Finite Group Databasebeta

← #61711

Subgroups — #61711

Order 256.

881 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic
2C2×#552C211yesmaximal, characteristic, complemented
3#3349C214yesmaximal, complemented
4#3348C214yesmaximal, complemented
5#3317C214yesmaximal, complemented
6#3315C214yesmaximal, complemented
7#3236C211yesmaximal, characteristic, complemented
8#3234C211yesmaximal, characteristic, complemented
9#3196C212yesmaximal, complemented
10#3190C212yesmaximal, complemented
11#3113C211yesmaximal, characteristic, complemented
12#3112C211yesmaximal, characteristic, complemented
13#3092C212yesmaximal, complemented
14#3091C212yesmaximal, complemented
15#2973C211yesmaximal, characteristic, complemented
16#559—22no—
17#556C2212yes—
18#555C2212yes—
19#553—22no—
20#552—22no—
21#552—22no—
22#552C2214yes—
23#552C2214yes—
24#550C2212yes—
25#547C2212yes—
26#538—21no—
27#538—21no—
28#535—21no—
29#533C2211yescharacteristic
30#532C2211yescharacteristic
31C2×#126—21no—
32C2×#125—21no—
33C2×#125C2211yescharacteristic
34#525C2212yes—
35#525C2212yes—
36C2×#123—21no—
37C2×C22⋊Q8C2211yescharacteristic
38C2×C22⋊Q8C2211yescharacteristic
39#520—21no—
40C2×C4×Q8C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×#117—21no—
43C2×#117C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2212yescomplemented
46#501C2212yescomplemented
47#491C2212yescomplemented
48#490C2212yescomplemented
49#485C2214yescomplemented
50#485C2214yescomplemented
51#485C2214yescomplemented
52#484C2212yes—
53#484C2214yescomplemented
54#483C2212yes—
55#483C2212yes—
56#482C2214yescomplemented
57#481C2212yes—
58#481C2214yescomplemented
59#480C2212yes—
60#480C2212yes—
61#479C2214yescomplemented
62#478C2214yescomplemented
63#477C2212yescomplemented
64#475C2212yescomplemented
65#470C2214yescomplemented
66#468C2214yescomplemented
67#467C2214yescomplemented
68#465C2214yescomplemented
69#462C2212yescomplemented
70#460C2212yescomplemented
71#452C2214yescomplemented
72#452C2214yescomplemented
73#451C2212yes—
74#448C2212yes—
75#441C2214yescomplemented
76#440C2214yescomplemented
77#435C2214yes—
78#435C2214yes—
79#432C2211yescharacteristic
80#432C2211yescharacteristic
81#428C2214yescomplemented
82#427C2214yescomplemented
83#417C2214yescomplemented
84#416C2214yescomplemented
85#407C2212yes—
86C2×C22⋊C8C2212yes—
87C2×#41—21no—
88C2×#41C2312yes—
89C2×#41C2312yes—
90C22×Q8C2311yescharacteristic
91C22×D4D411yescharacteristic
92C23×C4—21no—
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C2×Q16C2314yes—
96C2×SD16C2314yes—
97C2×M16C2314yes—
98C22×C8C2314yes—
99C4⋊Q8C2311yescharacteristic
100C4⋊Q8C2311yescharacteristic
101C4⋊Q8C2311yescharacteristic
102#127—22no—
103#127C2311yescharacteristic
104#126—24no—
105#126—24no—
106#126—22no—
107#126—22no—
108#125—24no—
109#125—24no—
110#125—21no—
111#125—21no—
112#125—21no—
113#125C2312yes—
114#125C2312yes—
115#124—22no—
116#124—22no—
117#124—22no—
118#124—21no—
119#123—24no—
120#123—24no—
121#123—22no—
122#123—22no—
123#123—22no—
124#123—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—21no—
128C22⋊Q8C2314yes—
129C22⋊Q8C2314yes—
130C22⋊Q8C2312yes—
131C22⋊Q8C2312yes—
132C22⋊Q8C2312yes—
133#121—22no—
134#121—22no—
135#121—21no—
136#121—22no—
137#121—22no—
138#121C2312yes—
139#121C2312yes—
140#121C2312yes—
141#120—22no—
142#120—22no—
143C4×Q8—22no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8C2312yes—
148C4×Q8C2312yes—
149C4×Q8C2312yes—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—21no—
156C4×D4D412yes—
157C4×D4D412yes—
158C4×D4C2312yes—
159C4×D4C2312yes—
160C4×D4C2312yes—
161#117—22no—
162#117—21no—
163#117—22no—
164#117—24no—
165#117—24no—
166#117C2314yes—
167#117C2312yes—
168#117C2312yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4D411yescharacteristic
178C2×C4⋊C4C2311yescharacteristic
179C2×C4⋊C4C2311yescharacteristic
180C2×C4⋊C4C2314yes—
181C2×C4⋊C4C2311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4D411yescharacteristic
186C2×C22⋊C4D411yescharacteristic
187C2×C42—21no—
188C2×C42—21no—
189C2×C42C2311yescharacteristic
190#107C2314yes—
191#107C2314yes—
192#107C2314yes—
193#106C2314yes—
194#106C2314yes—
195#106C2314yes—
196C4⋊C8C2312yes—
197C4⋊C8C2312yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#103C2314yes—
202#103C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208C22⋊C8C2314yes—
209C22⋊C8C2314yes—
210#97C2312yes—
211#97C2312yes—
212C24C2×D411yescharacteristic
213C24C2×D411yescharacteristic
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—21no—
218C2×Q8—21no—
219C2×Q8—21no—
220C2×Q8—42no—
221C2×Q8C2412yes—
222C2×Q8C2412yes—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—42no—
231C2×D4C2×D412yes—
232C2×D4C2×D412yes—
233C2×D4C2412yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2×D412yes—
250C22×C4C2×D412yes—
251C22×C4C2411yescharacteristic
252C22×C4C2412yes—
253C22×C4C2412yes—
254Q16—44no—
255SD16—44no—
256M16—44no—
257C2×C8—44no—
258C2×C8C2414yes—
259C2×C8C2414yes—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4C2×D412yes—
281C4⋊C4C2×D412yes—
282C4⋊C4C2412yes—
283C4⋊C4C2412yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C4⋊C4C2412yes—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4C2×D412yes—
300C22⋊C4C2×D412yes—
301C22⋊C4C2×D412yes—
302C22⋊C4C2×D412yes—
303C42—24no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42C2×D411yescharacteristic
310C42C2×D411yescharacteristic
311C42C2411yescharacteristic
312C42C2411yescharacteristic
313C23—22no—
314C23—22no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C22×D412yes—
321C23C22×D412yes—
322C23C22×D411yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—24no—
326Q8—42no—
327Q8—42no—
328Q8—81no—
329D4—24no—
330D4—42no—
331D4—42no—
332D4—42no—
333D4—44no—
334D4—81no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—82no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4C22×D412yes—
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D411yescharacteristic
366C2×C4C22×D411yescharacteristic
367C2×C4C22×D412yes—
368C2×C4C22×D411yescharacteristic
369C2×C4C22×D411yescharacteristic
370C2×C4C2511yescharacteristic
371C8—24no—
372C8—24no—
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22#57512yes—
387C22C23×D411yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4#58512yes—
403C2—22no—
404C2—42no—
405C2—42no—
406C2—82no—
407C2#348511yescharacteristic
408C2C2×#57511yescharacteristic
409C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3506C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#552C2semidirect
3. #3349C2semidirect
4. #3348C2semidirect
5. #3317C2semidirect
6. #3315C2semidirect
7. #3236C2semidirect
8. #3234C2semidirect
9. #3196C2semidirect
10. #3190C2semidirect
11. #3113C2semidirect
12. #3112C2semidirect
13. #3092C2semidirect
14. #3091C2semidirect
15. #2973C2semidirect
45. #501C22semidirect
46. #501C22semidirect
47. #491C22semidirect
48. #490C22semidirect
49. #485C22semidirect
50. #485C22semidirect
51. #485C22semidirect
53. #484C22semidirect
56. #482C22semidirect
58. #481C22semidirect
61. #479C22semidirect
62. #478C22semidirect
63. #477C22semidirect
64. #475C22semidirect
65. #470C22semidirect
66. #468C22semidirect
67. #467C22semidirect
68. #465C22semidirect
69. #462C22semidirect
70. #460C22semidirect
71. #452C22semidirect
72. #452C22semidirect
75. #441C22semidirect
76. #440C22semidirect
81. #428C22semidirect
82. #427C22semidirect
83. #417C22semidirect
84. #416C22semidirect

Maximal subgroups

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

subgroupindex
#35062
C2×#5522
#33492
#33482
#33172
#33152
#32362
#32342
#31962
#31902
#31132
#31122
#30922
#30912
#29732