Finite Group Databasebeta

← #62592

Subgroups — #62592

Order 256.

895 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514C211yesmaximal, characteristic, complemented
2#3511C211yesmaximal, characteristic, complemented
3#3361C212yesmaximal, complemented
4#3360C212yesmaximal, complemented
5#3329C212yesmaximal, complemented
6#3326C214yesmaximal, complemented
7#3325C212yesmaximal, complemented
8#3322C214yesmaximal, complemented
9#3240C212yesmaximal, complemented
10#3238C211yesmaximal, characteristic, complemented
11#3214C211yesmaximal, characteristic, complemented
12#3213C212yesmaximal, complemented
13#3171C211yesmaximal, characteristic, complemented
14#3171C212yesmaximal, complemented
15#3141C211yesmaximal, characteristic, complemented
16#3140C212yesmaximal, complemented
17#2971C211yesmaximal, characteristic, complemented
18#559C2211yescharacteristic
19#556—21no—
20#556C2211yescharacteristic
21#555—21no—
22#555C2212yes—
23#554C2212yescomplemented
24#553C2214yescomplemented
25#551—22no—
26D4×Q8C2211yescharacteristic
27#548—22no—
28#548C2212yes—
29#547—21no—
30#546—21no—
31#546C2212yes—
32D42C2211yescharacteristic
33#542—21no—
34#542—22no—
35#538—22no—
36#537—21no—
37#533C2212yescomplemented
38#532C2211yescharacteristic
39#532C2212yescomplemented
40#532C2211yescharacteristic, complemented
41C2×C4⋊Q8C2211yescharacteristic, complemented
42#529C2211yescharacteristic
43#529C2212yes—
44C2×#124C2212yes—
45#520—21no—
46#518—21no—
47C4×#41C2211yescharacteristic
48#501C2212yescomplemented
49#501C2211yescharacteristic, complemented
50#496C2211yescharacteristic
51#496C2212yescomplemented
52#490C2211yescharacteristic, complemented
53#489C2211yescharacteristic
54#489C2212yes—
55#489C2212yescomplemented
56#489C2212yescomplemented
57#489C2212yescomplemented
58#488C2214yes—
59#488C2214yescomplemented
60#487C2212yescomplemented
61#486C2212yes—
62#485C2214yescomplemented
63#483C2212yescomplemented
64#482C2214yes—
65#480C2212yes—
66#479C2214yescomplemented
67#478C2214yescomplemented
68#477C2212yescomplemented
69#476C2212yes—
70#469C2212yes—
71#469C2214yes—
72#468C2214yescomplemented
73#468C2212yescomplemented
74#463C2214yes—
75#463C2214yescomplemented
76#461C2212yescomplemented
77#459C2212yes—
78#449C2214yescomplemented
79#448C2214yescomplemented
80#447C2214yes—
81C8×Q8C2212yescomplemented
82C4×SD16C2212yescomplemented
83C4×SD16C2214yescomplemented
84C4×D8C2212yes—
85C4×D8C2214yescomplemented
86#436C2212yes—
87#436C2214yescomplemented
88#432C2212yescomplemented
89#432C2211yescharacteristic, complemented
90#419C2214yescomplemented
91#418C2214yes—
92#417C2214yescomplemented
93C4×M16C2212yescomplemented
94C4×M16C2211yescharacteristic, complemented
95C2×#41—21no—
96C2×#41—22no—
97C2×#41D411yescharacteristic
98C2×#41C2312yes—
99C22×Q8—21no—
100C22×Q8C2312yes—
101C22×D4—21no—
102C22×D4C2312yes—
103C2×SD16C2312yes—
104C2×SD16C2314yes—
105C2×D8C2312yes—
106C2×D8C2314yes—
107C2×M16C2312yes—
108C2×M16C2314yes—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2312yes—
114C4⋊Q8C2312yes—
115C4⋊Q8C2311yescharacteristic
116#127C2311yescharacteristic
117#127C2311yescharacteristic
118#126—22no—
119#126—24no—
120#126—22no—
121#126—21no—
122#125C2311yescharacteristic
123#125C2314yes—
124#124—21no—
125#124—22no—
126#124—24no—
127#124—22no—
128#124—22no—
129#124—21no—
130#124—22no—
131#124C2311yescharacteristic
132#124C2312yes—
133#124C2312yes—
134#124C2312yes—
135#123—21no—
136#123—22no—
137#123—22no—
138#123—22no—
139#123—24no—
140C22⋊Q8—22no—
141C22⋊Q8—21no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—24no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8C2312yes—
148C22⋊Q8C2311yescharacteristic
149#121—22no—
150#121—24no—
151#121—22no—
152#121—21no—
153#121—21no—
154#121—22no—
155#121—21no—
156#121—21no—
157#121—22no—
158#121C2312yes—
159#121C2311yescharacteristic
160#121C2312yes—
161#121C2314yes—
162#120—22no—
163#120—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8D411yescharacteristic
167C4×Q8C2312yes—
168C4×Q8C2311yescharacteristic
169C4×Q8C2311yescharacteristic
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—22no—
175C4×D4—24no—
176C4×D4—21no—
177C4×D4—22no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4D411yescharacteristic
181C4×D4D412yes—
182C4×D4C2311yescharacteristic
183C4×D4C2312yes—
184C4×D4C2314yes—
185C4×D4C2312yes—
186C4×D4C2311yescharacteristic
187C4×D4C2312yes—
188#117—21no—
189#117—22no—
190#117—21no—
191#117—22no—
192#117D411yescharacteristic
193#117D412yes—
194#117C2314yes—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42C2312yes—
202C2×C42C2311yescharacteristic
203#107C2312yes—
204#107C2314yes—
205#106C2314yes—
206#106C2312yes—
207C4⋊C8C2312yes—
208C4⋊C8C2314yes—
209#103C2314yes—
210#103C2314yes—
211#103C2312yes—
212#103C2314yes—
213#103C2314yes—
214#102C2314yes—
215#102C2314yes—
216#102C2312yes—
217#102C2314yes—
218#102C2314yes—
219#102C2312yes—
220#102C2312yes—
221C22⋊C8C2314yes—
222C22⋊C8C2312yes—
223#97C2311yescharacteristic
224#97C2312yes—
225#97C2311yescharacteristic
226C4×C8C2312yes—
227C4×C8C2314yes—
228C24—22no—
229#41—42no—
230#41—44no—
231#41—44no—
232#41—84no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—82no—
239C2×Q8C2×D411yescharacteristic
240C2×Q8C2411yescharacteristic
241C2×Q8C2412yes—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4C2×D411yescharacteristic
255C2×D4C2×D412yes—
256C2×D4C2411yescharacteristic
257C2×D4C2412yes—
258C2×D4C2412yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D412yes—
268C22×C4C2×D412yes—
269C22×C4C2×D411yescharacteristic
270C22×C4C2411yescharacteristic
271C22×C4C2412yes—
272SD16—42no—
273SD16—44no—
274D8—44no—
275D8—42no—
276M16—44no—
277M16—42no—
278C2×C8C2412yes—
279C2×C8C2412yes—
280C2×C8C2414yes—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4C2×D411yescharacteristic
296C4⋊C4C2×D411yescharacteristic
297C4⋊C4C2×D412yes—
298C4⋊C4C2×D412yes—
299C4⋊C4C2414yes—
300C4⋊C4C2412yes—
301C4⋊C4C2411yescharacteristic
302C4⋊C4C2411yescharacteristic
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—21no—
306C22⋊C4—22no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—21no—
319C22⋊C4—22no—
320C22⋊C4—24no—
321C22⋊C4—22no—
322C22⋊C4C2×D412yes—
323C22⋊C4C2×D412yes—
324C22⋊C4C2×D412yes—
325C42—22no—
326C42—24no—
327C42—21no—
328C42C2×D412yes—
329C42C2×D411yescharacteristic
330C42C2×D411yescharacteristic
331C42C2×D412yes—
332C42C2412yes—
333C42C2411yescharacteristic
334C42C2411yescharacteristic
335C23—21no—
336C23—22no—
337C23—22no—
338C23—44no—
339C23—82no—
340C23—82no—
341C23C22×D412yes—
342C23C22×D411yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—24no—
346Q8—44no—
347Q8—42no—
348Q8—82no—
349Q8—82no—
350D4—22no—
351D4—22no—
352D4—22no—
353D4—22no—
354D4—24no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—44no—
360D4—42no—
361D4—81no—
362D4—82no—
363D4—81no—
364D4—82no—
365D4—84no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4C22×D412yes—
386C2×C4C22×D412yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D411yescharacteristic
390C2×C4C22×D412yes—
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C2511yescharacteristic
394C8—22no—
395C8—22no—
396C8—24no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—42no—
412C22—164no—
413C22C23×D411yescharacteristic
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—44no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4#57512yes—
428C4#58512yes—
429C2—41no—
430C2—42no—
431C2—82no—
432C2—82no—
433C2—81no—
434C2#350011yescharacteristic
435C2C2×#57511yescharacteristic
436C2#358511yescharacteristic
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. #3514C2semidirect
2. #3511C2semidirect
3. #3361C2semidirect
4. #3360C2semidirect
5. #3329C2semidirect
6. #3326C2semidirect
7. #3325C2semidirect
8. #3322C2semidirect
9. #3240C2semidirect
10. #3238C2semidirect
11. #3214C2semidirect
12. #3213C2semidirect
13. #3171C2semidirect
14. #3171C2semidirect
15. #3141C2semidirect
16. #3140C2semidirect
17. #2971C2semidirect
23. #554C22semidirect
24. #553C22semidirect
37. #533C22semidirect
39. #532C22semidirect
40. #532C22semidirect
41. C2×C4⋊Q8C22semidirect
48. #501C22semidirect
49. #501C22semidirect
51. #496C22semidirect
52. #490C22semidirect
55. #489C22semidirect
56. #489C22semidirect
57. #489C22semidirect
59. #488C22semidirect
60. #487C22semidirect
62. #485C22semidirect
63. #483C22semidirect
66. #479C22semidirect
67. #478C22semidirect
68. #477C22semidirect
72. #468C22semidirect
73. #468C22semidirect
75. #463C22semidirect
76. #461C22semidirect
78. #449C22semidirect
79. #448C22semidirect
81. C8×Q8C22semidirect
82. C4×SD16C22semidirect
83. C4×SD16C22semidirect
85. C4×D8C22semidirect
87. #436C22semidirect
88. #432C22semidirect
89. #432C22semidirect
90. #419C22semidirect
92. #417C22semidirect
93. C4×M16C22semidirect
94. C4×M16C22semidirect

Maximal subgroups

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

subgroupindex
#35142
#35112
#33612
#33602
#33292
#33262
#33252
#33222
#32402
#32382
#32142
#32132
#31712
#31412
#31402
#29712