Finite Group Databasebeta

← #17513

Subgroups — #17513

Order 256.

479 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2#2965C211yesmaximal, characteristic, complemented
3#2925C211yesmaximal, characteristic, complemented
4#1497C211yesmaximal, characteristic, complemented
5#1495C211yesmaximal, characteristic, complemented
6#1494C211yesmaximal, characteristic, complemented
7#1490C211yesmaximal, characteristic, complemented
8#1489C211yesmaximal, characteristic, complemented
9#1488C211yesmaximal, characteristic, complemented
10#1485C211yesmaximal, characteristic, complemented
11#1483C211yesmaximal, characteristic, complemented
12#1482C211yesmaximal, characteristic, complemented
13#1478C211yesmaximal, characteristic, complemented
14#1477C211yesmaximal, characteristic, complemented
15#1476C211yesmaximal, characteristic, complemented
16#556C411yescharacteristic
17#555C411yescharacteristic
18#554C411yescharacteristic
19#553—21no—
20#552—21no—
21#552C411yescharacteristic
22#552C411yescharacteristic
23#551—21no—
24#551—21no—
25#547C411yescharacteristic
26#546C411yescharacteristic
27#546C411yescharacteristic
28#533C2211yescharacteristic
29#529—21no—
30#529—21no—
31#529C2211yescharacteristic
32#529C2211yescharacteristic
33#525—21no—
34C4×#41C2211yescharacteristic
35C4×#41C2211yescharacteristic
36C4×#41C2211yescharacteristic
37C2×C4×D4C2211yescharacteristic
38C2×#117—21no—
39#446—21no—
40#436—21no—
41#435—21no—
42#435—21no—
43#433—21no—
44#433—21no—
45#432C2211yescharacteristic
46#432C2211yescharacteristic, complemented
47#431C2211yescharacteristic
48#431C2211yescharacteristic
49#431C2211yescharacteristic, complemented
50#431C2211yescharacteristic, complemented
51#431C2211yescharacteristic, complemented
52#423C2211yescharacteristic, complemented
53C2×C4⋊C8C2211yescharacteristic, complemented
54#408—21no—
55#405—21no—
56C4×M16C2211yescharacteristic
57C4×M16C2211yescharacteristic, complemented
58C2×C4×C8C2211yescharacteristic, complemented
59Q8⋊C8C2211yescharacteristic, complemented
60Q8⋊C8C2211yescharacteristic, complemented
61Q8⋊C8C2211yescharacteristic, complemented
62Q8⋊C8C2211yescharacteristic, complemented
63Q8⋊C8C2211yescharacteristic, complemented
64Q8⋊C8C2211yescharacteristic, complemented
65Q8⋊C8C2211yescharacteristic, complemented
66Q8⋊C8C2211yescharacteristic, complemented
67D4⋊C8C2211yescharacteristic
68D4⋊C8C2211yescharacteristic
69D4⋊C8C2211yescharacteristic
70D4⋊C8C2211yescharacteristic
71D4⋊C8C2211yescharacteristic
72D4⋊C8C2211yescharacteristic
73D4⋊C8C2211yescharacteristic
74D4⋊C8C2211yescharacteristic
75C2×#41C2×C411yescharacteristic
76C2×#41C2×C411yescharacteristic
77C2×#41D411yescharacteristic
78C22×D4C2×C411yescharacteristic
79C23×C4—21no—
80C2×M16—21no—
81C2×M16—21no—
82C2×M16—21no—
83C22×C8—21no—
84C22×C8—21no—
85C4⋊Q8C2×C411yescharacteristic
86C4⋊Q8C2×C411yescharacteristic
87#126—21no—
88#126—21no—
89#126—21no—
90#126—21no—
91#126—21no—
92#126—21no—
93#125—21no—
94#125—21no—
95#125—21no—
96#125C2×C411yescharacteristic
97#125C2×C411yescharacteristic
98#124—21no—
99#124—21no—
100#124C2×C411yescharacteristic
101#124C2×C411yescharacteristic
102#123—21no—
103#123—21no—
104#123—21no—
105#123—21no—
106#123—21no—
107#123—21no—
108#123—21no—
109#123—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C22⋊Q8—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C22⋊Q8C2×C411yescharacteristic
116C22⋊Q8C2×C411yescharacteristic
117C22⋊Q8C2×C411yescharacteristic
118C22⋊Q8C2×C411yescharacteristic
119#121—21no—
120#121—21no—
121#121—21no—
122#121—21no—
123#121C2×C411yescharacteristic
124#121C2×C411yescharacteristic
125#120—21no—
126#120—21no—
127C4×Q8—21no—
128C4×Q8C2×C411yescharacteristic
129C4×Q8C2×C411yescharacteristic
130C4×Q8C2×C411yescharacteristic
131C4×Q8D411yescharacteristic
132C4×Q8C2311yescharacteristic
133C4×Q8C2311yescharacteristic
134C4×D4—21no—
135C4×D4—21no—
136C4×D4—21no—
137C4×D4—21no—
138C4×D4—21no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—21no—
142C4×D4C2×C411yescharacteristic
143C4×D4C2×C411yescharacteristic
144C4×D4C2×C411yescharacteristic
145C4×D4C2×C411yescharacteristic
146C4×D4C2×C411yescharacteristic
147C4×D4C2×C411yescharacteristic
148C4×D4C2×C411yescharacteristic
149C4×D4D411yescharacteristic
150C4×D4D411yescharacteristic
151C4×D4D411yescharacteristic
152C4×D4C2311yescharacteristic
153C4×D4C2311yescharacteristic
154#117—21no—
155#117—21no—
156#117—21no—
157#117—21no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—21no—
162#117—21no—
163#117C2×C411yescharacteristic
164#117C2×C411yescharacteristic
165#117D411yescharacteristic
166#117D411yescharacteristic
167#117D411yescharacteristic
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×C411yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2311yescharacteristic
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8C2311yescharacteristic
185C4⋊C8C2311yescharacteristic
186C4⋊C8C2311yescharacteristic
187C4⋊C8C2311yescharacteristic
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195#97—21no—
196#97—21no—
197#97—21no—
198#97—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—21no—
202C4×C8C2311yescharacteristic
203C4×C8C2311yescharacteristic
204C4×C8C2311yescharacteristic
205C4×C8C2311yescharacteristic
206C24—21no—
207#41—22no—
208#41—22no—
209#41—41no—
210#41—41no—
211#41—41no—
212#41—41no—
213C2×Q8—21no—
214C2×Q8C22⋊C411yescharacteristic
215C2×Q8C22×C411yescharacteristic
216C2×Q8C22×C411yescharacteristic
217C2×D4—22no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4C22⋊C411yescharacteristic
225C2×D4C22⋊C411yescharacteristic
226C2×D4C22⋊C411yescharacteristic
227C2×D4C22×C411yescharacteristic
228C2×D4C22×C411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C22×C411yescharacteristic
241C22×C4C22×C411yescharacteristic
242C22×C4C22×C411yescharacteristic
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249M16—42no—
250M16—42no—
251M16—42no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
279C4⋊C4C22⋊C411yescharacteristic
280C4⋊C4C22⋊C411yescharacteristic
281C4⋊C4C22⋊C411yescharacteristic
282C4⋊C4C22⋊C411yescharacteristic
283C4⋊C4C22⋊C411yescharacteristic
284C4⋊C4C22×C411yescharacteristic
285C4⋊C4C22×C411yescharacteristic
286C4⋊C4C22×C411yescharacteristic
287C4⋊C4C22×C411yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
307C22⋊C4C22⋊C411yescharacteristic
308C22⋊C4C22⋊C411yescharacteristic
309C22⋊C4C22⋊C411yescharacteristic
310C22⋊C4C22⋊C411yescharacteristic
311C22⋊C4C22⋊C411yescharacteristic
312C42—21no—
313C42—21no—
314C42—21no—
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—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊C411yescharacteristic
334C23C2×C22⋊C411yescharacteristic
335C23C2×C22⋊C411yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—41no—
341Q8—81no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353D4#13112yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
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—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D411yescharacteristic
398C8—22no—
399C8—41no—
400C8—41no—
401C8—41no—
402C8—41no—
403C8—41no—
404C8—41no—
405C8—41no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22C22×C22⋊C411yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4C2×#13111yescharacteristic
440C4#56811yescharacteristic
441C4#57511yescharacteristic
442C4#57511yescharacteristic
443C2—41no—
444C2—41no—
445C2—41no—
446C2—42no—
447C2—81no—
448C2#288011yescharacteristic
449C2#289611yescharacteristic
450C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3475C2
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. #2965C2semidirect
3. #2925C2semidirect
4. #1497C2semidirect
5. #1495C2semidirect
6. #1494C2semidirect
7. #1490C2semidirect
8. #1489C2semidirect
9. #1488C2semidirect
10. #1485C2semidirect
11. #1483C2semidirect
12. #1482C2semidirect
13. #1478C2semidirect
14. #1477C2semidirect
15. #1476C2semidirect
46. #432C22semidirect
49. #431C22semidirect
50. #431C22semidirect
51. #431C22semidirect
52. #423C22semidirect
53. C2×C4⋊C8C22semidirect
57. C4×M16C22semidirect
58. C2×C4×C8C22semidirect
59. Q8⋊C8C22semidirect
60. Q8⋊C8C22semidirect
61. Q8⋊C8C22semidirect
62. Q8⋊C8C22semidirect
63. Q8⋊C8C22semidirect
64. Q8⋊C8C22semidirect
65. Q8⋊C8C22semidirect
66. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34752
#29652
#29252
#14972
#14952
#14942
#14902
#14892
#14882
#14852
#14832
#14822
#14782
#14772
#14762