Finite Group Databasebeta

← #18866

Subgroups — #18866

Order 256.

510 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3519C211yesmaximal, characteristic, complemented
2#3212C211yesmaximal, characteristic, complemented
3#3207C211yesmaximal, characteristic, complemented
4#3100C211yesmaximal, characteristic, complemented
5#1646C211yesmaximal, characteristic, complemented
6#1644C211yesmaximal, characteristic, complemented
7#1641C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1624C211yesmaximal, characteristic, complemented
10#1622C211yesmaximal, characteristic, complemented
11#1621C211yesmaximal, characteristic, complemented
12#1619C211yesmaximal, characteristic, complemented
13#1543C211yesmaximal, characteristic, complemented
14#1482C211yesmaximal, characteristic, complemented
15#1478C211yesmaximal, characteristic, complemented
16#555C2211yescharacteristic
17#554C2211yescharacteristic
18#553—21no—
19#553—21no—
20#550C2211yescharacteristic
21#546—22no—
22#546—21no—
23D42—21no—
24D42C2211yescharacteristic
25#540—21no—
26#538—22no—
27#532C2211yescharacteristic
28#532C2211yescharacteristic, complemented
29#529C2211yescharacteristic
30#525—21no—
31C2×#121—21no—
32#518—21no—
33C2×C4×D4C2211yescharacteristic
34#495—21no—
35#493C2211yescharacteristic, complemented
36#492C2211yescharacteristic, complemented
37#488—21no—
38#487C2211yescharacteristic, complemented
39#486C2211yescharacteristic, complemented
40#480—21no—
41#474C2211yescharacteristic
42#474C2211yescharacteristic, complemented
43#463—22no—
44#462C2211yescharacteristic, complemented
45#461C2211yescharacteristic, complemented
46#460C2211yescharacteristic, complemented
47#459C2211yescharacteristic
48#459C2211yescharacteristic, complemented
49#459C2211yescharacteristic, complemented
50#452—21no—
51#449—22no—
52#449—21no—
53#447—21no—
54C4×D8—21no—
55C4×D8—21no—
56#431C2211yescharacteristic, complemented
57#431C2211yescharacteristic, complemented
58#431C2211yescharacteristic, complemented
59#333C2211yescharacteristic, complemented
60#333C2211yescharacteristic, complemented
61#331C2211yescharacteristic, complemented
62#331C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic, complemented
68D4⋊C8C2211yescharacteristic, complemented
69D4⋊C8C2211yescharacteristic, complemented
70D4⋊C8C2211yescharacteristic, complemented
71C2×#41—21no—
72C2×#41—21no—
73C2×#41D412yes—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4D411yescharacteristic
78C23×C4—21no—
79C2×Q16—21no—
80C2×SD16—22no—
81C2×SD16—21no—
82C2×SD16—21no—
83C2×D8—21no—
84C2×D8—21no—
85C2×D8—21no—
86C2×D8—22no—
87C4⋊Q8C2311yescharacteristic
88C4⋊Q8C2311yescharacteristic
89#127—21no—
90#127C2311yescharacteristic
91#127C2311yescharacteristic
92#126—22no—
93#126—21no—
94#125—21no—
95#125D411yescharacteristic
96#124—22no—
97#124—21no—
98#124D411yescharacteristic
99#124D412yes—
100#123—22no—
101#123—22no—
102#123—22no—
103#123—21no—
104C22⋊Q8—21no—
105C22⋊Q8—22no—
106C22⋊Q8D411yescharacteristic
107#121—22no—
108#121—22no—
109#121—22no—
110#121—22no—
111#121—21no—
112#121—22no—
113#121—21no—
114#121—21no—
115#121—21no—
116#121—21no—
117#121—21no—
118#121—21no—
119#121—21no—
120#121—21no—
121#121D411yescharacteristic
122#120—22no—
123#120—22no—
124#120—22no—
125C4×Q8C2311yescharacteristic
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—22no—
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
142C4×D4C2311yescharacteristic
143#117—21no—
144#117—21no—
145#117—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4D411yescharacteristic
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C42C2311yescharacteristic
152#107—21no—
153#107—21no—
154C4⋊C8—21no—
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159#103—22no—
160#103—21no—
161#103—21no—
162#102—21no—
163#102—21no—
164#102—22no—
165#102—21no—
166#102—21no—
167#102—21no—
168#102—21no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174C4×C8C2311yescharacteristic
175C4×C8C2311yescharacteristic
176C24—22no—
177C24—21no—
178#41—81no—
179#41—81no—
180#41—81no—
181#41—81no—
182#41—82no—
183C2×Q8—21no—
184C2×Q8C2×D411yescharacteristic
185C2×Q8C2×D412yes—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—41no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—84no—
207C2×D4C2×D411yescharacteristic
208C2×D4C2×D411yescharacteristic
209C2×D4C2×D411yescharacteristic
210C2×D4C2×D412yes—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226Q16—81no—
227SD16—81no—
228SD16—81no—
229SD16—82no—
230D8—42no—
231D8—42no—
232D8—81no—
233D8—82no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4C2×D411yescharacteristic
252C4⋊C4C2×D411yescharacteristic
253C4⋊C4C2×D411yescharacteristic
254C4⋊C4C2×D411yescharacteristic
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—22no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—22no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C42—22no—
274C42—21no—
275C42C2×D411yescharacteristic
276C42C2×D411yescharacteristic
277C42C2×D411yescharacteristic
278C42C2411yescharacteristic
279C23—21no—
280C23—21no—
281C23—21no—
282C23—22no—
283C23—42no—
284C23—42no—
285C23—44no—
286C23—41no—
287C23—81no—
288C23—82no—
289C23—82no—
290C23#12011yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—41no—
295Q8—41no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—44no—
305D4—42no—
306D4—44no—
307D4—41no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—82no—
316D4—82no—
317D4—82no—
318D4—82no—
319D4—82no—
320D4#13512yes—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#12011yescharacteristic
347C2×C4#12011yescharacteristic
348C2×C4#12011yescharacteristic
349C2×C4C22×D411yescharacteristic
350C2×C4C22×D411yescharacteristic
351C2×C4C22×D411yescharacteristic
352C8—41no—
353C8—41no—
354C8—41no—
355C8—41no—
356C8—41no—
357C8—41no—
358C22—21no—
359C22—22no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—42no—
369C22—41no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—84no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—164no—
378C22C2×#12011yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—42no—
389C4—81no—
390C4C2×#13511yescharacteristic
391C4#57511yescharacteristic
392C4#57611yescharacteristic
393C4#57611yescharacteristic
394C2—41no—
395C2—42no—
396C2—81no—
397C2—82no—
398C2—81no—
399C2—161no—
400C2#300211yescharacteristic
401C2#300811yescharacteristic
402C2#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. #3519C2semidirect
2. #3212C2semidirect
3. #3207C2semidirect
4. #3100C2semidirect
5. #1646C2semidirect
6. #1644C2semidirect
7. #1641C2semidirect
8. #1639C2semidirect
9. #1624C2semidirect
10. #1622C2semidirect
11. #1621C2semidirect
12. #1619C2semidirect
13. #1543C2semidirect
14. #1482C2semidirect
15. #1478C2semidirect
28. #532C22semidirect
35. #493C22semidirect
36. #492C22semidirect
38. #487C22semidirect
39. #486C22semidirect
42. #474C22semidirect
44. #462C22semidirect
45. #461C22semidirect
46. #460C22semidirect
48. #459C22semidirect
49. #459C22semidirect
56. #431C22semidirect
57. #431C22semidirect
58. #431C22semidirect
59. #333C22semidirect
60. #333C22semidirect
61. #331C22semidirect
62. #331C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
67. D4⋊C8C22semidirect
68. D4⋊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
#35192
#32122
#32072
#31002
#16462
#16442
#16412
#16392
#16242
#16222
#16212
#16192
#15432
#14822
#14782