Finite Group Databasebeta

← #60836

Subgroups — #60836

Order 256.

1036 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3499C211yesmaximal, characteristic, complemented
2#3460C211yesmaximal, characteristic, complemented
3#3225C214yesmaximal, complemented
4#3221C214yesmaximal, complemented
5#3218C212yesmaximal, complemented
6#3214C212yesmaximal, complemented
7#3212C212yesmaximal, complemented
8#3206C212yesmaximal, complemented
9#3154C212yesmaximal, complemented
10#3137C212yesmaximal, complemented
11#3039C211yesmaximal, characteristic, complemented
12#3036C211yesmaximal, characteristic, complemented
13#3035C211yesmaximal, characteristic, complemented
14C2×#461C211yesmaximal, characteristic, complemented
15#3013C212yesmaximal, complemented
16#3010C212yesmaximal, complemented
17#2925C211yesmaximal, characteristic, complemented
18C2×2−1+4C2212yes—
19C2×#137C2212yes—
20C2×AGL(1,ℤ/8)C2212yes—
21C2×#135C2212yes—
22C22×SD16C2212yes—
23D4×Q8—21no—
24D4×Q8—21no—
25#547—22no—
26#547—22no—
27#547—21no—
28#547—21no—
29#544C2211yescharacteristic
30#542—21no—
31#542C2212yes—
32#540—21no—
33#540C2212yes—
34#538—22no—
35#537C2211yescharacteristic
36#536C2212yes—
37#536C2214yescomplemented
38#535C2211yescharacteristic
39#535C2211yescharacteristic
40#532—21no—
41#532C2211yescharacteristic
42#532C2212yes—
43#532C2212yescomplemented
44C2×C4⋊Q8C2211yescharacteristic
45C2×C4⋊Q8C2212yescomplemented
46#529—21no—
47#529C2212yes—
48#525—22no—
49C4×#41C2211yescharacteristic
50C4×#41C2211yescharacteristic
51C2×C4×D4C2211yescharacteristic
52#497C2214yes—
53#496C2214yes—
54#495C2214yes—
55#492C2214yes—
56#491C2214yescomplemented
57#489C2214yescomplemented
58#488C2214yescomplemented
59#487C2214yescomplemented
60#464C2212yescomplemented
61#463C2214yes—
62#463C2212yescomplemented
63#462C2212yescomplemented
64#462C2211yescharacteristic, complemented
65#462C2212yescomplemented
66#461C2212yes—
67#461C2212yes—
68#461C2212yes—
69#461C2214yes—
70#461C2211yescharacteristic, complemented
71#461C2212yescomplemented
72#461C2212yescomplemented
73#460C2211yescharacteristic, complemented
74#459C2212yes—
75#459C2211yescharacteristic, complemented
76#452C2214yescomplemented
77#451C2214yescomplemented
78#449C2214yes—
79#449C2214yes—
80#449C2214yescomplemented
81#448C2214yes—
82#448C2214yes—
83#448C2214yescomplemented
84#433C2214yescomplemented
85#432C2212yescomplemented
86#431C2212yescomplemented
87#423C2211yescharacteristic
88C2×C4⋊C8C2211yescharacteristic
89#419C2212yes—
90#417C2212yes—
91#416C2212yes—
92C2×#103C2212yes—
93#408C2212yes—
94#405C2212yes—
952−1+4—44no—
962−1+4—44no—
97C2×#41—24no—
98C2×#41—22no—
99C2×#41—22no—
100C2×#41C2311yescharacteristic
101C2×#41C2312yes—
102C2×#41C2314yes—
103C2×#41C2311yescharacteristic
104C22×Q8—22no—
105C22×Q8C2312yes—
106C22×Q8C2314yes—
107C22×D4C2311yescharacteristic
108C23×C4—21no—
109#137—44no—
110AGL(1,ℤ/8)—44no—
111#135—44no—
112C2×Q16C2312yes—
113C2×Q16C2314yes—
114C2×SD16—44no—
115C2×SD16C2314yes—
116C2×SD16C2314yes—
117C2×SD16C2312yes—
118C2×SD16C2312yes—
119C2×SD16C2314yes—
120C2×SD16C2314yes—
121C2×D8C2312yes—
122C2×D8C2314yes—
123C2×M16C2312yes—
124C22×C8C2312yes—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8C2312yes—
131C4⋊Q8C2312yes—
132C4⋊Q8C2311yescharacteristic
133C4⋊Q8C2314yes—
134C4⋊Q8C2312yes—
135#127—21no—
136#127C2311yescharacteristic
137#127C2312yes—
138#126—22no—
139#126—22no—
140#125—21no—
141#125C2312yes—
142#124—21no—
143#124—22no—
144#124—22no—
145#124C2312yes—
146#124C2312yes—
147#124C2314yes—
148#123—22no—
149#123—22no—
150#123—22no—
151#123—22no—
152#123—24no—
153#123—24no—
154#123—24no—
155C22⋊Q8—21no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—22no—
162C22⋊Q8C2312yes—
163#121—22no—
164#121—21no—
165#121—22no—
166#121—21no—
167#121—21no—
168#121—22no—
169#121—22no—
170#121—21no—
171#121—21no—
172#121—22no—
173#121—22no—
174#121C2312yes—
175#121C2312yes—
176#121C2312yes—
177#120—22no—
178#120—22no—
179C4×Q8D411yescharacteristic
180C4×Q8D411yescharacteristic
181C4×Q8C2311yescharacteristic
182C4×Q8C2311yescharacteristic
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4D412yes—
194C4×D4D411yescharacteristic
195C4×D4D412yes—
196C4×D4D411yescharacteristic
197C4×D4C2312yes—
198C4×D4C2312yes—
199C4×D4C2311yescharacteristic
200C4×D4C2311yescharacteristic
201#117—21no—
202#117—22no—
203#117C2311yescharacteristic
204#117C2311yescharacteristic
205#117C2312yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4C2311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C42—21no—
213C2×C42C2311yescharacteristic
214C2×C42C2312yes—
215C4⋊C8C2311yescharacteristic
216C4⋊C8C2311yescharacteristic
217C4⋊C8C2312yes—
218C4⋊C8C2312yes—
219C4⋊C8C2312yes—
220#103C2314yes—
221#103C2314yes—
222#103C2312yes—
223#103C2312yes—
224#103C2314yes—
225#103C2314yes—
226#103C2314yes—
227#102C2312yes—
228#102C2312yes—
229#102C2314yes—
230C22⋊C8C2314yes—
231C22⋊C8C2314yes—
232#97C2314yes—
233C4×C8C2314yes—
234C24—21no—
235#41—22no—
236#41—22no—
237#41—22no—
238#41—24no—
239#41—24no—
240#41—22no—
241#41—44no—
242#41—44no—
243#41—416no—
244#41—48no—
245#41—84no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—48no—
254C2×Q8C2×D411yescharacteristic
255C2×Q8C2414yes—
256C2×Q8C2412yes—
257C2×Q8C2411yescharacteristic
258C2×D4—21no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4C2×D412yes—
272C2×D4C2×D411yescharacteristic
273C2×D4C2412yes—
274C2×D4C2412yes—
275C2×D4C2411yescharacteristic
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—82no—
287C22×C4C2×D411yescharacteristic
288C22×C4C2×D412yes—
289C22×C4C2×D411yescharacteristic
290C22×C4C2×D412yes—
291C22×C4C2412yes—
292C22×C4C2411yescharacteristic
293Q16—42no—
294Q16—44no—
295SD16—44no—
296SD16—44no—
297SD16—42no—
298SD16—42no—
299SD16—44no—
300SD16—44no—
301D8—42no—
302D8—44no—
303M16—24no—
304C2×C8—24no—
305C2×C8C2412yes—
306C2×C8C2414yes—
307C2×C8C2412yes—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4C2×D411yescharacteristic
324C4⋊C4C2×D411yescharacteristic
325C4⋊C4C2×D412yes—
326C4⋊C4C2×D412yes—
327C4⋊C4C2412yes—
328C4⋊C4C2411yescharacteristic
329C4⋊C4C2411yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—24no—
343C22⋊C4—22no—
344C22⋊C4—24no—
345C22⋊C4C2×D412yes—
346C22⋊C4C2×D412yes—
347C22⋊C4C2×D412yes—
348C42—21no—
349C42—21no—
350C42—21no—
351C42C2×D411yescharacteristic
352C42C2×D411yescharacteristic
353C42C2×D412yes—
354C42C2×D412yes—
355C42C2412yes—
356C42C2411yescharacteristic
357C42C2411yescharacteristic
358C23—22no—
359C23—22no—
360C23—21no—
361C23—42no—
362C23—81no—
363C23—81no—
364C23C22×D412yes—
365C23C22×D411yescharacteristic
366Q8—22no—
367Q8—21no—
368Q8—24no—
369Q8—22no—
370Q8—24no—
371Q8—22no—
372Q8—21no—
373Q8—48no—
374Q8—44no—
375Q8—82no—
376D4—24no—
377D4—22no—
378D4—22no—
379D4—21no—
380D4—22no—
381D4—22no—
382D4—22no—
383D4—21no—
384D4—42no—
385D4—48no—
386D4—42no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—82no—
391D4—82no—
392D4—82no—
393D4—82no—
394D4—82no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4C22×D411yescharacteristic
418C2×C4C22×D411yescharacteristic
419C2×C4C22×D412yes—
420C2×C4C22×D412yes—
421C2×C4C22×D411yescharacteristic
422C2×C4C22×D412yes—
423C2×C4C22×D411yescharacteristic
424C2×C4C22×D411yescharacteristic
425C2×C4H(5,𝔽2)12yes—
426C2×C4H(5,𝔽2)12yes—
427C2×C4C2511yescharacteristic
428C8—22no—
429C8—22no—
430C8—24no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—42no—
442C22—42no—
443C22—41no—
444C22—42no—
445C22—84no—
446C22#53512yes—
447C22C23×D411yescharacteristic
448C4—21no—
449C4—21no—
450C4—22no—
451C4—22no—
452C4—21no—
453C4—22no—
454C4—41no—
455C4—44no—
456C4—42no—
457C4—41no—
458C4—42no—
459C4—41no—
460C4#53512yes—
461C4#57712yes—
462C4C2×H(5,𝔽2)11yescharacteristic
463C4C2×H(5,𝔽2)11yescharacteristic
464C2—22no—
465C2—42no—
466C2—81no—
467C2—82no—
468C2—82no—
469C2C2×#53511yescharacteristic
470C2C2×#57711yescharacteristic
471C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3499C2semidirect
2. #3460C2semidirect
3. #3225C2semidirect
4. #3221C2semidirect
5. #3218C2semidirect
6. #3214C2semidirect
7. #3212C2semidirect
8. #3206C2semidirect
9. #3154C2semidirect
10. #3137C2semidirect
11. #3039C2semidirect
12. #3036C2semidirect
13. #3035C2semidirect
14. C2×#461C2semidirect
15. #3013C2semidirect
16. #3010C2semidirect
17. #2925C2semidirect
37. #536C22semidirect
43. #532C22semidirect
45. C2×C4⋊Q8C22semidirect
56. #491C22semidirect
57. #489C22semidirect
58. #488C22semidirect
59. #487C22semidirect
60. #464C22semidirect
62. #463C22semidirect
63. #462C22semidirect
64. #462C22semidirect
65. #462C22semidirect
70. #461C22semidirect
71. #461C22semidirect
72. #461C22semidirect
73. #460C22semidirect
75. #459C22semidirect
76. #452C22semidirect
77. #451C22semidirect
80. #449C22semidirect
83. #448C22semidirect
84. #433C22semidirect
85. #432C22semidirect
86. #431C22semidirect

Maximal subgroups

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

subgroupindex
#34992
#34602
#32252
#32212
#32182
#32142
#32122
#32062
#31542
#31372
#30392
#30362
#30352
C2×#4612
#30132
#30102
#29252