Finite Group Databasebeta

← #409378

Subgroups — #409378

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1807C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#14343C211yesmaximal, characteristic
4#14343C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11316C212yesmaximal, complemented
7#11315C212yesmaximal, complemented
8#11257C212yesmaximal, complemented
9#10722C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21#1807—24no—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30#1729—24no—
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1445—21no—
34#1445—21no—
35#1445—21no—
36#1445—21no—
37#1440—22no—
38#1440—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—42no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—42no—
53C23×C8D414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42C2312yes—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—24no—
80C2×#106—42no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—24no—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4#4112yes—
129C23×C4#4112yes—
130C23×C4#4112yes—
131C23×C4C2411yescharacteristic
132C2×M16—28no—
133C2×M16—48no—
134C2×M16—48no—
135C2×M16—44no—
136C2×M16—44no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—28no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—28no—
152C22×C8—44no—
153C22×C8—48no—
154C22×C8—48no—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8#4114yes—
158C22×C8#4114yes—
159#117—44no—
160#117—44no—
161#117—44no—
162#117—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—81no—
205C2×C22⋊C4—81no—
206C2×C22⋊C4—81no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216#106—44no—
217#106—44no—
218#95—42no—
219#95—42no—
220#95—42no—
221#95—42no—
222#95—82no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229#95—82no—
230C24—22no—
231C24—22no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24#12112yes—
239C24#12312yes—
240C24#12312yes—
241C24C22×D411yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4#12112yes—
307C22×C4#12314yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12414yes—
311C22×C4#12614yes—
312C22×C4#12614yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C42−1+412yes—
323M16—416no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—416no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8#12314yes—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C22⋊C4—48no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23#39714yes—
396C23C2×#12111yescharacteristic
397C23C2×#12311yescharacteristic
398C23C2×#12311yescharacteristic
399C23#53212yes—
400C23#53712yes—
401C23#57611yescharacteristic
402C23#57711yescharacteristic
403C23#57711yescharacteristic
404C23#57712yes—
405C23#57712yes—
406C23#57811yescharacteristic
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4#39714yes—
466C2×C4C2×#12311yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C2×C4#52511yescharacteristic
469C2×C4C2×#12411yescharacteristic
470C2×C4C2×#12611yescharacteristic
471C2×C4C2×#12611yescharacteristic
472C2×C4#52912yes—
473C2×C4#53411yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53912yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55112yes—
479C2×C4#55112yes—
480C2×C4#55212yes—
481C2×C4#55212yes—
482C2×C4#55212yes—
483C2×C4#55212yes—
484C8—24no—
485C8—44no—
486C8—42no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22C2×#39711yescharacteristic
499C22#249812yes—
500C22#249812yes—
501C22#261811yescharacteristic
502C22#285411yescharacteristic
503C22#305711yescharacteristic
504C22#305711yescharacteristic
505C22#306011yescharacteristic
506C22#306111yescharacteristic
507C22#309611yescharacteristic
508C22#309611yescharacteristic
509C22#309711yescharacteristic
510C22#309711yescharacteristic
511C22#309912yes—
512C22#309912yes—
513C22#317312yes—
514C22#317412yes—
515C22#317812yes—
516C22#317912yes—
517C22#325712yes—
518C22#325812yes—
519C22#326012yes—
520C22#326312yes—
521C4—24no—
522C4—82no—
523C4—82no—
524C4—161no—
525C4—161no—
526C4—161no—
527C4—161no—
528C4—162no—
529C4—162no—
530C4#248311yescharacteristic
531C4#248411yescharacteristic
532C4#249312yes—
533C4#249312yes—
534C4#250311yescharacteristic
535C4#256111yescharacteristic
536C2—44no—
537C2#1501911yescharacteristic
538C2#2364011yescharacteristic
539C2#2364111yescharacteristic
540C2#2786712yes—
541C2#2786912yes—
542C2#2787112yes—
543C2#2787312yes—
544C2#2978011yescharacteristic
545C2#2978111yescharacteristic
546C2#3313111yescharacteristic
547C2#3313711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11378C2semidirect
6. #11316C2semidirect
7. #11315C2semidirect
8. #11257C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18072
C2×#17292
#143432
#113782
#113162
#113152
#112572
#107222
#107212