Finite Group Databasebeta

← #211069

Subgroups — #211069

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#473C211yesmaximal, characteristic
2#17872C211yesmaximal, characteristic
3C2×#1587C211yesmaximal, characteristic, complemented
4C8×#137C211yesmaximal, characteristic
5#17320C211yesmaximal, characteristic
6#9671C211yesmaximal, characteristic, complemented
7#9609C211yesmaximal, characteristic, complemented
8#9551C211yesmaximal, characteristic, complemented
9#9551C211yesmaximal, characteristic, complemented
10#9096C211yesmaximal, characteristic
11#9096C211yesmaximal, characteristic
12#8604C211yesmaximal, characteristic
13#8296C211yesmaximal, characteristic, complemented
14#8167C211yesmaximal, characteristic, complemented
15#8165C211yesmaximal, characteristic, complemented
16C2×#473C411yescharacteristic
17C8×#41C2211yescharacteristic
18C8×#41C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C4×#137C2211yescharacteristic
22C4×#137C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1979C411yescharacteristic
26#1952C411yescharacteristic
27#1930C411yescharacteristic
28#1930C411yescharacteristic
29#1908C411yescharacteristic
30#1908C411yescharacteristic
31#1816C411yescharacteristic
32C4×#108C2211yescharacteristic
33C4×#101C2211yescharacteristic
34C4×#101C2211yescharacteristic
35#1587—21no—
36#1587—21no—
37#1587—21no—
38#1587—21no—
39#1587—21no—
40#1585—21no—
41#1584—21no—
42#1582—21no—
43#1579—21no—
44SD16⋊C8—21no—
45C8×Q16—21no—
46C8×SD16—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#1452—21no—
60#1452C2211yescharacteristic
61#1451—21no—
62#1451—21no—
63#1451C2211yescharacteristic
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C2×#322—21no—
68C2×#322C2211yescharacteristic
69#1283C2211yescharacteristic
70#1283C2211yescharacteristic
71#1283C2211yescharacteristic
72#1283C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#137C2×C411yescharacteristic
76C2×#137C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131C2×C411yescharacteristic
80C4×#41C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83#473—24no—
84#446—21no—
85#446—21no—
86#446—21no—
87#446—21no—
88C8×Q8—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92C8×Q8—21no—
93C8×Q8D411yescharacteristic
94C8×Q8D411yescharacteristic
95#441—21no—
96#441—21no—
97#440—21no—
98#440—21no—
99C4×Q16—21no—
100C4×Q16—21no—
101C4×Q16—21no—
102C4×Q16—21no—
103C4×Q16—21no—
104C4×Q16—21no—
105C4×SD16—21no—
106C4×SD16—21no—
107C8×D4—21no—
108C8×D4—21no—
109C8×D4—21no—
110C8×D4—22no—
111C8×D4D411yescharacteristic
112C8×D4D411yescharacteristic
113#431—21no—
114#431—21no—
115#431C2311yescharacteristic
116#431C2311yescharacteristic
117#431C2311yescharacteristic
118#431C2311yescharacteristic
119C2×#108C2×C411yescharacteristic
120C2×#108C2×C411yescharacteristic
121#428C2×C411yescharacteristic
122#428C2×C411yescharacteristic
123C2×#107C2×C411yescharacteristic
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128#419C2×C411yescharacteristic
129#419C2×C411yescharacteristic
130#417C2×C411yescharacteristic
131#417C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#101C2×C411yescharacteristic
135C2×#101C2×C411yescharacteristic
136C2×#101C2×C411yescharacteristic
137C2×#101C2×C411yescharacteristic
138#405—21no—
139#405—21no—
140#405C2×C411yescharacteristic
141#405C2×C411yescharacteristic
142C4×M16—21no—
143C4×M16C2311yescharacteristic
144C4×M16C2311yescharacteristic
145C4×M16C2311yescharacteristic
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#343C2×C411yescharacteristic
155#343C2×C411yescharacteristic
156#343C2×C411yescharacteristic
157#343C2×C411yescharacteristic
158#341C2×C411yescharacteristic
159#341C2×C411yescharacteristic
160#335—21no—
161#335—21no—
162#335—21no—
163#334—21no—
164#334—21no—
165#334—21no—
166#334—21no—
167#334—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—
177Q8⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180#324—22no—
181#324—22no—
182#322—21no—
183#322—21no—
184#322—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191#322D411yescharacteristic
192#322D411yescharacteristic
193#322D411yescharacteristic
194#322D411yescharacteristic
195C82—21no—
196C2×#41C22×C411yescharacteristic
197C22×Q8C22×C411yescharacteristic
198C22×Q8C22×C411yescharacteristic
199#137—22no—
200#137—22no—
201#137—22no—
202#137M1614yes—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—22no—
206C2×Q16—21no—
207C2×Q16—21no—
208C2×Q16—41no—
209C2×SD16—21no—
210C2×SD16—21no—
211#131—22no—
212#131—22no—
213#131—41no—
214#131—41no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16C22×C411yescharacteristic
218C2×M16C22×C411yescharacteristic
219C2×M16C22×C411yescharacteristic
220C2×M16C22×C411yescharacteristic
221C2×M16C22×C411yescharacteristic
222C2×M16C22×C411yescharacteristic
223C22×C8—22no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8#4111yescharacteristic
234C22×C8#4111yescharacteristic
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8C2×D411yescharacteristic
244C4×Q8#4111yescharacteristic
245C4×D4—21no—
246C4×D4—21no—
247C4×D4C2×D411yescharacteristic
248C4×D4#4111yescharacteristic
249#117—21no—
250#117C22×C411yescharacteristic
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42C2411yescharacteristic
259#108—22no—
260#108—41no—
261#107—21no—
262#107—21no—
263#107—21no—
264#107—21no—
265#106—21no—
266#106—21no—
267C4⋊C8—22no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8C2×D411yescharacteristic
282C4⋊C8C2×D411yescharacteristic
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295#101—22no—
296#101—22no—
297#101—41no—
298#101—41no—
299C22⋊C8—22no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8#4112yes—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—21no—
310#97—21no—
311#97—41no—
312#97—41no—
313C4×C8—22no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—41no—
327C4×C8—41no—
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8#4111yescharacteristic
337C4×C8#4111yescharacteristic
338#41—21no—
339#41—21no—
340#41C2×M1612yes—
341#41#13112yes—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8C4×D411yescharacteristic
354C2×Q8C2×M1612yes—
355C2×D4—21no—
356C2×D4C4×D411yescharacteristic
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4C23×C411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367Q16—24no—
368Q16—42no—
369Q16—42no—
370Q16—42no—
371SD16—24no—
372SD16—42no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—42no—
385M16—42no—
386M16C2×M1612yes—
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—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—44no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—81no—
413C2×C8—81no—
414C2×C8—81no—
415C2×C8—81no—
416C2×C8C4×D411yescharacteristic
417C2×C8C4×D411yescharacteristic
418C2×C8C4×D411yescharacteristic
419C2×C8C4×D411yescharacteristic
420C2×C8#12111yescharacteristic
421C2×C8#12111yescharacteristic
422C2×C8#12111yescharacteristic
423C2×C8#12111yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4C4×D411yescharacteristic
435C4⋊C4C4×D411yescharacteristic
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42C22×D411yescharacteristic
447C42C22×D411yescharacteristic
448C42C2×#4111yescharacteristic
449C42C2×#4111yescharacteristic
450C23—21no—
451C23C4×#4111yescharacteristic
452Q8—21no—
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8#43512yes—
462D4—22no—
463D4—22no—
464D4—21no—
465D4#43512yes—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4#52911yescharacteristic
492C2×C4#53211yescharacteristic
493C2×C4C22×M1611yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4#56811yescharacteristic
496C2×C4#56811yescharacteristic
497C8—22no—
498C8—22no—
499C8—21no—
500C8—21no—
501C8—21no—
502C8—21no—
503C8—21no—
504C8—21no—
505C8—21no—
506C8—21no—
507C8—42no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—81no—
517C8—81no—
518C8—81no—
519C8—81no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22C4×#12111yescharacteristic
526C22#296811yescharacteristic
527C22#297011yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—22no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4C2×#43511yescharacteristic
539C4#293211yescharacteristic
540C4#293311yescharacteristic
541C4D4×M1611yescharacteristic
542C4#306911yescharacteristic
543C4#306911yescharacteristic
544C2—21no—
545C2—21no—
546C2—42no—
547C2#2148911yescharacteristic
548C2#2155611yescharacteristic
549C2#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
3. C2×#1587C2semidirect
6. #9671C2semidirect
7. #9609C2semidirect
8. #9551C2semidirect
9. #9551C2semidirect
13. #8296C2semidirect
14. #8167C2semidirect
15. #8165C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4732
#178722
C2×#15872
C8×#1372
#173202
#96712
#96092
#95512
#90962
#86042
#82962
#81672
#81652