Finite Group Databasebeta

← #61831

Subgroups — #61831

Order 256.

875 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3507C211yesmaximal, characteristic, complemented
2C2×#554C211yesmaximal, characteristic, complemented
3#3389C214yesmaximal, complemented
4#3388C214yesmaximal, complemented
5#3376C214yesmaximal, complemented
6#3370C214yesmaximal, complemented
7#3239C212yesmaximal, complemented
8#3236C211yesmaximal, characteristic, complemented
9#3234C211yesmaximal, characteristic, complemented
10#3232C212yesmaximal, complemented
11#3114C212yesmaximal, complemented
12#3110C211yesmaximal, characteristic, complemented
13#3109C211yesmaximal, characteristic
14#3105C212yesmaximal, complemented
15#2983C211yesmaximal, characteristic, complemented
16#559C2212yescomplemented
17#557C2212yescomplemented
18#556—22no—
19#556C2212yescomplemented
20#554—22no—
21#554—22no—
22#554C2214yes—
23#554C2214yes—
24#554C2212yescomplemented
25#552—22no—
26#539—21no—
27#539—21no—
28#536—21no—
29#533C2211yescharacteristic
30#532C2211yescharacteristic
31C2×C4⋊Q8C2211yescharacteristic
32#529C2212yes—
33#529C2212yes—
34C2×#125C2211yescharacteristic
35C2×#125C2211yescharacteristic
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C22⋊Q8—21no—
39#520—21no—
40C2×C4×Q8C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×C4×D4C2211yescharacteristic
43C2×C4×D4C2211yescharacteristic
44C22×C4⋊C4—21no—
45#501C2212yescomplemented
46#501C2214yescomplemented
47#501C2212yescomplemented
48#499C2214yescomplemented
49#491C2212yescomplemented
50#490C2212yescomplemented
51#489C2214yescomplemented
52#488C2214yescomplemented
53#484C2212yes—
54#484C2212yes—
55#483C2212yes—
56#481C2212yes—
57#481C2212yes—
58#480C2212yes—
59#479C2214yescomplemented
60#479C2214yescomplemented
61#478C2214yes—
62#478C2214yes—
63#478C2214yescomplemented
64#478C2214yescomplemented
65#476C2212yes—
66#475C2214yescomplemented
67#474C2212yes—
68#474C2214yescomplemented
69#463C2214yes—
70#463C2214yes—
71#461C2212yes—
72#459C2212yes—
73#450C2212yes—
74#447C2212yes—
75#446C2214yescomplemented
76#446C2214yescomplemented
77#442C2214yes—
78#442C2214yes—
79#440C2214yes—
80#440C2214yes—
81C4×SD16C2214yes—
82C4×D8C2214yes—
83#432C2211yescharacteristic
84#432C2211yescharacteristic
85#432C2212yes—
86#431C2212yes—
87C2×#41—21no—
88C22×Q8—21no—
89C22×Q8D411yescharacteristic
90C22×D4C2311yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C2×SD16C2314yes—
95C2×D8C2314yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8C2311yescharacteristic
99C4⋊Q8C2314yes—
100C4⋊Q8C2311yescharacteristic
101C4⋊Q8C2311yescharacteristic
102#127C2311yescharacteristic
103#126—22no—
104#126—22no—
105#126—22no—
106#126—22no—
107#125—22no—
108#125—22no—
109#125—22no—
110#125—21no—
111#125—21no—
112#125—21no—
113#125C2312yes—
114#125C2314yes—
115#125C2312yes—
116#125C2312yes—
117#125C2312yes—
118#124—22no—
119#124—21no—
120#124C2312yes—
121#124C2312yes—
122#123—22no—
123#123—22no—
124#123—22no—
125#123—22no—
126C22⋊Q8—24no—
127C22⋊Q8—24no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—21no—
131C22⋊Q8—24no—
132C22⋊Q8—24no—
133C22⋊Q8—24no—
134C22⋊Q8—24no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8C2312yes—
138#121—21no—
139#121C2312yes—
140C4×Q8—22no—
141C4×Q8—21no—
142C4×Q8D412yes—
143C4×Q8D412yes—
144C4×Q8C2312yes—
145C4×Q8C2312yes—
146C4×Q8C2312yes—
147C4×Q8C2312yes—
148C4×Q8C2312yes—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4C2314yes—
158C4×D4C2312yes—
159C4×D4C2314yes—
160C4×D4C2312yes—
161C4×D4C2312yes—
162C4×D4C2312yes—
163C4×D4C2312yes—
164#117—21no—
165#117—22no—
166#117—22no—
167#117—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4D411yescharacteristic
177C2×C4⋊C4D411yescharacteristic
178C2×C4⋊C4D411yescharacteristic
179C2×C4⋊C4C2311yescharacteristic
180C2×C4⋊C4C2311yescharacteristic
181C2×C4⋊C4C2311yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42C2311yescharacteristic
186C2×C42C2311yescharacteristic
187C2×C42C2311yescharacteristic
188#107C2314yes—
189#107C2314yes—
190#107C2314yes—
191#106C2314yes—
192#106C2314yes—
193#106C2314yes—
194C4⋊C8C2314yes—
195C4⋊C8C2312yes—
196C4⋊C8C2312yes—
197C4⋊C8C2314yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#102C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208#97C2312yes—
209#97C2314yes—
210#97C2312yes—
211C4×C8C2314yes—
212C24—21no—
213#41—44no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—24no—
217C2×Q8—42no—
218C2×Q8—82no—
219C2×Q8C2×D412yes—
220C2×Q8C2×D412yes—
221C2×Q8C2412yes—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4C2412yes—
228C2×D4C2412yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—44no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4C2×D411yescharacteristic
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2411yescharacteristic
249SD16—28no—
250D8—28no—
251C2×C8C2414yes—
252C2×C8C2414yes—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—21no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4C2×D412yes—
276C4⋊C4C2×D412yes—
277C4⋊C4C2×D412yes—
278C4⋊C4C2×D412yes—
279C4⋊C4C2×D412yes—
280C4⋊C4C2×D412yes—
281C4⋊C4C2412yes—
282C4⋊C4C2412yes—
283C4⋊C4C2412yes—
284C4⋊C4C2412yes—
285C4⋊C4C2412yes—
286C22⋊C4—22no—
287C22⋊C4—24no—
288C22⋊C4—22no—
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—
299C42—22no—
300C42—22no—
301C42—22no—
302C42C2×D412yes—
303C42C2×D412yes—
304C42C2×D411yescharacteristic
305C42C2×D411yescharacteristic
306C42C2412yes—
307C42C2411yescharacteristic
308C42C2412yes—
309C42C2411yescharacteristic
310C23—24no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23C22×D411yescharacteristic
316Q8—24no—
317Q8—42no—
318Q8—42no—
319Q8—42no—
320Q8—81no—
321Q8—84no—
322D4—24no—
323D4—22no—
324D4—24no—
325D4—42no—
326D4—44no—
327D4—81no—
328D42−1+414yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4C22×D411yescharacteristic
353C2×C4C22×D412yes—
354C2×C4C22×D412yes—
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D412yes—
357C2×C4C22×D412yes—
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C22×D411yescharacteristic
361C2×C4C2511yescharacteristic
362C8—24no—
363C8—24no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—48no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22#57512yes—
374C22C23×D411yescharacteristic
375C4—22no—
376C4—21no—
377C4—21no—
378C4—22no—
379C4—21no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4C2×2−1+412yes—
391C2—22no—
392C2—44no—
393C2—82no—
394C2#348611yescharacteristic
395C2C2×#57511yescharacteristic
396C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3109C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3507C2semidirect
2. C2×#554C2semidirect
3. #3389C2semidirect
4. #3388C2semidirect
5. #3376C2semidirect
6. #3370C2semidirect
7. #3239C2semidirect
8. #3236C2semidirect
9. #3234C2semidirect
10. #3232C2semidirect
11. #3114C2semidirect
12. #3110C2semidirect
14. #3105C2semidirect
15. #2983C2semidirect
16. #559C22semidirect
17. #557C22semidirect
19. #556C22semidirect
24. #554C22semidirect
45. #501C22semidirect
46. #501C22semidirect
47. #501C22semidirect
48. #499C22semidirect
49. #491C22semidirect
50. #490C22semidirect
51. #489C22semidirect
52. #488C22semidirect
59. #479C22semidirect
60. #479C22semidirect
63. #478C22semidirect
64. #478C22semidirect
66. #475C22semidirect
68. #474C22semidirect
75. #446C22semidirect
76. #446C22semidirect

Maximal subgroups

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

subgroupindex
#35072
C2×#5542
#33892
#33882
#33762
#33702
#32392
#32362
#32342
#32322
#31142
#31102
#31092
#31052
#29832