Finite Group Databasebeta

← #211113

Subgroups — #211113

Order 512.

577 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23717C211yesmaximal, characteristic
2#17892C211yesmaximal, characteristic
3#17870C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5#17330C211yesmaximal, characteristic
6#9669C211yesmaximal, characteristic, complemented
7#9607C211yesmaximal, characteristic, complemented
8#9551C211yesmaximal, characteristic, complemented
9#9551C211yesmaximal, characteristic, complemented
10#9096C211yesmaximal, characteristic
11#9096C211yesmaximal, characteristic
12#8607C211yesmaximal, characteristic
13#8296C211yesmaximal, characteristic, complemented
14#8167C211yesmaximal, characteristic, complemented
15C8×#101C211yesmaximal, characteristic, complemented
16#2965C2211yescharacteristic
17C8×#41C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C4×#137C2211yescharacteristic
21C4×#137C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23#2922C2211yescharacteristic
24#2089C411yescharacteristic
25#2078C411yescharacteristic
26#2064C411yescharacteristic
27#2064C411yescharacteristic
28#2052C411yescharacteristic
29#2052C411yescharacteristic
30#2047C411yescharacteristic
31#2020C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#101C2211yescharacteristic
34C4×#101C2211yescharacteristic
35#1593—21no—
36#1592—21no—
37#1591—21no—
38#1590—21no—
39#1588—21no—
40#1587—21no—
41#1587—21no—
42#1587—21no—
43#1587—21no—
44#1587—21no—
45#1584—21no—
46#1583—21no—
47#1565C2211yescharacteristic
48#1565C2211yescharacteristic
49#1564C2211yescharacteristic
50C2×#334C2211yescharacteristic
51#1479C2211yescharacteristic
52#1479C2211yescharacteristic
53#1477C2211yescharacteristic
54#1477C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57#1461C2211yescharacteristic
58#1461C2211yescharacteristic
59#1454—21no—
60#1453—21no—
61#1451—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450—21no—
65#1450C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#137C2×C411yescharacteristic
76C2×#137C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C4×#41C2311yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81#446—21no—
82#446—21no—
83#446—21no—
84#446—21no—
85#446D411yescharacteristic
86#446D411yescharacteristic
87C8×Q8—21no—
88C8×Q8—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92#441—21no—
93#441—21no—
94#440—21no—
95#440—21no—
96C4×Q16—21no—
97C4×Q16—21no—
98C4×Q16—21no—
99C4×Q16—21no—
100C4×Q16—21no—
101C4×Q16—21no—
102C4×SD16—21no—
103C4×SD16—21no—
104#436D411yescharacteristic
105#436D411yescharacteristic
106#435—21no—
107#435—21no—
108C8×D4—21no—
109C8×D4—22no—
110#433—21no—
111#433—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120C2×#108C2×C411yescharacteristic
121C2×#108C2×C411yescharacteristic
122#428C2×C411yescharacteristic
123#428C2×C411yescharacteristic
124C2×#107C2×C411yescharacteristic
125#424C2×C411yescharacteristic
126#424C2×C411yescharacteristic
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131#419C2×C411yescharacteristic
132#419C2×C411yescharacteristic
133#417C2×C411yescharacteristic
134#417C2×C411yescharacteristic
135C2×#103C2×C411yescharacteristic
136C2×#103C2×C411yescharacteristic
137C2×#101C2×C411yescharacteristic
138C2×#101C2×C411yescharacteristic
139C2×#101C2×C411yescharacteristic
140C2×#101C2×C411yescharacteristic
141#408C2×C411yescharacteristic
142#408C2×C411yescharacteristic
143C4×M16—21no—
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C4×M16C2311yescharacteristic
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153#343C2×C411yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#341C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#335—21no—
160#335—21no—
161#335—21no—
162#334—21no—
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167Q8⋊C8—21no—
168Q8⋊C8—21no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173Q8⋊C8—21no—
174Q8⋊C8—21no—
175Q8⋊C8—21no—
176Q8⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179#324—22no—
180#324—22no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
193C82—21no—
194C82—21no—
195C2×#41C22×C411yescharacteristic
196C22×Q8C22×C411yescharacteristic
197C22×Q8C22×C411yescharacteristic
198#137—22no—
199#137—22no—
200#137—41no—
201#137—41no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—22no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16—41no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16C22×C411yescharacteristic
213C2×M16C22×C411yescharacteristic
214C2×M16C22×C411yescharacteristic
215C2×M16C22×C411yescharacteristic
216C2×M16C22×C411yescharacteristic
217C2×M16C22×C411yescharacteristic
218C22×C8—22no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8#4111yescharacteristic
225C22×C8#4111yescharacteristic
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C2×D411yescharacteristic
235C4×Q8#4111yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C2×D411yescharacteristic
239C4×D4#4111yescharacteristic
240#117—21no—
241#117C22×C411yescharacteristic
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C2411yescharacteristic
250#108—22no—
251#108—41no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#106—21no—
257#106—21no—
258C4⋊C8—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—41no—
273C4⋊C8—41no—
274C4⋊C8—41no—
275C4⋊C8—41no—
276C4⋊C8C2×D411yescharacteristic
277C4⋊C8C2×D411yescharacteristic
278#103—21no—
279#103—21no—
280#103—21no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#102—21no—
289#102—21no—
290#101—22no—
291#101—22no—
292#101—41no—
293#101—41no—
294C22⋊C8—22no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8#4112yes—
305#97—21no—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—21no—
310#97—41no—
311#97—41no—
312C4×C8—22no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—41no—
322C4×C8—41no—
323C4×C8C2×D411yescharacteristic
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8#4111yescharacteristic
332C4×C8#4111yescharacteristic
333#41—21no—
334#41—21no—
335#41—21no—
336#41—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8C4×D411yescharacteristic
349C2×Q8#13112yes—
350C2×D4—21no—
351C2×D4C4×D411yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4C23×C411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362Q16—42no—
363Q16—42no—
364Q16—42no—
365Q16—42no—
366SD16—42no—
367SD16—42no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—81no—
380M16—81no—
381M16#13112yes—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—21no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—42no—
400C2×C8—44no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—81no—
406C2×C8—81no—
407C2×C8C4×D411yescharacteristic
408C2×C8C4×D411yescharacteristic
409C2×C8C4×D411yescharacteristic
410C2×C8C4×D411yescharacteristic
411C2×C8#12111yescharacteristic
412C2×C8#12111yescharacteristic
413C2×C8#12111yescharacteristic
414C2×C8#12111yescharacteristic
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4C4×D411yescharacteristic
426C4⋊C4C4×D411yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42C22×D411yescharacteristic
438C42C22×D411yescharacteristic
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C23—21no—
442C23C4×#4111yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—21no—
446Q8—21no—
447Q8—41no—
448Q8—41no—
449Q8—41no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453D4—21no—
454D4—21no—
455D4—41no—
456D4—41no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4#52511yescharacteristic
482C2×C4#52911yescharacteristic
483C2×C4#53211yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4C2×#13111yescharacteristic
486C2×C4#56811yescharacteristic
487C2×C4#56811yescharacteristic
488C8—22no—
489C8—21no—
490C8—21no—
491C8—22no—
492C8—21no—
493C8—21no—
494C8—21no—
495C8—21no—
496C8—21no—
497C8—21no—
498C8—42no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—41no—
506C8—41no—
507C8—81no—
508C8—81no—
509C8—81no—
510C8—81no—
511C22—21no—
512C22—21no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22C4×#12111yescharacteristic
517C22#296611yescharacteristic
518C22#296711yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4#293311yescharacteristic
530C4#293311yescharacteristic
531C4#293311yescharacteristic
532C4#293511yescharacteristic
533C4#306911yescharacteristic
534C4#306911yescharacteristic
535C2—21no—
536C2—21no—
537C2—81no—
538C2#2149611yescharacteristic
539C2#2155611yescharacteristic
540C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#41C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1587C2semidirect
6. #9669C2semidirect
7. #9607C2semidirect
8. #9551C2semidirect
9. #9551C2semidirect
13. #8296C2semidirect
14. #8167C2semidirect
15. C8×#101C2semidirect

Maximal subgroups

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

subgroupindex
#237172
#178922
#178702
C2×#15872
#173302
#96692
#96072
#95512
#90962
#86072
#82962
#81672
C8×#1012