Finite Group Databasebeta

← #18555

Subgroups — #18555

Order 256.

595 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#41C211yesmaximal, characteristic
2#3182C211yesmaximal, characteristic, complemented
3#3116C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1649C211yesmaximal, characteristic, complemented
6#1647C211yesmaximal, characteristic, complemented
7#1641C211yesmaximal, characteristic, complemented
8#1639C211yesmaximal, characteristic, complemented
9#1637C211yesmaximal, characteristic, complemented
10#1635C211yesmaximal, characteristic, complemented
11#1631C211yesmaximal, characteristic, complemented
12#1629C211yesmaximal, characteristic, complemented
13#1557C211yesmaximal, characteristic, complemented
14#1490C211yesmaximal, characteristic, complemented
15#1489C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17#550C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#548—21no—
20#547—21no—
21#547C2211yescharacteristic
22#546—22no—
23#546—21no—
24D42—21no—
25D42C2211yescharacteristic
26#533C2211yescharacteristic, complemented
27#532—21no—
28#532C2211yescharacteristic
29#525—22no—
30#525—21no—
31C4×#41C2211yescharacteristic
32C2×C4×D4—21no—
33C2×C4×D4C2211yescharacteristic
34#500C2211yescharacteristic, complemented
35#492C2211yescharacteristic, complemented
36#487C2211yescharacteristic, complemented
37#486C2211yescharacteristic, complemented
38#485—21no—
39#478—21no—
40#478—21no—
41#477C2211yescharacteristic, complemented
42#476C2211yescharacteristic
43#476C2211yescharacteristic, complemented
44#476C2211yescharacteristic, complemented
45#474C2211yescharacteristic
46#474C2211yescharacteristic, complemented
47#468—22no—
48#462C2211yescharacteristic, complemented
49#461C2211yescharacteristic, complemented
50#452—21no—
51#441—21no—
52#440—21no—
53#440—22no—
54#432C2211yescharacteristic, complemented
55#428—21no—
56#423C2211yescharacteristic, complemented
57#418—21no—
58C4×M16C2211yescharacteristic, complemented
59#332C2211yescharacteristic, complemented
60#332C2211yescharacteristic, 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—22no—
72C2×#41—22no—
73C2×#41—21no—
74C2×#41—21no—
75C2×#41—21no—
76C2×#41—44no—
77C2×#41—44no—
78C2×#41D411yescharacteristic
79C22×Q8—21no—
80C22×D4—22no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4D411yescharacteristic
84C23×C4—22no—
85C23×C4—21no—
86C2×Q16—21no—
87C2×SD16—21no—
88C2×SD16—22no—
89C2×M16—21no—
90C2×M16—21no—
91C4⋊Q8—21no—
92C4⋊Q8C2311yescharacteristic
93C4⋊Q8C2311yescharacteristic
94C4⋊Q8C2311yescharacteristic
95#127—21no—
96#127C2311yescharacteristic
97#125—21no—
98#124—22no—
99#124—21no—
100#123—22no—
101#123—22no—
102#123—22no—
103C22⋊Q8—22no—
104C22⋊Q8—21no—
105C22⋊Q8—22no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109#121—22no—
110#121—22no—
111#121—22no—
112#121—21no—
113#121—21no—
114#121—21no—
115#121—21no—
116#121—21no—
117#121D412yes—
118#120—22no—
119#120—22no—
120#120—22no—
121C4×Q8—21no—
122C4×Q8—21no—
123C4×Q8D411yescharacteristic
124C4×Q8C2311yescharacteristic
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—21no—
134C4×D4—21no—
135C4×D4D411yescharacteristic
136C4×D4D411yescharacteristic
137C4×D4D411yescharacteristic
138C4×D4D412yes—
139C4×D4C2311yescharacteristic
140C4×D4C2311yescharacteristic
141C4×D4C2311yescharacteristic
142#117—21no—
143#117—21no—
144#117D411yescharacteristic
145C2×C4⋊C4—21no—
146C2×C4⋊C4D411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C42—21no—
150C2×C42C2311yescharacteristic
151#107—21no—
152#107—21no—
153#107—22no—
154#107—21no—
155#106—21no—
156#106—21no—
157#106—21no—
158C4⋊C8—21no—
159C4⋊C8C2311yescharacteristic
160C4⋊C8C2311yescharacteristic
161C4⋊C8C2311yescharacteristic
162C4⋊C8C2311yescharacteristic
163#103—21no—
164#103—21no—
165#103—22no—
166#103—21no—
167#102—21no—
168#102—21no—
169#102—21no—
170#102—21no—
171#102—21no—
172#102—22no—
173C22⋊C8—21no—
174#97—21no—
175#97—21no—
176C4×C8C2311yescharacteristic
177C4×C8C2311yescharacteristic
178C24—22no—
179C24—21no—
180#41—22no—
181#41—22no—
182#41—48no—
183#41—42no—
184#41—42no—
185#41—48no—
186#41—42no—
187#41—42no—
188#41—42no—
189#41—42no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8C2×D411yescharacteristic
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—48no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4C2×D411yescharacteristic
215C2×D4C2×D411yescharacteristic
216C2×D4C2×D411yescharacteristic
217C2×D4C2×D412yes—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—48no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236Q16—42no—
237SD16—44no—
238SD16—42no—
239M16—42no—
240M16—42no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4C2×D411yescharacteristic
260C4⋊C4C2×D411yescharacteristic
261C4⋊C4C2×D412yes—
262C4⋊C4C2×D411yescharacteristic
263C4⋊C4C2×D411yescharacteristic
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—22no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—22no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—41no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—41no—
282C42C2×D411yescharacteristic
283C42C2×D411yescharacteristic
284C42C2×D411yescharacteristic
285C42C2411yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—42no—
291C23—42no—
292C23—44no—
293C23—44no—
294C23—81no—
295C23—82no—
296C23#12011yescharacteristic
297Q8—21no—
298Q8—44no—
299Q8—42no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8AGL(1,ℤ/8)12yes—
305D4—24no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—42no—
312D4—44no—
313D4—44no—
314D4—44no—
315D4—44no—
316D4—44no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—82no—
322D4AGL(1,ℤ/8)12yes—
323D4#13712yes—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4#12011yescharacteristic
353C2×C4#12011yescharacteristic
354C2×C4#12011yescharacteristic
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D411yescharacteristic
357C2×C4C22×D411yescharacteristic
358C8—22no—
359C8—41no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—41no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—22no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—84no—
380C22—88no—
381C22C2×#12011yescharacteristic
382C4—21no—
383C4—22no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4C2×AGL(1,ℤ/8)11yescharacteristic
395C4C2×AGL(1,ℤ/8)11yescharacteristic
396C4C2×#13711yescharacteristic
397C4#57511yescharacteristic
398C2—41no—
399C2—42no—
400C2—42no—
401C2—81no—
402C2—82no—
403C2#300311yescharacteristic
404C2#300711yescharacteristic
405C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalD4×#41C2
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
2. #3182C2semidirect
3. #3116C2semidirect
4. #3077C2semidirect
5. #1649C2semidirect
6. #1647C2semidirect
7. #1641C2semidirect
8. #1639C2semidirect
9. #1637C2semidirect
10. #1635C2semidirect
11. #1631C2semidirect
12. #1629C2semidirect
13. #1557C2semidirect
14. #1490C2semidirect
15. #1489C2semidirect
26. #533C22semidirect
34. #500C22semidirect
35. #492C22semidirect
36. #487C22semidirect
37. #486C22semidirect
41. #477C22semidirect
43. #476C22semidirect
44. #476C22semidirect
46. #474C22semidirect
48. #462C22semidirect
49. #461C22semidirect
54. #432C22semidirect
56. #423C22semidirect
58. C4×M16C22semidirect
59. #332C22semidirect
60. #332C22semidirect
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
D4×#412
#31822
#31162
#30772
#16492
#16472
#16412
#16392
#16372
#16352
#16312
#16292
#15572
#14902
#14892