Finite Group Databasebeta

← #17505

Subgroups — #17505

Order 256.

477 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3460C211yesmaximal, characteristic
2#2989C211yesmaximal, characteristic, complemented
3#2965C211yesmaximal, characteristic, complemented
4#1496C211yesmaximal, characteristic, complemented
5#1495C211yesmaximal, characteristic, complemented
6#1494C211yesmaximal, characteristic, complemented
7#1492C211yesmaximal, characteristic, complemented
8#1490C211yesmaximal, characteristic, complemented
9#1489C211yesmaximal, characteristic, complemented
10#1488C211yesmaximal, characteristic, complemented
11#1486C211yesmaximal, characteristic, complemented
12#1484C211yesmaximal, characteristic, complemented
13#1483C211yesmaximal, characteristic, complemented
14#1482C211yesmaximal, characteristic, complemented
15#1480C211yesmaximal, characteristic, complemented
16#544C411yescharacteristic
17#542—21no—
18#542C411yescharacteristic
19#542C411yescharacteristic
20#540—21no—
21#540C411yescharacteristic
22#540C411yescharacteristic
23#538—21no—
24#538—21no—
25#537C411yescharacteristic
26#535C411yescharacteristic
27#535C411yescharacteristic
28#532—21no—
29#532C2211yescharacteristic
30#532C2211yescharacteristic
31#529—21no—
32#529C2211yescharacteristic
33#529C2211yescharacteristic
34#525—21no—
35#525—21no—
36C4×#41C2211yescharacteristic
37C4×#41C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39#446—21no—
40#446—21no—
41#436—21no—
42#436—21no—
43#435—21no—
44#435—21no—
45#435—21no—
46#435—21no—
47#432C2211yescharacteristic
48#432C2211yescharacteristic, complemented
49#431C2211yescharacteristic
50#431C2211yescharacteristic, complemented
51#431C2211yescharacteristic, complemented
52#431C2211yescharacteristic, complemented
53#423C2211yescharacteristic
54#423C2211yescharacteristic, complemented
55C2×C4⋊C8C2211yescharacteristic, complemented
56C4×M16C2211yescharacteristic
57C4×M16C2211yescharacteristic, complemented
58C2×C4×C8C2211yescharacteristic, complemented
59Q8⋊C8C2211yescharacteristic, complemented
60Q8⋊C8C2211yescharacteristic, complemented
61Q8⋊C8C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66D4⋊C8C2211yescharacteristic
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71D4⋊C8C2211yescharacteristic, complemented
72D4⋊C8C2211yescharacteristic, complemented
73D4⋊C8C2211yescharacteristic, complemented
74D4⋊C8C2211yescharacteristic, complemented
75C2×#41—21no—
76C2×#41—21no—
77C2×#41C2×C411yescharacteristic
78C2×#41C2×C411yescharacteristic
79C22×D4C2×C411yescharacteristic
80C23×C4—21no—
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C4⋊Q8—21no—
88C4⋊Q8C2×C411yescharacteristic
89C4⋊Q8C2×C411yescharacteristic
90#127—21no—
91#127C2×C411yescharacteristic
92#127C2×C411yescharacteristic
93#126—21no—
94#126—21no—
95#126—21no—
96#126—21no—
97#125—21no—
98#125C2×C411yescharacteristic
99#125C2×C411yescharacteristic
100#124—21no—
101#124—21no—
102#124—21no—
103#124—21no—
104#124—21no—
105#124C2×C411yescharacteristic
106#124C2×C411yescharacteristic
107#123—21no—
108#123—21no—
109#123—21no—
110#123—21no—
111#123—21no—
112#123—21no—
113#123—21no—
114#123—21no—
115C22⋊Q8—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8C2×C411yescharacteristic
121C22⋊Q8C2×C411yescharacteristic
122#121—21no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—21no—
127#121—21no—
128#121—21no—
129#121—21no—
130#121—21no—
131#121—21no—
132#121—21no—
133#121C2×C411yescharacteristic
134#121C2×C411yescharacteristic
135#121C2×C411yescharacteristic
136#121C2×C411yescharacteristic
137#121C2×C411yescharacteristic
138#121C2×C411yescharacteristic
139#120—21no—
140#120—21no—
141#120—21no—
142#120—21no—
143C4×Q8C2×C411yescharacteristic
144C4×Q8D411yescharacteristic
145C4×Q8D411yescharacteristic
146C4×Q8C2311yescharacteristic
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4C2×C411yescharacteristic
156C4×D4C2×C411yescharacteristic
157C4×D4C2×C411yescharacteristic
158C4×D4D411yescharacteristic
159C4×D4D411yescharacteristic
160C4×D4D411yescharacteristic
161C4×D4D411yescharacteristic
162C4×D4D411yescharacteristic
163C4×D4D411yescharacteristic
164C4×D4C2311yescharacteristic
165C4×D4C2311yescharacteristic
166C4×D4C2311yescharacteristic
167#117—21no—
168#117—21no—
169#117—21no—
170#117C2×C411yescharacteristic
171#117C2×C411yescharacteristic
172#117C2×C411yescharacteristic
173#117C2×C411yescharacteristic
174C2×C4⋊C4C2×C411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C42—21no—
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C2×C42C2311yescharacteristic
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8C2311yescharacteristic
187C4⋊C8C2311yescharacteristic
188C4⋊C8C2311yescharacteristic
189C4⋊C8C2311yescharacteristic
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196#97—21no—
197#97—21no—
198#97—21no—
199#97—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8C2311yescharacteristic
203C4×C8C2311yescharacteristic
204C4×C8C2311yescharacteristic
205C4×C8C2311yescharacteristic
206C24—21no—
207#41—42no—
208#41—42no—
209#41—81no—
210#41—81no—
211#41—81no—
212#41—81no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8C22⋊C411yescharacteristic
216C2×Q8C22×C411yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊C411yescharacteristic
231C2×D4C22⋊C411yescharacteristic
232C2×D4C22⋊C411yescharacteristic
233C2×D4C22×C411yescharacteristic
234C2×D4C22×C411yescharacteristic
235C2×D4C22×C411yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C22×C411yescharacteristic
247C22×C4C22×C411yescharacteristic
248C22×C4C22×C411yescharacteristic
249C22×C4C2×D411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255M16—42no—
256M16—42no—
257M16—81no—
258M16—81no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—42no—
268C2×C8—81no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4C22⋊C411yescharacteristic
281C4⋊C4C22⋊C411yescharacteristic
282C4⋊C4C22⋊C411yescharacteristic
283C4⋊C4C22⋊C411yescharacteristic
284C4⋊C4C22⋊C411yescharacteristic
285C4⋊C4C22⋊C411yescharacteristic
286C4⋊C4C22×C411yescharacteristic
287C4⋊C4C22×C411yescharacteristic
288C4⋊C4C22×C411yescharacteristic
289C4⋊C4C22×C411yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4C22⋊C411yescharacteristic
310C22⋊C4C22⋊C411yescharacteristic
311C22⋊C4C22⋊C411yescharacteristic
312C22⋊C4C22⋊C411yescharacteristic
313C22⋊C4C22⋊C411yescharacteristic
314C22⋊C4C22⋊C411yescharacteristic
315C42—21no—
316C42—21no—
317C42—21no—
318C42C22×C411yescharacteristic
319C42C22×C411yescharacteristic
320C42C22×C411yescharacteristic
321C42C2×D411yescharacteristic
322C42C2×D411yescharacteristic
323C42C2×D411yescharacteristic
324C42C2×D411yescharacteristic
325C42C2×D411yescharacteristic
326C42C2×D411yescharacteristic
327C42C2411yescharacteristic
328C23—21no—
329C23—21no—
330C23—21no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23C2×C22⋊C411yescharacteristic
335C23C2×C22⋊C411yescharacteristic
336C23C2×C22⋊C411yescharacteristic
337Q8—22no—
338Q8—41no—
339Q8—81no—
340Q8—81no—
341D4—22no—
342D4—22no—
343D4—22no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—42no—
348D4—42no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4C2×C22⋊C411yescharacteristic
388C2×C4C2×C22⋊C411yescharacteristic
389C2×C4C2×C22⋊C411yescharacteristic
390C2×C4C2×C22⋊C411yescharacteristic
391C2×C4C2×C22⋊C411yescharacteristic
392C2×C4C2×C22⋊C411yescharacteristic
393C2×C4C2×C22⋊C411yescharacteristic
394C2×C4C2×C22⋊C411yescharacteristic
395C2×C4C2×C22⋊C411yescharacteristic
396C2×C4C23×C411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C2×C4C22×D411yescharacteristic
399C8—22no—
400C8—41no—
401C8—41no—
402C8—41no—
403C8—41no—
404C8—41no—
405C8—41no—
406C8—41no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—84no—
426C22C22×C22⋊C411yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4#56812yes—
443C4#57611yescharacteristic
444C4#57711yescharacteristic
445C2—41no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—81no—
450C2—81no—
451C2#287911yescharacteristic
452C2#289711yescharacteristic
453C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3460C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2989C2semidirect
3. #2965C2semidirect
4. #1496C2semidirect
5. #1495C2semidirect
6. #1494C2semidirect
7. #1492C2semidirect
8. #1490C2semidirect
9. #1489C2semidirect
10. #1488C2semidirect
11. #1486C2semidirect
12. #1484C2semidirect
13. #1483C2semidirect
14. #1482C2semidirect
15. #1480C2semidirect
48. #432C22semidirect
50. #431C22semidirect
51. #431C22semidirect
52. #431C22semidirect
54. #423C22semidirect
55. C2×C4⋊C8C22semidirect
57. C4×M16C22semidirect
58. C2×C4×C8C22semidirect
59. Q8⋊C8C22semidirect
60. Q8⋊C8C22semidirect
61. Q8⋊C8C22semidirect
62. Q8⋊C8C22semidirect
71. D4⋊C8C22semidirect
72. D4⋊C8C22semidirect
73. D4⋊C8C22semidirect
74. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34602
#29892
#29652
#14962
#14952
#14942
#14922
#14902
#14892
#14882
#14862
#14842
#14832
#14822
#14802