Finite Group Databasebeta

← #205236

Subgroups — #205236

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3185C211yesmaximal, characteristic
2#21889C211yesmaximal, characteristic
3#21837C211yesmaximal, characteristic
4#20482C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic
6C2×#1544C211yesmaximal, characteristic
7C2×#1542C211yesmaximal, characteristic
8#9748C211yesmaximal, characteristic, complemented
9#9745C211yesmaximal, characteristic, complemented
10#9742C211yesmaximal, characteristic, complemented
11#9741C211yesmaximal, characteristic, complemented
12#9737C211yesmaximal, characteristic, complemented
13#9736C211yesmaximal, characteristic, complemented
14#9733C211yesmaximal, characteristic, complemented
15#9729C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3185—24no—
19#3185C418yescomplemented
20C2×#498C2211yescharacteristic
21C2×#494C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#470—21no—
25C2×#470—21no—
26C2×#441—21no—
27C2×#441—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2283C2211yescharacteristic
33#1991—22no—
34#1988C2211yescharacteristic
35#1988C2211yescharacteristic
36#1988C2211yescharacteristic
37#1988C2211yescharacteristic
38#1987C2211yescharacteristic
39#1987C2211yescharacteristic
40#1981C2211yescharacteristic
41#1981C2211yescharacteristic
42#1892—22no—
43#1825—22no—
44#1805—22no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1753—21no—
50#1753—21no—
51#1746—21no—
52#1726—21no—
53#1565—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1544—24no—
59#1542—24no—
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65C2×#332C2211yescharacteristic
66C2×#332C2211yescharacteristic
67C2×#332C2211yescharacteristic
68C22×Q16—21no—
69C22×Q16—21no—
70C22×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72C23×C8—21no—
73#533—22no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533C2×C414yes—
80#533C2×C414yes—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×#125—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×#117—21no—
96C2×#117—21no—
97C22×C42C2311yescharacteristic
98#498—22no—
99#498C2×C414yes—
100#494—22no—
101#494C2×C414yes—
102#487—22no—
103#487—22no—
104#487C2×C414yes—
105#487C2×C414yes—
106#470—28no—
107#470—28no—
108#441—28no—
109#441—28no—
110#431—22no—
111#431—22no—
112#431—22no—
113#431D414yes—
114C2×#106—21no—
115C2×#106—21no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—22no—
130C2×#103—22no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C4×M16C2×C414yes—
134C4×M16D414yes—
135C2×#97—21no—
136C2×#97—21no—
137C2×#97D411yescharacteristic
138C2×#97D411yescharacteristic
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#383—21no—
146#382D412yes—
147#381—21no—
148#380D412yes—
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—21no—
152#376—21no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#335—22no—
164#335—22no—
165#334—22no—
166#334—22no—
167#333—22no—
168#333—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#332—22no—
174#332—22no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×Q16—24no—
183C2×Q16—24no—
184C2×Q16—42no—
185C2×Q16—42no—
186C2×Q16—42no—
187C2×Q16—42no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—41no—
191C2×M16—41no—
192C2×M16C22⋊C414yes—
193C2×M16C22⋊C414yes—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—42no—
201C22×C8—82no—
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8C22×C412yes—
219C4⋊Q8C22×C412yes—
220C4⋊Q8C22×C412yes—
221C4⋊Q8C22×C412yes—
222#125—24no—
223#125—24no—
224#125—42no—
225#125—42no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×Q8—42no—
241C4×Q8—42no—
242#117—24no—
243#117—24no—
244#117—42no—
245#117—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42C22×C412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D412yes—
272C2×C42#4111yescharacteristic
273C2×C42#4112yes—
274C2×C42C2411yescharacteristic
275#106—24no—
276#106—24no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—42no—
288C4⋊C8C2×D412yes—
289C4⋊C8C2×D412yes—
290#103—24no—
291#103—24no—
292#103—24no—
293#103—24no—
294#103—82no—
295#103—82no—
296C22⋊C8—24no—
297C22⋊C8—42no—
298#97—42no—
299#97—42no—
300#97C22⋊C414yes—
301#97C22⋊C414yes—
302C4×C8—24no—
303C4×C8—42no—
304C4×C8—42no—
305C4×C8—42no—
306C4×C8C22×C412yes—
307C4×C8C22×C412yes—
308C4×C8C2×D412yes—
309C4×C8C2×D412yes—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—21no—
314#95—21no—
315C24#12111yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—82no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C2×C22⋊C412yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#12112yes—
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12412yes—
356C22×C4#12712yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360Q16—48no—
361Q16—48no—
362M16—28no—
363M16—44no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—84no—
385C2×C8C2×C22⋊C412yes—
386C2×C8C2×C22⋊C412yes—
387C2×C8C2×C22⋊C412yes—
388C2×C8C2×C22⋊C412yes—
389C2×C8#13512yes—
390C2×C8#13512yes—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—44no—
423C42—44no—
424C42C2×C22⋊C412yes—
425C42C4×D412yes—
426C42C4×D412yes—
427C42#12112yes—
428C42#12412yes—
429C42#12712yes—
430C42C23×C411yescharacteristic
431C42C22×D411yescharacteristic
432C42C22×D411yescharacteristic
433C42C2×#4111yescharacteristic
434C23—21no—
435C23—21no—
436C23—22no—
437C23—22no—
438C23#39012yes—
439C23C2×#12111yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—42no—
444Q8—44no—
445Q8—82no—
446Q8—82no—
447Q8—81no—
448Q8—81no—
449Q8—81no—
450Q8—81no—
451Q8—81no—
452Q8—81no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4#39012yes—
484C2×C4#39012yes—
485C2×C4#39012yes—
486C2×C4C22×C22⋊C411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12411yescharacteristic
491C2×C4C2×#12711yescharacteristic
492C2×C4C2×#13511yescharacteristic
493C2×C4C2×#13511yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57511yescharacteristic
496C2×C4#57712yes—
497C2×C4#57812yes—
498C8—22no—
499C8—22no—
500C8—22no—
501C8—42no—
502C8—42no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—81no—
507C8#41614yes—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—42no—
512C22—42no—
513C22C2×#39011yescharacteristic
514C22#303511yescharacteristic
515C22#303511yescharacteristic
516C22#306312yes—
517C22#306912yes—
518C4—21no—
519C4—21no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4C2×#41611yescharacteristic
530C4#289611yescharacteristic
531C4#294712yes—
532C4#303511yescharacteristic
533C4#303511yescharacteristic
534C4#315411yescharacteristic
535C4#315611yescharacteristic
536C2—42no—
537C2#1924912yes—
538C2#1925112yes—
539C2#2123411yescharacteristic
540C2#2125011yescharacteristic
541C2#2125711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9748C2semidirect
9. #9745C2semidirect
10. #9742C2semidirect
11. #9741C2semidirect
12. #9737C2semidirect
13. #9736C2semidirect
14. #9733C2semidirect
15. #9729C2semidirect
19. #3185C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31852
#218892
#218372
#204822
C2×#15652
C2×#15442
C2×#15422
#97482
#97452
#97422
#97412
#97372
#97362
#97332
#97292