Finite Group Databasebeta

← #62593

Subgroups — #62593

Order 256.

867 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3514C211yesmaximal, characteristic
2#3509C211yesmaximal, characteristic, complemented
3#3365C212yesmaximal, complemented
4#3364C212yesmaximal, complemented
5#3328C212yesmaximal, complemented
6#3327C214yesmaximal, complemented
7#3324C212yesmaximal, complemented
8#3323C214yesmaximal, complemented
9#3240C212yesmaximal, complemented
10#3239C211yesmaximal, characteristic, complemented
11#3216C211yesmaximal, characteristic, complemented
12#3215C212yesmaximal, complemented
13#3172C211yesmaximal, characteristic, complemented
14#3172C212yesmaximal, complemented
15#3141C211yesmaximal, characteristic, complemented
16#3140C212yesmaximal, complemented
17#2971C211yesmaximal, characteristic, complemented
18#559C2211yescharacteristic
19#557—21no—
20#557C2212yescomplemented
21#556—21no—
22#556C2211yescharacteristic
23#556C2212yes—
24#555—21no—
25#552—22no—
26#552C2214yescomplemented
27#551—22no—
28D4×Q8C2211yescharacteristic
29#548C2212yes—
30#546—21no—
31#546C2212yes—
32D42C2211yescharacteristic
33#542—21no—
34#541—22no—
35#538—22no—
36#536—21no—
37#533C2212yescomplemented
38#533C2211yescharacteristic, complemented
39#533C2212yescomplemented
40#532C2211yescharacteristic
41C2×C4⋊Q8C2211yescharacteristic, complemented
42#529C2211yescharacteristic
43#529C2212yes—
44C2×#124C2212yes—
45#519—21no—
46#518—21no—
47C4×#41C2211yescharacteristic
48#501C2212yescomplemented
49#501C2211yescharacteristic, complemented
50#497C2211yescharacteristic
51#497C2212yescomplemented
52#491C2211yescharacteristic, complemented
53#489C2211yescharacteristic
54#489C2212yes—
55#489C2212yescomplemented
56#489C2212yescomplemented
57#489C2212yescomplemented
58#488C2214yes—
59#488C2214yescomplemented
60#487C2212yescomplemented
61#486C2212yes—
62#485C2214yescomplemented
63#484C2212yescomplemented
64#482C2214yes—
65#481C2212yes—
66#479C2214yescomplemented
67#478C2214yescomplemented
68#475C2212yescomplemented
69#474C2212yes—
70#470C2214yescomplemented
71#470C2212yescomplemented
72#468C2212yes—
73#468C2214yes—
74#464C2214yes—
75#464C2214yescomplemented
76#462C2212yescomplemented
77#460C2212yes—
78#452C2214yescomplemented
79#451C2214yescomplemented
80#450C2214yes—
81C8×Q8C2212yescomplemented
82C4×Q16C2212yescomplemented
83C4×Q16C2214yescomplemented
84C4×SD16C2212yes—
85C4×SD16C2214yescomplemented
86#436C2212yes—
87#436C2214yescomplemented
88#432C2212yescomplemented
89#432C2211yescharacteristic, complemented
90#419C2214yescomplemented
91#418C2214yes—
92#417C2214yescomplemented
93C4×M16C2212yescomplemented
94C4×M16C2211yescharacteristic, complemented
95C2×#41D411yescharacteristic
96C2×#41C2312yes—
97C22×Q8—21no—
98C22×Q8C2312yes—
99C22×D4—21no—
100C22×D4C2312yes—
101C2×Q16C2312yes—
102C2×Q16C2314yes—
103C2×SD16C2312yes—
104C2×SD16C2314yes—
105C2×M16C2312yes—
106C2×M16C2314yes—
107C4⋊Q8—22no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8C2311yescharacteristic
112C4⋊Q8C2311yescharacteristic
113C4⋊Q8C2312yes—
114C4⋊Q8C2312yes—
115C4⋊Q8C2311yescharacteristic
116#127C2311yescharacteristic
117#126—22no—
118#126—24no—
119#126—24no—
120#126—22no—
121#126—21no—
122#125—24no—
123#125—24no—
124#125—21no—
125#125—22no—
126#125C2311yescharacteristic
127#125C2314yes—
128#124—21no—
129#124—22no—
130#124—22no—
131#124—22no—
132#124—22no—
133#124C2311yescharacteristic
134#124C2312yes—
135#124C2312yes—
136#124C2312yes—
137#123—24no—
138#123—21no—
139#123—22no—
140#123—22no—
141#123—22no—
142C22⋊Q8—21no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—21no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8C2312yes—
150C22⋊Q8C2311yescharacteristic
151C22⋊Q8C2312yes—
152C22⋊Q8C2314yes—
153#121—22no—
154#121—21no—
155#121—21no—
156#121—21no—
157#121—22no—
158#121C2312yes—
159#121C2311yescharacteristic
160#120—22no—
161#120—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—21no—
168C4×Q8D411yescharacteristic
169C4×Q8C2312yes—
170C4×Q8C2311yescharacteristic
171C4×Q8C2314yes—
172C4×Q8C2312yes—
173C4×Q8C2311yescharacteristic
174C4×D4—21no—
175C4×D4—22no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4D411yescharacteristic
182C4×D4D412yes—
183C4×D4C2311yescharacteristic
184C4×D4C2312yes—
185C4×D4C2311yescharacteristic
186C4×D4C2312yes—
187#117—21no—
188#117—22no—
189#117—21no—
190#117—24no—
191#117—21no—
192#117—22no—
193#117D411yescharacteristic
194#117D412yes—
195#117C2314yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C42C2312yes—
202C2×C42C2311yescharacteristic
203#107C2314yes—
204#107C2312yes—
205#106C2312yes—
206#106C2314yes—
207C4⋊C8C2312yes—
208C4⋊C8C2314yes—
209#103C2314yes—
210#103C2314yes—
211#103C2312yes—
212#103C2314yes—
213#103C2314yes—
214#103C2312yes—
215#103C2312yes—
216#102C2314yes—
217#102C2314yes—
218#102C2312yes—
219#102C2314yes—
220#102C2314yes—
221C22⋊C8C2314yes—
222C22⋊C8C2312yes—
223#97C2311yescharacteristic
224#97C2312yes—
225#97C2311yescharacteristic
226C4×C8C2312yes—
227C4×C8C2314yes—
228C24—22no—
229#41—42no—
230#41—44no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8C2×D411yescharacteristic
238C2×Q8C2411yescharacteristic
239C2×Q8C2412yes—
240C2×Q8C2412yes—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4C2×D411yescharacteristic
250C2×D4C2×D412yes—
251C2×D4C2411yescharacteristic
252C2×D4C2412yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D412yes—
260C22×C4C2×D412yes—
261C22×C4C2×D411yescharacteristic
262C22×C4C2411yescharacteristic
263C22×C4C2412yes—
264Q16—42no—
265Q16—44no—
266SD16—42no—
267SD16—44no—
268M16—44no—
269M16—42no—
270C2×C8C2412yes—
271C2×C8C2412yes—
272C2×C8C2414yes—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4C2×D411yescharacteristic
294C4⋊C4C2×D412yes—
295C4⋊C4C2×D411yescharacteristic
296C4⋊C4C2×D412yes—
297C4⋊C4C2414yes—
298C4⋊C4C2412yes—
299C4⋊C4C2411yescharacteristic
300C4⋊C4C2411yescharacteristic
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—24no—
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—22no—
316C22⋊C4—21no—
317C22⋊C4—22no—
318C22⋊C4C2×D412yes—
319C22⋊C4C2×D412yes—
320C22⋊C4C2×D412yes—
321C42—22no—
322C42—24no—
323C42—24no—
324C42—21no—
325C42—22no—
326C42C2×D412yes—
327C42C2×D411yescharacteristic
328C42C2×D411yescharacteristic
329C42C2×D412yes—
330C42C2412yes—
331C42C2411yescharacteristic
332C42C2411yescharacteristic
333C23—21no—
334C23—22no—
335C23—44no—
336C23—82no—
337C23—82no—
338C23C22×D412yes—
339C23C22×D411yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—22no—
343Q8—22no—
344Q8—24no—
345Q8—42no—
346Q8—44no—
347Q8—44no—
348Q8—82no—
349D4—22no—
350D4—22no—
351D4—24no—
352D4—42no—
353D4—44no—
354D4—42no—
355D4—81no—
356D4—82no—
357D4—81no—
358D4—82no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4C22×D412yes—
378C2×C4C22×D412yes—
379C2×C4C22×D411yescharacteristic
380C2×C4C22×D411yescharacteristic
381C2×C4C22×D411yescharacteristic
382C2×C4C22×D412yes—
383C2×C4C22×D411yescharacteristic
384C2×C4C22×D411yescharacteristic
385C2×C4C2511yescharacteristic
386C8—22no—
387C8—22no—
388C8—24no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—42no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—164no—
402C22C23×D411yescharacteristic
403C4—21no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—22no—
408C4—21no—
409C4—22no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—44no—
417C4#57512yes—
418C4#58512yes—
419C2—41no—
420C2—42no—
421C2—82no—
422C2—81no—
423C2#350011yescharacteristic
424C2C2×#57511yescharacteristic
425C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3514C2
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
2. #3509C2semidirect
3. #3365C2semidirect
4. #3364C2semidirect
5. #3328C2semidirect
6. #3327C2semidirect
7. #3324C2semidirect
8. #3323C2semidirect
9. #3240C2semidirect
10. #3239C2semidirect
11. #3216C2semidirect
12. #3215C2semidirect
13. #3172C2semidirect
14. #3172C2semidirect
15. #3141C2semidirect
16. #3140C2semidirect
17. #2971C2semidirect
20. #557C22semidirect
26. #552C22semidirect
37. #533C22semidirect
38. #533C22semidirect
39. #533C22semidirect
41. C2×C4⋊Q8C22semidirect
48. #501C22semidirect
49. #501C22semidirect
51. #497C22semidirect
52. #491C22semidirect
55. #489C22semidirect
56. #489C22semidirect
57. #489C22semidirect
59. #488C22semidirect
60. #487C22semidirect
62. #485C22semidirect
63. #484C22semidirect
66. #479C22semidirect
67. #478C22semidirect
68. #475C22semidirect
70. #470C22semidirect
71. #470C22semidirect
75. #464C22semidirect
76. #462C22semidirect
78. #452C22semidirect
79. #451C22semidirect
81. C8×Q8C22semidirect
82. C4×Q16C22semidirect
83. C4×Q16C22semidirect
85. C4×SD16C22semidirect
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
#35092
#33652
#33642
#33282
#33272
#33242
#33232
#32402
#32392
#32162
#32152
#31722
#31412
#31402
#29712