Finite Group Databasebeta

← #61220

Subgroups — #61220

Order 256.

1041 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic, complemented
2#3497C211yesmaximal, characteristic, complemented
3#3321C214yesmaximal, complemented
4#3320C214yesmaximal, complemented
5#3309C214yesmaximal, complemented
6#3308C214yesmaximal, complemented
7#3135C211yesmaximal, characteristic, complemented
8#3098C212yesmaximal, complemented
9#3078C211yesmaximal, characteristic, complemented
10#3062C212yesmaximal, complemented
11#3036C211yesmaximal, characteristic, complemented
12#3010C212yesmaximal, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14#2932C211yesmaximal, characteristic, complemented
15#2898C212yesmaximal, complemented
16C2×2−1+4C2212yes—
17C2×AGL(1,ℤ/8)C2212yes—
18C22×SD16C2211yescharacteristic
19C2×#131C2212yes—
20#559—22no—
21#556—21no—
22#555C2212yes—
23#553C2214yes—
24#550—21no—
25#550—21no—
26#550C2212yescomplemented
27D4×Q8—21no—
28D4×Q8C2212yescomplemented
29#548—22no—
30#548C2214yescomplemented
31#547C2212yes—
32#538—22no—
33#537—22no—
34#537C2212yes—
35#535C2212yes—
36#533C2211yescharacteristic
37#532—22no—
38#532C2211yescharacteristic
39#532C2211yescharacteristic
40#525—22no—
41#520C2212yes—
42C2×C4×Q8C2211yescharacteristic
43C2×C4×D4C2211yescharacteristic
44#491C2212yescomplemented
45#490C2212yescomplemented
46#489C2212yescomplemented
47#489C2212yescomplemented
48#485C2214yescomplemented
49#484C2214yescomplemented
50#482C2214yescomplemented
51#480C2214yescomplemented
52#479C2214yescomplemented
53#477C2212yescomplemented
54#475C2212yescomplemented
55#469C2214yes—
56#468C2214yes—
57#466C2214yes—
58#465C2214yes—
59#463C2214yes—
60#461C2212yes—
61#459C2212yes—
62#449C2214yes—
63#449C2214yes—
64#448C2214yes—
65#448C2214yes—
66#440C2214yes—
67#440C2214yes—
68#435C2214yescomplemented
69#435C2214yescomplemented
70#428C2212yes—
71C2×#107C2211yescharacteristic
72#423C2211yescharacteristic
73#419C2212yes—
74#419C2214yescomplemented
75#417C2212yes—
76#417C2214yescomplemented
77#416C2212yes—
78#416C2212yes—
79#416C2214yescomplemented
80C2×#103C2211yescharacteristic
81C2×#103C2214yescomplemented
82C2×#102C2211yescharacteristic
83#408C2212yes—
84C2×#97C2211yescharacteristic
852−1+4—28no—
862−1+4—44no—
87C2×#41—24no—
88C2×#41—22no—
89C2×#41—22no—
90C2×#41C2312yes—
91C2×#41C2314yes—
92C2×#41C2312yes—
93C22×Q8—22no—
94C22×Q8C2314yes—
95C22×Q8C2311yescharacteristic
96C22×D4C2311yescharacteristic
97C23×C4—21no—
98AGL(1,ℤ/8)—44no—
99C2×SD16—24no—
100C2×SD16C2314yes—
101C2×SD16C2314yes—
102C2×D8C2314yes—
103#131—44no—
104C2×M16C2312yes—
105C2×M16C2314yes—
106C22×C8C2311yescharacteristic
107C22×C8C2311yescharacteristic
108C22×C8C2314yes—
109C4⋊Q8—22no—
110C4⋊Q8D411yescharacteristic
111C4⋊Q8C2311yescharacteristic
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2311yescharacteristic
114#127—22no—
115#127—22no—
116#127D411yescharacteristic
117#127C2311yescharacteristic
118#126—24no—
119#125C2312yes—
120#124—22no—
121#124—24no—
122#124—24no—
123#124D412yes—
124#124C2312yes—
125#123—24no—
126#123—24no—
127C22⋊Q8—22no—
128C22⋊Q8—21no—
129C22⋊Q8—21no—
130C22⋊Q8—22no—
131C22⋊Q8—24no—
132C22⋊Q8—24no—
133C22⋊Q8C2314yes—
134#121—22no—
135#121—24no—
136#121—24no—
137#121—22no—
138#121—24no—
139#121—21no—
140#121—21no—
141#121—24no—
142#121D412yes—
143#121D412yes—
144#121C2314yes—
145#121C2314yes—
146#121C2314yes—
147#120—24no—
148C4×Q8—22no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8C2312yes—
154C4×Q8C2312yes—
155C4×D4—24no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4C2312yes—
166C4×D4C2312yes—
167C4×D4C2312yes—
168C4×D4C2312yes—
169#117—22no—
170#117C2312yes—
171#117C2314yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C2311yescharacteristic
174C2×C4⋊C4C2311yescharacteristic
175C2×C4⋊C4C2314yes—
176C2×C22⋊C4—21no—
177C2×C42—21no—
178C2×C42C2311yescharacteristic
179#107C2314yes—
180#107C2314yes—
181#106C2314yes—
182C4⋊C8C2312yes—
183C4⋊C8C2312yes—
184#103C2314yes—
185#103C2312yes—
186#103C2314yes—
187#103C2314yes—
188#103C2312yes—
189#103C2312yes—
190#103C2312yes—
191#103C2314yes—
192#102C2314yes—
193#102C2312yes—
194#102C2312yes—
195#102C2312yes—
196#102C2312yes—
197#102C2314yes—
198C22⋊C8C2314yes—
199C22⋊C8C2314yes—
200#97C2312yes—
201#97C2312yes—
202C24—21no—
203#41—24no—
204#41—24no—
205#41—28no—
206#41—24no—
207#41—24no—
208#41—44no—
209#41—416no—
210#41—48no—
211#41—84no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—24no—
215C2×Q8—21no—
216C2×Q8—22no—
217C2×Q8—28no—
218C2×Q8—44no—
219C2×Q8—48no—
220C2×Q8C2×D412yes—
221C2×Q8#4114yescomplemented
222C2×Q8C2412yes—
223C2×Q8C2412yes—
224C2×D4—22no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4C2×D414yes—
235C2×D4C2×D412yes—
236C2×D4C2×D412yes—
237C2×D4C2×D412yes—
238C2×D4C2412yes—
239C2×D4C2412yes—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—82no—
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2414yes—
253C22×C4C2411yescharacteristic
254SD16—28no—
255SD16—44no—
256D8—44no—
257M16—24no—
258M16—44no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—44no—
262C2×C8C2412yes—
263C2×C8C2412yes—
264C2×C8C2414yes—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4C2×D412yes—
278C4⋊C4C2×D412yes—
279C4⋊C4C2412yes—
280C4⋊C4C2412yes—
281C4⋊C4C2412yes—
282C4⋊C4C2412yes—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—24no—
289C22⋊C4—22no—
290C22⋊C4—24no—
291C22⋊C4—22no—
292C22⋊C4—24no—
293C22⋊C4C2×D414yes—
294C22⋊C4C2×D414yes—
295C42—22no—
296C42—22no—
297C42—24no—
298C42—22no—
299C42C2×D411yescharacteristic
300C42C2×D411yescharacteristic
301C42C2411yescharacteristic
302C42C2411yescharacteristic
303C23—22no—
304C23—22no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23C22×D414yes—
309C23C22×D411yescharacteristic
310Q8—24no—
311Q8—24no—
312Q8—22no—
313Q8—22no—
314Q8—22no—
315Q8—48no—
316Q8—42no—
317Q8—81no—
318Q8C2×#4114yes—
319D4—28no—
320D4—24no—
321D4—22no—
322D4—22no—
323D4—22no—
324D4—22no—
325D4—42no—
326D4—42no—
327D4—48no—
328D4—42no—
329D4—44no—
330D4—42no—
331D4—81no—
332D4—84no—
333D4—84no—
334D4—84no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—84no—
354C2×C4C22×D411yescharacteristic
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D414yes—
357C2×C4C22×D411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4112yes—
361C2×C4H(5,𝔽2)12yes—
362C2×C4C2511yescharacteristic
363C8—22no—
364C8—22no—
365C8—24no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—24no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—84no—
378C22#54612yes—
379C22C23×D411yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—24no—
385C4—24no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4#54612yes—
392C4C22×#4111yescharacteristic
393C4C2×H(5,𝔽2)11yescharacteristic
394C2—22no—
395C2—44no—
396C2—82no—
397C2—82no—
398C2C2×#54611yescharacteristic
399C2#358511yescharacteristic
400C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #532 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3506C2semidirect
2. #3497C2semidirect
3. #3321C2semidirect
4. #3320C2semidirect
5. #3309C2semidirect
6. #3308C2semidirect
7. #3135C2semidirect
8. #3098C2semidirect
9. #3078C2semidirect
10. #3062C2semidirect
11. #3036C2semidirect
12. #3010C2semidirect
13. C2×#440C2semidirect
14. #2932C2semidirect
15. #2898C2semidirect
26. #550C22semidirect
28. D4×Q8C22semidirect
30. #548C22semidirect
44. #491C22semidirect
45. #490C22semidirect
46. #489C22semidirect
47. #489C22semidirect
48. #485C22semidirect
49. #484C22semidirect
50. #482C22semidirect
51. #480C22semidirect
52. #479C22semidirect
53. #477C22semidirect
54. #475C22semidirect
68. #435C22semidirect
69. #435C22semidirect
74. #419C22semidirect
76. #417C22semidirect
79. #416C22semidirect
81. C2×#103C22semidirect
221. C2×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#35062
#34972
#33212
#33202
#33092
#33082
#31352
#30982
#30782
#30622
#30362
#30102
C2×#4402
#29322
#28982