411yescharacteristic
140C4×D4C2311yescharacteristic
141C4×D4C2311yescharacteristic
142#117—22no—
143#117—21no—
144#117—21no—
145#117D411yescharacteristic
146#117D411yescharacteristic
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42C2311yescharacteristic
154#107—21no—
155#106—21no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162#103—21no—
163#103—22no—
164#103—21no—
165#103—21no—
166#102—21no—
167#102—22no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—21no—
172C22⋊C8—21no—
173C22⋊C8—21no—
174#97—21no—
175C4×C8C2311yescharacteristic
176C4×C8C2311yescharacteristic
177C24—22no—
178#41—22no—
179#41—22no—
180#41—41no—
181#41—41no—
182#41—42no—
183#41—42no—
184#41—82no—
185C2×Q8—21no—
186C2×Q8C2×D411yescharacteristic
187C2×Q8C2×D411yescharacteristic
188C2×Q8C2×D412yes—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4C2×D411yescharacteristic
205C2×D4C2×D411yescharacteristic
206C2×D4C2×D412yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—84no—
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222Q16—42no—
223SD16—42no—
224SD16—42no—
225SD16—42no—
226SD16—82no—
227D8—42no—
228D8—42no—
229D8—82no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4C2×D411yescharacteristic
250C4⋊C4C2×D411yescharacteristic
251C4⋊C4C2×D411yescharacteristic
252C4⋊C4C2×D411yescharacteristic
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42C2×D411yescharacteristic
276C42C2×D411yescharacteristic
277C42C2×D411yescharacteristic
278C42C2411yescharacteristic
279C23—21no—
280C23—21no—
281C23—22no—
282C23—44no—
283C23—44no—
284C23—41no—
285C23—82no—
286C23#12011yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—41no—
293Q8—41no—
294D4—21no—
295D4—21no—
296D4—42no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—44no—
301D4—44no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—82no—
311D4—82no—
312D4#13512yescomplemented
313D4AGL(1,ℤ/8)12yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#12011yescharacteristic
341C2×C4#12011yescharacteristic
342C2×C4#12011yescharacteristic
343C2×C4C22×D411yescharacteristic
344C2×C4C22×D411yescharacteristic
345C2×C4C22×D411yescharacteristic
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—41no—
351C8—41no—
352C22—21no—
353C22—22no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—88no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#12011yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—81no—
381C4C2×#13511yescharacteristic
382C4C2×AGL(1,ℤ/8)11yescharacteristic
383C4#57511yescharacteristic
384C4#57511yescharacteristic
385C2—41no—
386C2—42no—
387C2—42no—
388C2—82no—
389C2—161no—
390C2#300211yescharacteristic
391C2#300611yescharacteristic
392C2#301911yescharacteristic
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. #3481C2semidirect
2. #3207C2semidirect
3. #3115C2semidirect
4. #3035C2semidirect
5. #1644C2semidirect
6. #1643C2semidirect
7. #1640C2semidirect
8. #1639C2semidirect
9. #1622C2semidirect
10. #1622C2semidirect
11. #1620C2semidirect
12. #1619C2semidirect
13. #1551C2semidirect
14. #1488C2semidirect
15. #1477C2semidirect
27. #532C22semidirect
36. #492C22semidirect
37. #487C22semidirect
40. #475C22semidirect
45. #462C22semidirect
47. #460C22semidirect
48. #460C22semidirect
55. #432C22semidirect
56. #431C22semidirect
57. C2×C4⋊C8C22semidirect
59. #333C22semidirect
60. #333C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
65. Q8⋊C8C22semidirect
66. Q8⋊C8C22semidirect
312. D4#135semidirect

Maximal subgroups

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

subgroupindex
#34812
#32072
#31152
#30352
#16442
#16432
#16402
#16392
#16222
#16202
#16192
#15512
#14882
#14772