Finite Group Databasebeta

← #62543

Subgroups — #62543

Order 256.

888 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic, complemented
2#3511C211yesmaximal, characteristic, complemented
3#3376C214yesmaximal, complemented
4#3326C214yesmaximal, complemented
5#3321C214yesmaximal, complemented
6#3307C214yesmaximal, complemented
7#3231C211yesmaximal, characteristic, complemented
8#3229C211yesmaximal, characteristic, complemented
9#3228C211yesmaximal, characteristic, complemented
10#3207C211yesmaximal, characteristic, complemented
11#3207C211yesmaximal, characteristic, complemented
12#3206C211yesmaximal, characteristic, complemented
13#3184C211yesmaximal, characteristic, complemented
14#3182C211yesmaximal, characteristic, complemented
15#3167C211yesmaximal, characteristic, complemented
16#3141C211yesmaximal, characteristic, complemented
17#3139C211yesmaximal, characteristic, complemented
18#3131C211yesmaximal, characteristic, complemented
19#2965C211yesmaximal, characteristic, complemented
20#556—21no—
21#556C2212yes—
22#555—21no—
23#555C2212yes—
24#554—21no—
25#554C2212yes—
26#553—21no—
27#553—21no—
28#553C2212yes—
29#553C2212yes—
30#548—21no—
31#548—21no—
32#548C2212yes—
33#548C2212yes—
34#547—21no—
35#547C2212yes—
36#542—21no—
37#542—21no—
38#542—21no—
39#542—21no—
40#537—21no—
41#537—21no—
42#533C2211yescharacteristic
43#533C2211yescharacteristic
44#533C2211yescharacteristic
45#533C2211yescharacteristic
46#532C2211yescharacteristic
47#532C2211yescharacteristic
48#532C2211yescharacteristic
49#532C2211yescharacteristic
50#532C2211yescharacteristic
51#532C2211yescharacteristic
52C2×C4⋊Q8C2211yescharacteristic
53C2×C4⋊Q8C2211yescharacteristic
54#520—21no—
55#520—21no—
56C4×#41C2211yescharacteristic
57#500C2211yescharacteristic
58#498C2211yescharacteristic
59#498C2211yescharacteristic
60#493C2211yescharacteristic
61#492C2211yescharacteristic
62#492C2211yescharacteristic
63#491C2214yes—
64#489C2214yes—
65#488C2214yes—
66#487C2211yescharacteristic
67#487C2211yescharacteristic
68#487C2211yescharacteristic
69#487C2211yescharacteristic
70#487C2211yescharacteristic
71#487C2211yescharacteristic
72#486C2211yescharacteristic
73#486C2211yescharacteristic
74#486C2211yescharacteristic
75#486C2211yescharacteristic
76#485C2214yes—
77#483C2214yes—
78#482C2214yes—
79#479C2214yes—
80#478C2214yes—
81#475C2214yes—
82#469C2214yes—
83#468C2214yes—
84#465C2214yes—
85#463C2214yes—
86#463C2214yes—
87#461C2214yes—
88#449C2214yes—
89#449C2214yes—
90#448C2214yes—
91#446C2212yes—
92#442C2214yes—
93#440C2214yes—
94C4×SD16C2214yes—
95#436C2212yes—
96#435C2212yes—
97#435C2212yes—
98#431C2211yescharacteristic
99#431C2211yescharacteristic
100#431C2211yescharacteristic
101#419C2214yes—
102#417C2214yes—
103#416C2214yes—
104C4×M16C2211yescharacteristic
105C4×M16C2211yescharacteristic
106C2×C4×C8C2211yescharacteristic
107C2×#41—21no—
108C2×#41—21no—
109C2×#41—21no—
110C2×#41—21no—
111C2×#41D411yescharacteristic
112C2×#41C2312yes—
113C2×#41C2312yes—
114C22×Q8—21no—
115C22×Q8C2312yes—
116C2×SD16C2314yes—
117C2×SD16C2314yes—
118C2×D8C2314yes—
119C2×M16C2312yes—
120C2×M16C2312yes—
121C22×C8C2312yes—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C2311yescharacteristic
127C4⋊Q8C2311yescharacteristic
128C4⋊Q8C2312yes—
129C4⋊Q8C2311yescharacteristic
130C4⋊Q8C2311yescharacteristic
131C4⋊Q8C2312yes—
132C4⋊Q8C2311yescharacteristic
133C4⋊Q8C2311yescharacteristic
134#127C2311yescharacteristic
135#127C2311yescharacteristic
136#126—22no—
137#126—22no—
138#126—22no—
139#126—22no—
140#125—21no—
141#125—21no—
142#125C2312yes—
143#125C2312yes—
144#124—22no—
145#124—22no—
146#124—22no—
147#124—22no—
148#124—21no—
149#124—21no—
150#124—21no—
151#124—21no—
152#124C2312yes—
153#124C2312yes—
154#123—22no—
155#123—22no—
156#123—22no—
157#123—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8C2312yes—
171C22⋊Q8C2312yes—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—21no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—21no—
183#121—21no—
184#121—21no—
185#121—21no—
186#121C2312yes—
187#121C2312yes—
188#121C2312yes—
189#121C2312yes—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8D411yescharacteristic
197C4×Q8C2312yes—
198C4×Q8C2312yes—
199C4×Q8C2311yescharacteristic
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4D411yescharacteristic
213C4×D4D411yescharacteristic
214C4×D4D411yescharacteristic
215C4×D4C2312yes—
216C4×D4C2312yes—
217C4×D4C2312yes—
218C4×D4C2312yes—
219C4×D4C2311yescharacteristic
220C4×D4C2311yescharacteristic
221C4×D4C2311yescharacteristic
222#117—21no—
223#117—21no—
224#117—22no—
225#117—22no—
226#117D411yescharacteristic
227#117D411yescharacteristic
228#117D411yescharacteristic
229#117C2312yes—
230#117C2312yes—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2312yes—
237C2×C42C2311yescharacteristic
238C2×C42C2311yescharacteristic
239C2×C42C2311yescharacteristic
240#107C2314yes—
241#106C2314yes—
242#106C2314yes—
243C4⋊C8C2312yes—
244C4⋊C8C2312yes—
245C4⋊C8C2312yes—
246#103C2314yes—
247#103C2314yes—
248#103C2314yes—
249#103C2314yes—
250#103C2314yes—
251#103C2314yes—
252#102C2314yes—
253#102C2314yes—
254#102C2314yes—
255#102C2314yes—
256C22⋊C8C2312yes—
257C22⋊C8C2312yes—
258C22⋊C8C2312yes—
259#97C2312yes—
260#97C2312yes—
261C4×C8C2311yescharacteristic
262C4×C8C2311yescharacteristic
263C4×C8C2312yes—
264C4×C8C2311yescharacteristic
265C4×C8C2311yescharacteristic
266#41—42no—
267#41—44no—
268#41—44no—
269#41—44no—
270#41—44no—
271#41—82no—
272#41—82no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—44no—
280C2×Q8—82no—
281C2×Q8C2×D411yescharacteristic
282C2×Q8C2412yes—
283C2×Q8C2412yes—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C2×D411yescharacteristic
295C2×D4C2×D411yescharacteristic
296C2×D4C2×D411yescharacteristic
297C2×D4C2412yes—
298C2×D4C2412yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4C2×D411yescharacteristic
308C22×C4C2×D411yescharacteristic
309C22×C4C2×D411yescharacteristic
310C22×C4C2×D411yescharacteristic
311C22×C4C2×D411yescharacteristic
312C22×C4C2×D411yescharacteristic
313C22×C4C2411yescharacteristic
314C22×C4C2411yescharacteristic
315C22×C4C2411yescharacteristic
316SD16—44no—
317SD16—44no—
318D8—44no—
319M16—42no—
320M16—42no—
321C2×C8—42no—
322C2×C8C2412yes—
323C2×C8C2412yes—
324C2×C8C2412yes—
325C2×C8C2412yes—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4C2×D411yescharacteristic
347C4⋊C4C2×D411yescharacteristic
348C4⋊C4C2×D411yescharacteristic
349C4⋊C4C2×D411yescharacteristic
350C4⋊C4C2×D411yescharacteristic
351C4⋊C4C2×D411yescharacteristic
352C4⋊C4C2412yes—
353C4⋊C4C2412yes—
354C4⋊C4C2412yes—
355C4⋊C4C2412yes—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4C2×D411yescharacteristic
373C22⋊C4C2×D411yescharacteristic
374C22⋊C4C2×D411yescharacteristic
375C22⋊C4C2×D411yescharacteristic
376C22⋊C4C2×D411yescharacteristic
377C22⋊C4C2×D411yescharacteristic
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42C2×D411yescharacteristic
383C42C2×D411yescharacteristic
384C42C2×D411yescharacteristic
385C42C2×D411yescharacteristic
386C42C2×D411yescharacteristic
387C42C2×D411yescharacteristic
388C42C2411yescharacteristic
389C42C2411yescharacteristic
390C42C2411yescharacteristic
391C42C2411yescharacteristic
392C23—22no—
393C23—22no—
394C23C22×D411yescharacteristic
395C23C22×D411yescharacteristic
396C23C22×D411yescharacteristic
397Q8—24no—
398Q8—24no—
399Q8—41no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404Q8—82no—
405Q8—82no—
406D4—24no—
407D4—24no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—44no—
412D4—44no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—82no—
419D4—82no—
420D4—82no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4C22×D411yescharacteristic
445C2×C4C22×D411yescharacteristic
446C2×C4C22×D411yescharacteristic
447C2×C4C22×D411yescharacteristic
448C2×C4C22×D411yescharacteristic
449C2×C4C22×D411yescharacteristic
450C2×C4C22×D411yescharacteristic
451C2×C4C22×D411yescharacteristic
452C2×C4C22×D411yescharacteristic
453C2×C4C22×D411yescharacteristic
454C2×C4C22×D411yescharacteristic
455C2×C4C2511yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22C23×D411yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4#57512yes—
492C4#57712yes—
493C4#58512yes—
494C2—41no—
495C2—41no—
496C2—41no—
497C2—82no—
498C2—82no—
499C2#350011yescharacteristic
500C2C2×#57511yescharacteristic
501C2C2×#57711yescharacteristic
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. #3511C2semidirect
2. #3511C2semidirect
3. #3376C2semidirect
4. #3326C2semidirect
5. #3321C2semidirect
6. #3307C2semidirect
7. #3231C2semidirect
8. #3229C2semidirect
9. #3228C2semidirect
10. #3207C2semidirect
11. #3207C2semidirect
12. #3206C2semidirect
13. #3184C2semidirect
14. #3182C2semidirect
15. #3167C2semidirect
16. #3141C2semidirect
17. #3139C2semidirect
18. #3131C2semidirect
19. #2965C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#33762
#33262
#33212
#33072
#32312
#32292
#32282
#32072
#32062
#31842
#31822
#31672
#31412
#31392
#31312
#29652