Finite Group Databasebeta

← #61841

Subgroups — #61841

Order 256.

872 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3523C211yesmaximal, characteristic, complemented
2C2×#554C211yesmaximal, characteristic, complemented
3#3388C214yesmaximal, complemented
4#3384C212yesmaximal, complemented
5#3381C212yesmaximal, complemented
6#3376C214yesmaximal, complemented
7#3372C212yesmaximal, complemented
8#3360C212yesmaximal, complemented
9#3238C211yesmaximal, characteristic, complemented
10#3236C212yesmaximal, complemented
11#3232C212yesmaximal, complemented
12#3229C211yesmaximal, characteristic, complemented
13#3114C212yesmaximal
14#3110C211yesmaximal, characteristic, complemented
15#3105C211yesmaximal, characteristic
16#3101C212yesmaximal, complemented
17#2984C211yesmaximal, characteristic, complemented
18#557C2211yescharacteristic, complemented
19#557C2211yescharacteristic, complemented
20#557C2212yescomplemented
21#556C2211yescharacteristic, complemented
22#556C2212yescomplemented
23#554—21no—
24#554—21no—
25#554—22no—
26#554C2212yes—
27#554C2212yes—
28#554C2214yes—
29#552—21no—
30#552—22no—
31D4×Q8—21no—
32D4×Q8C2211yescharacteristic, complemented
33#541—22no—
34#541—21no—
35#533C2212yes—
36#533C2211yescharacteristic
37C2×C4⋊Q8C2211yescharacteristic
38#529C2211yescharacteristic
39#529C2212yes—
40C2×#125C2212yes—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—22no—
43#519—21no—
44C2×C4×Q8C2211yescharacteristic
45C2×C4×D4C2212yes—
46C2×C4×D4C2211yescharacteristic
47C22×C4⋊C4—21no—
48#501C2214yescomplemented
49#501C2212yescomplemented
50#499C2214yescomplemented
51#498C2212yescomplemented
52#491C2214yescomplemented
53#489C2212yescomplemented
54#488C2214yescomplemented
55#487C2212yescomplemented
56#484C2211yescharacteristic
57#484C2212yes—
58#483C2212yes—
59#483C2212yes—
60#483C2212yes—
61#481C2211yescharacteristic
62#480C2212yes—
63#479C2214yescomplemented
64#479C2214yescomplemented
65#479C2214yescomplemented
66#478C2212yes—
67#478C2214yes—
68#478C2214yescomplemented
69#477C2212yescomplemented
70#476C2214yes—
71#476C2212yescomplemented
72#475C2214yescomplemented
73#474C2212yes—
74#463C2214yes—
75#463C2212yes—
76#461C2214yes—
77#461C2212yes—
78#450C2211yescharacteristic
79#450C2212yes—
80#447C2211yescharacteristic
81#446C2212yescomplemented
82#446C2214yescomplemented
83C8×Q8C2212yescomplemented
84#442C2214yes—
85#440C2214yes—
86#440C2214yes—
87C4×SD16C2214yes—
88C4×SD16C2214yes—
89C4×SD16C2212yes—
90C4×D8C2212yes—
91#432C2211yescharacteristic
92#432C2212yes—
93#431C2212yes—
94#431C2211yescharacteristic
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8D411yescharacteristic
98C22×D4C2311yescharacteristic
99C23×C4—21no—
100C23×C4—22no—
101C2×SD16C2312yes—
102C2×SD16C2314yes—
103C2×D8C2312yes—
104C4⋊Q8—21no—
105C4⋊Q8—22no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8C2311yescharacteristic
109C4⋊Q8C2312yes—
110C4⋊Q8C2312yes—
111C4⋊Q8C2312yes—
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2312yes—
114#126—22no—
115#126—22no—
116#126—22no—
117#126—22no—
118#126—21no—
119#125—22no—
120#125—22no—
121#125—22no—
122#125—21no—
123#125—22no—
124#125—22no—
125#125—22no—
126#125—22no—
127#125—21no—
128#125C2314yes—
129#125C2314yes—
130#125C2312yes—
131#125C2311yescharacteristic
132#124C2312yes—
133#124C2311yescharacteristic
134#123—21no—
135#123—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—24no—
139C22⋊Q8—21no—
140C22⋊Q8—22no—
141C22⋊Q8—21no—
142C22⋊Q8—22no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—21no—
148C22⋊Q8—22no—
149C22⋊Q8C2311yescharacteristic
150C22⋊Q8C2312yes—
151#121C2311yescharacteristic
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—21no—
157C4×Q8D411yescharacteristic
158C4×Q8D412yes—
159C4×Q8D411yescharacteristic
160C4×Q8C2311yescharacteristic
161C4×Q8C2312yes—
162C4×Q8C2312yes—
163C4×Q8C2312yes—
164C4×Q8C2312yes—
165C4×Q8C2311yescharacteristic
166C4×Q8C2311yescharacteristic
167C4×Q8C2312yes—
168C4×D4—21no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—21no—
172C4×D4—22no—
173C4×D4—21no—
174C4×D4C2312yes—
175C4×D4C2314yes—
176C4×D4C2312yes—
177C4×D4C2314yes—
178C4×D4C2311yescharacteristic
179C4×D4C2312yes—
180#117—21no—
181#117—22no—
182#117—21no—
183#117—22no—
184#117—22no—
185#117—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4D411yescharacteristic
193C2×C4⋊C4D412yes—
194C2×C4⋊C4C2311yescharacteristic
195C2×C4⋊C4C2312yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C42C2312yes—
199C2×C42C2311yescharacteristic
200#107C2312yes—
201#107C2314yes—
202#107C2314yes—
203#106C2314yes—
204#106C2314yes—
205#106C2312yes—
206#106C2314yes—
207C4⋊C8C2314yes—
208C4⋊C8C2312yes—
209C4⋊C8C2312yes—
210C4⋊C8C2314yes—
211#103C2314yes—
212#103C2314yes—
213#103C2312yes—
214#103C2314yes—
215#103C2314yes—
216#102C2312yes—
217#102C2312yes—
218#102C2312yes—
219#102C2314yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2311yescharacteristic
223C22⋊C8C2312yes—
224#97C2312yes—
225#97C2314yes—
226C4×C8C2312yes—
227C4×C8C2314yes—
228C24—21no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—82no—
236C2×Q8C2×D411yescharacteristic
237C2×Q8C2×D411yescharacteristic
238C2×Q8C2×D412yes—
239C2×Q8C2412yes—
240C2×Q8C2411yescharacteristic
241C2×D4—24no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4C2411yescharacteristic
245C2×D4C2412yes—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—44no—
255C22×C4—84no—
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D412yes—
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D412yes—
260C22×C4C2411yescharacteristic
261SD16—24no—
262SD16—28no—
263D8—24no—
264C2×C8C2412yes—
265C2×C8C2414yes—
266C2×C8C2412yes—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4C2×D411yescharacteristic
293C4⋊C4C2×D411yescharacteristic
294C4⋊C4C2×D412yes—
295C4⋊C4C2×D412yes—
296C4⋊C4C2×D412yes—
297C4⋊C4C2×D412yes—
298C4⋊C4C2×D412yes—
299C4⋊C4C2411yescharacteristic
300C4⋊C4C2412yes—
301C4⋊C4C2411yescharacteristic
302C4⋊C4C2414yes—
303C4⋊C4C2412yes—
304C22⋊C4—22no—
305C22⋊C4—21no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—21no—
322C42C2×D412yes—
323C42C2×D411yescharacteristic
324C42C2×D412yes—
325C42C2×D411yescharacteristic
326C42C2411yescharacteristic
327C42C2412yes—
328C42C2412yes—
329C42C2411yescharacteristic
330C23—24no—
331C23—22no—
332C23—21no—
333C23—42no—
334C23—42no—
335C23C22×D411yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—22no—
339Q8—24no—
340Q8—42no—
341Q8—44no—
342Q8—42no—
343Q8—82no—
344Q8—82no—
345D4—24no—
346D4—22no—
347D4—22no—
348D4—42no—
349D42−1+414yes—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4C22×D411yescharacteristic
371C2×C4C22×D412yes—
372C2×C4C22×D412yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D412yes—
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D412yes—
377C2×C4C22×D411yescharacteristic
378C2×C4C22×D411yescharacteristic
379C2×C4C2511yescharacteristic
380C8—22no—
381C8—22no—
382C8—24no—
383C22—21no—
384C22—21no—
385C22—24no—
386C22—48no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#57712yes—
393C22C23×D411yescharacteristic
394C4—21no—
395C4—21no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—41no—
404C4—42no—
405C4—42no—
406C4—41no—
407C4—44no—
408C4—41no—
409C4#57512yes—
410C4C2×2−1+412yes—
411C2—22no—
412C2—44no—
413C2—81no—
414C2#348611yescharacteristic
415C2C2×#57511yescharacteristic
416C2C2×#57711yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#447C22
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. #3523C2semidirect
2. C2×#554C2semidirect
3. #3388C2semidirect
4. #3384C2semidirect
5. #3381C2semidirect
6. #3376C2semidirect
7. #3372C2semidirect
8. #3360C2semidirect
9. #3238C2semidirect
10. #3236C2semidirect
11. #3232C2semidirect
12. #3229C2semidirect
14. #3110C2semidirect
16. #3101C2semidirect
17. #2984C2semidirect
18. #557C22semidirect
19. #557C22semidirect
20. #557C22semidirect
21. #556C22semidirect
22. #556C22semidirect
32. D4×Q8C22semidirect
48. #501C22semidirect
49. #501C22semidirect
50. #499C22semidirect
51. #498C22semidirect
52. #491C22semidirect
53. #489C22semidirect
54. #488C22semidirect
55. #487C22semidirect
63. #479C22semidirect
64. #479C22semidirect
65. #479C22semidirect
68. #478C22semidirect
69. #477C22semidirect
71. #476C22semidirect
72. #475C22semidirect
81. #446C22semidirect
82. #446C22semidirect
83. C8×Q8C22semidirect

Maximal subgroups

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

subgroupindex
#35232
C2×#5542
#33882
#33842
#33812
#33762
#33722
#33602
#32382
#32362
#32322
#32292
#31142
#31102
#31052
#31012
#29842