Finite Group Databasebeta

← #60867

Subgroups — #60867

Order 256.

843 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
1C2×#544C211yesmaximal, characteristic, complemented
2#3455C211yesmaximal, characteristic
3#3244C214yesmaximal, complemented
4#3243C214yesmaximal, complemented
5#3242C214yesmaximal, complemented
6#3241C214yesmaximal, complemented
7#3166C211yesmaximal, characteristic, complemented
8#3165C211yesmaximal, characteristic, complemented
9#3142C211yesmaximal, characteristic, complemented
10#3141C211yesmaximal, characteristic, complemented
11#3093C212yesmaximal, complemented
12#3092C212yesmaximal, complemented
13#3080C212yesmaximal, complemented
14#3079C212yesmaximal, complemented
15#2921C211yesmaximal, characteristic, complemented
16#544—22no—
17#544—22no—
18#544C2214yes—
19#544C2214yes—
20#542C2212yes—
21#542C2212yes—
22#541C2212yes—
23#540C2212yes—
24#539—21no—
25#539—21no—
26#538—21no—
27#538—21no—
28#533C2211yescharacteristic
29#532C2211yescharacteristic
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8C2211yescharacteristic
32C2×C4⋊Q8C2211yescharacteristic
33#529—22no—
34#529—22no—
35C2×#125—21no—
36C2×#125C2211yescharacteristic
37C2×#125C2211yescharacteristic
38#525C2212yes—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C4×#41C2212yes—
42C2×#117C2211yescharacteristic
43C2×#117C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2214yescomplemented
46#501C2212yescomplemented
47#501C2212yescomplemented
48#500C2212yescomplemented
49#499C2214yescomplemented
50#498C2212yescomplemented
51#491C2212yescomplemented
52#490C2212yescomplemented
53#489C2214yescomplemented
54#488C2214yescomplemented
55#487C2212yescomplemented
56#486C2212yescomplemented
57#485C2214yes—
58#485C2214yes—
59#484C2214yes—
60#483C2214yes—
61#482C2214yes—
62#482C2214yes—
63#481C2214yes—
64#480C2214yes—
65#479C2214yescomplemented
66#479C2214yescomplemented
67#478C2214yescomplemented
68#478C2214yescomplemented
69#477C2214yescomplemented
70#476C2214yescomplemented
71#475C2214yescomplemented
72#474C2214yescomplemented
73#433C2214yes—
74#433C2214yes—
75#428C2212yes—
76#428C2212yes—
77#428C2212yes—
78#428C2212yes—
79#423C2212yes—
80#417C2212yes—
81#417C2212yes—
82#417C2212yes—
83#417C2212yes—
84#407C2212yes—
85C4×M16C2211yescharacteristic
86C4×M16C2211yescharacteristic
87C2×#41C2312yes—
88C22×Q8—21no—
89C22×Q8—21no—
90C23×C4—21no—
91C23×C4C2311yescharacteristic
92C2×M16C2312yes—
93C2×M16C2312yes—
94C4⋊Q8—24no—
95C4⋊Q8—24no—
96C4⋊Q8—22no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8C2311yescharacteristic
100C4⋊Q8C2314yes—
101C4⋊Q8C2312yes—
102C4⋊Q8C2312yes—
103C4⋊Q8C2311yescharacteristic
104C4⋊Q8C2311yescharacteristic
105#127C2311yescharacteristic
106#126—22no—
107#126—22no—
108#126—22no—
109#126—22no—
110#125—24no—
111#125—24no—
112#125—22no—
113#125—22no—
114#125—22no—
115#125—21no—
116#125—21no—
117#125C2314yes—
118#125C2312yes—
119#125C2312yes—
120#125C2312yes—
121#124—22no—
122#124—22no—
123#124C2312yes—
124#123—22no—
125#123—22no—
126#123—22no—
127#123—22no—
128#123—22no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—24no—
134C22⋊Q8—24no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8C2312yes—
138C22⋊Q8C2312yes—
139#121—21no—
140#121—21no—
141#121—22no—
142#121—22no—
143#121C2312yes—
144#121C2312yes—
145#120—22no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8C2312yes—
149C4×Q8C2312yes—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4C2312yes—
157C4×D4C2312yes—
158#117—22no—
159#117—22no—
160#117D412yes—
161#117D412yes—
162#117C2314yes—
163#117C2312yes—
164#117C2312yes—
165#117C2312yes—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4D411yescharacteristic
175C2×C4⋊C4D411yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C4⋊C4C2311yescharacteristic
178C2×C4⋊C4C2314yes—
179C2×C4⋊C4C2311yescharacteristic
180C2×C4⋊C4C2311yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4D411yescharacteristic
183C2×C22⋊C4D411yescharacteristic
184C2×C42—21no—
185C2×C42—22no—
186C2×C42C2311yescharacteristic
187C2×C42C2311yescharacteristic
188#107C2314yes—
189#107C2314yes—
190#107C2314yes—
191#107C2314yes—
192#106C2314yes—
193#106C2314yes—
194#106C2314yes—
195#106C2314yes—
196C4⋊C8C2314yes—
197C4⋊C8C2314yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#103C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2314yes—
207C22⋊C8C2314yes—
208#97C2312yes—
209#97C2312yes—
210C4×C8C2312yes—
211C4×C8C2312yes—
212C24C2×D411yescharacteristic
213#41—28no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8C2412yes—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4C2412yes—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—82no—
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D412yes—
240C22×C4C2×D411yescharacteristic
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243C22×C4C2×D411yescharacteristic
244C22×C4C2412yes—
245C22×C4C2411yescharacteristic
246M16—24no—
247M16—24no—
248C2×C8C2414yes—
249C2×C8C2414yes—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4C2×D412yes—
275C4⋊C4C2×D412yes—
276C4⋊C4C2×D412yes—
277C4⋊C4C2×D412yes—
278C4⋊C4C2412yes—
279C4⋊C4C2412yes—
280C4⋊C4C2412yes—
281C4⋊C4C2412yes—
282C4⋊C4C2412yes—
283C4⋊C4C2412yes—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4C2×D412yes—
295C22⋊C4C2×D412yes—
296C22⋊C4C2×D412yes—
297C22⋊C4C2×D412yes—
298C42—22no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42C2×D412yes—
303C42C2×D411yescharacteristic
304C42C2×D411yescharacteristic
305C42C2411yescharacteristic
306C42C2411yescharacteristic
307C42C2412yes—
308C23—22no—
309C23—22no—
310C23—41no—
311C23—41no—
312C23C22×D412yes—
313C23C22×D411yescharacteristic
314Q8—24no—
315Q8—81no—
316Q8—81no—
317Q8—84no—
318Q8—82no—
319Q8—82no—
320D4—24no—
321D4—44no—
322D4—81no—
323D4—81no—
324D4—82no—
325D4—82no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—84no—
348C2×C4C22×D412yes—
349C2×C4C22×D412yes—
350C2×C4C22×D412yes—
351C2×C4C22×D411yescharacteristic
352C2×C4C22×D411yescharacteristic
353C2×C4C22×D411yescharacteristic
354C2×C4C22×D411yescharacteristic
355C2×C4C22×D411yescharacteristic
356C2×C42−1+414yes—
357C2×C4C2511yescharacteristic
358C8—24no—
359C8—24no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22#53612yes—
370C22C23×D411yescharacteristic
371C4—22no—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—22no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4#53612yes—
386C4C2×2−1+412yes—
387C2—22no—
388C2—42no—
389C2—82no—
390C2C2×#53611yescharacteristic
391C2#358511yescharacteristic
392C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3455C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
3. #3244C2semidirect
4. #3243C2semidirect
5. #3242C2semidirect
6. #3241C2semidirect
7. #3166C2semidirect
8. #3165C2semidirect
9. #3142C2semidirect
10. #3141C2semidirect
11. #3093C2semidirect
12. #3092C2semidirect
13. #3080C2semidirect
14. #3079C2semidirect
15. #2921C2semidirect
45. #501C22semidirect
46. #501C22semidirect
47. #501C22semidirect
48. #500C22semidirect
49. #499C22semidirect
50. #498C22semidirect
51. #491C22semidirect
52. #490C22semidirect
53. #489C22semidirect
54. #488C22semidirect
55. #487C22semidirect
56. #486C22semidirect
65. #479C22semidirect
66. #479C22semidirect
67. #478C22semidirect
68. #478C22semidirect
69. #477C22semidirect
70. #476C22semidirect
71. #475C22semidirect
72. #474C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#34552
#32442
#32432
#32422
#32412
#31662
#31652
#31422
#31412
#30932
#30922
#30802
#30792
#29212