Finite Group Databasebeta

← #62107

Subgroups — #62107

Order 256.

979 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3499C211yesmaximal, characteristic, complemented
2#3475C211yesmaximal, characteristic, complemented
3#3384C212yesmaximal, complemented
4#3367C212yesmaximal, complemented
5#3354C212yesmaximal, complemented
6#3343C214yesmaximal, complemented
7#3330C212yesmaximal, complemented
8#3306C214yesmaximal, complemented
9#3251C212yesmaximal, complemented
10#3246C211yesmaximal, characteristic, complemented
11#3239C212yesmaximal, complemented
12#3229C211yesmaximal, characteristic, complemented
13#3115C212yesmaximal, complemented
14#3100C211yesmaximal, characteristic, complemented
15#3079C212yesmaximal, complemented
16C2×#474C211yesmaximal, characteristic, complemented
17#2989C211yesmaximal, characteristic, complemented
18C2×2−1+4—21no—
19#556C2211yescharacteristic
20#555C2211yescharacteristic
21#554C2211yescharacteristic
22#553—21no—
23#552—21no—
24#552C2212yes—
25#551—22no—
26D4×Q8—21no—
27D4×Q8C2212yescomplemented
28#547—22no—
29#547—21no—
30#547C2211yescharacteristic
31#547C2214yes—
32#547C2212yescomplemented
33#546C2212yes—
34#536—21no—
35#536—22no—
36#533C2211yescharacteristic
37#532C2211yescharacteristic
38#532C2212yescomplemented
39C2×C4⋊Q8C2211yescharacteristic
40C2×C4⋊Q8C2212yescomplemented
41#529—22no—
42#529C2212yes—
43#525—21no—
44C4×#41C2212yes—
45C4×#41C2211yescharacteristic
46C2×C4×D4C2211yescharacteristic
47C2×#117—21no—
48#501C2214yescomplemented
49#498C2212yescomplemented
50#489C2214yes—
51#487C2212yescomplemented
52#484C2214yescomplemented
53#483C2214yes—
54#482C2214yescomplemented
55#481C2214yes—
56#480C2212yes—
57#478C2214yescomplemented
58#477C2214yescomplemented
59#476C2214yes—
60#476C2212yescomplemented
61#475C2212yescomplemented
62#475C2212yescomplemented
63#475C2212yescomplemented
64#474C2212yes—
65#474C2211yescharacteristic
66#474C2212yes—
67#474C2212yes—
68#474C2211yescharacteristic
69#474C2212yes—
70#469C2214yes—
71#468C2214yescomplemented
72#465C2214yes—
73#463C2214yes—
74#461C2212yescomplemented
75#460C2212yescomplemented
76#459C2211yescharacteristic
77#459C2211yescharacteristic
78#449C2214yes—
79#447C2212yes—
80#446C2212yescomplemented
81#442C2214yes—
82#440C2214yescomplemented
83C4×D8C2212yes—
84C4×D8C2212yes—
85#436C2212yescomplemented
86#435C2214yes—
87#432C2212yescomplemented
88#431C2211yescharacteristic
89#428C2214yescomplemented
90C2×#107C2212yes—
91#423C2212yescomplemented
92C2×C4⋊C8C2211yescharacteristic
93#417C2214yes—
94C2×#102C2212yes—
952−1+4—48no—
96C2×#41—24no—
97C2×#41—22no—
98C2×#41—21no—
99C2×#41—22no—
100C2×#41D411yescharacteristic
101C2×#41C2312yes—
102C22×Q8—22no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×D4C2311yescharacteristic
106C23×C4—21no—
107C2×SD16C2314yes—
108C2×D8C2312yes—
109C2×D8C2312yes—
110C2×M16C2314yes—
111C22×C8C2312yes—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8C2311yescharacteristic
117C4⋊Q8C2312yes—
118C4⋊Q8C2312yes—
119C4⋊Q8C2311yescharacteristic
120C4⋊Q8C2314yes—
121C4⋊Q8C2311yescharacteristic
122#127C2311yescharacteristic
123#126—22no—
124#126—22no—
125#126—22no—
126#125—22no—
127#125—21no—
128#125C2312yes—
129#124—22no—
130#124—21no—
131#124—22no—
132#124C2312yes—
133#123—22no—
134#123—22no—
135#123—22no—
136#123—22no—
137#123—24no—
138#123—24no—
139#123—24no—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—22no—
148C22⋊Q8C2312yes—
149C22⋊Q8C2312yes—
150#121—21no—
151#121—21no—
152#121—21no—
153#121—22no—
154#121—22no—
155#121C2312yes—
156#121C2312yes—
157#121C2314yes—
158#120—22no—
159C4×Q8—21no—
160C4×Q8D411yescharacteristic
161C4×Q8C2312yes—
162C4×Q8C2312yes—
163C4×Q8C2311yescharacteristic
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—22no—
171C4×D4D412yes—
172C4×D4D411yescharacteristic
173C4×D4C2311yescharacteristic
174C4×D4C2311yescharacteristic
175C4×D4C2311yescharacteristic
176C4×D4C2311yescharacteristic
177C4×D4C2312yes—
178C4×D4C2312yes—
179C4×D4C2314yes—
180C4×D4C2312yes—
181C4×D4C2311yescharacteristic
182#117—21no—
183#117—21no—
184#117—21no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117D411yescharacteristic
189#117D412yes—
190#117C2312yes—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2311yescharacteristic
196C2×C4⋊C4C2312yes—
197C2×C4⋊C4C2314yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C42—22no—
201C2×C42C2311yescharacteristic
202C2×C42C2312yes—
203#107C2314yes—
204#107C2314yes—
205#107C2312yes—
206#107C2312yes—
207#106C2314yes—
208#106C2314yes—
209#106C2314yes—
210C4⋊C8C2312yes—
211C4⋊C8C2312yes—
212C4⋊C8C2311yescharacteristic
213C4⋊C8C2314yes—
214C4⋊C8C2311yescharacteristic
215#103C2314yes—
216#103C2314yes—
217#103C2314yes—
218#102C2314yes—
219#102C2314yes—
220#102C2312yes—
221#102C2312yes—
222#102C2312yes—
223#102C2312yes—
224#102C2314yes—
225C22⋊C8C2312yes—
226C22⋊C8C2314yes—
227#97C2314yes—
228C4×C8C2312yes—
229C24—21no—
230#41—24no—
231#41—24no—
232#41—42no—
233#41—48no—
234#41—416no—
235#41—48no—
236#41—44no—
237C2×Q8—24no—
238C2×Q8—22no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—44no—
245C2×Q8C2×D411yescharacteristic
246C2×Q8C2412yes—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4C2×D412yes—
257C2×D4C2×D411yescharacteristic
258C2×D4C2411yescharacteristic
259C2×D4C2412yes—
260C2×D4C2411yescharacteristic
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—44no—
273C22×C4C2×D412yes—
274C22×C4C2×D411yescharacteristic
275C22×C4C2×D412yes—
276C22×C4C2×D411yescharacteristic
277C22×C4C2411yescharacteristic
278C22×C4C2412yes—
279SD16—44no—
280D8—24no—
281D8—42no—
282M16—44no—
283C2×C8—42no—
284C2×C8C2412yes—
285C2×C8C2412yes—
286C2×C8C2414yes—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4C2×D411yescharacteristic
304C4⋊C4C2×D411yescharacteristic
305C4⋊C4C2×D412yes—
306C4⋊C4C2×D412yes—
307C4⋊C4C2414yes—
308C4⋊C4C2412yes—
309C4⋊C4C2412yes—
310C4⋊C4C2411yescharacteristic
311C4⋊C4C2411yescharacteristic
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—21no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—24no—
326C22⋊C4—22no—
327C22⋊C4C2×D412yes—
328C22⋊C4C2×D412yes—
329C22⋊C4C2×D412yes—
330C42—22no—
331C42—22no—
332C42—22no—
333C42C2×D412yes—
334C42C2×D412yes—
335C42C2×D411yescharacteristic
336C42C2×D411yescharacteristic
337C42C2411yescharacteristic
338C42C2412yes—
339C42C2411yescharacteristic
340C23—21no—
341C23—21no—
342C23—22no—
343C23—42no—
344C23—42no—
345C23—81no—
346C23C22×D412yes—
347C23C22×D411yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—22no—
351Q8—48no—
352Q8—42no—
353Q8—44no—
354Q8—44no—
355Q8—81no—
356D4—24no—
357D4—22no—
358D4—22no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—22no—
363D4—42no—
364D4—42no—
365D4—44no—
366D4—48no—
367D4—44no—
368D4—42no—
369D4—82no—
370D42−1+412yescomplemented
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—82no—
395C2×C4AGL(1,ℤ/8)14yes—
396C2×C4C22×D412yes—
397C2×C4C22×D412yes—
398C2×C4C22×D411yescharacteristic
399C2×C4C22×D411yescharacteristic
400C2×C4C22×D411yescharacteristic
401C2×C4C22×D412yes—
402C2×C4C22×D411yescharacteristic
403C2×C4C22×D411yescharacteristic
404C2×C4C2511yescharacteristic
405C8—22no—
406C8—24no—
407C8—22no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—41no—
415C22—41no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—84no—
423C22C2×AGL(1,ℤ/8)12yes—
424C22C23×D411yescharacteristic
425C4—21no—
426C4—22no—
427C4—21no—
428C4—22no—
429C4—21no—
430C4—22no—
431C4—42no—
432C4—42no—
433C4—41no—
434C4—41no—
435C4—44no—
436C4—42no—
437C4C2×AGL(1,ℤ/8)12yes—
438C4C2×AGL(1,ℤ/8)12yes—
439C4C2×2−1+411yescharacteristic
440C4#58511yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—82no—
445C2—81no—
446C2#348911yescharacteristic
447C2C22×AGL(1,ℤ/8)11yescharacteristic
448C2#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. #3475C2semidirect
3. #3384C2semidirect
4. #3367C2semidirect
5. #3354C2semidirect
6. #3343C2semidirect
7. #3330C2semidirect
8. #3306C2semidirect
9. #3251C2semidirect
10. #3246C2semidirect
11. #3239C2semidirect
12. #3229C2semidirect
13. #3115C2semidirect
14. #3100C2semidirect
15. #3079C2semidirect
16. C2×#474C2semidirect
17. #2989C2semidirect
27. D4×Q8C22semidirect
32. #547C22semidirect
38. #532C22semidirect
40. C2×C4⋊Q8C22semidirect
48. #501C22semidirect
49. #498C22semidirect
51. #487C22semidirect
52. #484C22semidirect
54. #482C22semidirect
57. #478C22semidirect
58. #477C22semidirect
60. #476C22semidirect
61. #475C22semidirect
62. #475C22semidirect
63. #475C22semidirect
71. #468C22semidirect
74. #461C22semidirect
75. #460C22semidirect
80. #446C22semidirect
82. #440C22semidirect
85. #436C22semidirect
87. #432C22semidirect
89. #428C22semidirect
91. #423C22semidirect
370. D42−1+4semidirect

Maximal subgroups

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

subgroupindex
#34992
#34752
#33842
#33672
#33542
#33432
#33302
#33062
#32512
#32462
#32392
#32292
#31152
#31002
#30792
C2×#4742
#29892