Finite Group Databasebeta

← #409269

Subgroups — #409269

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1807C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#14264C211yesmaximal, characteristic
4#14264C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11327C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#10722C211yesmaximal, characteristic
10#10722C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1807—24no—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32#1729—24no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1444—21no—
36#1444—21no—
37#1442—21no—
38#1442—21no—
39#1439—22no—
40#1439—22no—
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—42no—
60C2×#117—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—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#389—42no—
82#389—42no—
83#389—42no—
84#389—42no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380Q814yes—
92#380Q814yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—24no—
113C25C2×D411yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×Q812yes—
129C23×C4C2×Q812yes—
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C2×M16—28no—
135C2×M16—48no—
136C2×M16—48no—
137C2×M16—44no—
138C2×M16—44no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—28no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—28no—
154C22×C8—44no—
155C22×C8—48no—
156C22×C8—48no—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8#4114yes—
160C22×C8#4114yes—
161#117—44no—
162#117—44no—
163#117—48no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
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—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×Q814yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—81no—
210C2×C22⋊C4—81no—
211C2×C22⋊C4—81no—
212C2×C22⋊C4—81no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221#106—44no—
222#106—44no—
223#95—24no—
224#95—24no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—82no—
230#95—82no—
231#95—82no—
232#95—82no—
233C24—22no—
234C24—22no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—21no—
241C24#12112yes—
242C24C22⋊Q812yes—
243C24C22⋊Q812yes—
244C24C22×D411yescharacteristic
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
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—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4#12112yes—
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4#12314yes—
313C22×C4#12514yes—
314C22×C4#12514yes—
315C22×C4C4⋊Q814yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C22×Q811yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326M16—416no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—416no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8#12314yes—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—88no—
368C4⋊C4—88no—
369C22⋊C4—48no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23#39814yes—
401C23C2×#12111yescharacteristic
402C23C2×C22⋊Q811yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23#53212yes—
405C23#53412yes—
406C23#57611yescharacteristic
407C23#57711yescharacteristic
408C23#57711yescharacteristic
409C23#57712yes—
410C23#57712yes—
411C23#57811yescharacteristic
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
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—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4#39814yes—
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4#52511yescharacteristic
474C2×C4C2×#12511yescharacteristic
475C2×C4C2×#12511yescharacteristic
476C2×C4#52912yes—
477C2×C4C2×C4⋊Q811yescharacteristic
478C2×C4#53411yescharacteristic
479C2×C4#53511yescharacteristic
480C2×C4#53812yes—
481C2×C4#55112yes—
482C2×C4#55112yes—
483C2×C4#55112yes—
484C2×C4#55112yes—
485C2×C4#55412yes—
486C2×C4#55412yes—
487C2×C4#55412yes—
488C2×C4#55412yes—
489C8—24no—
490C8—44no—
491C8—42no—
492C22—24no—
493C22—24no—
494C22—24no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22C2×#39811yescharacteristic
504C22#256412yes—
505C22#256412yes—
506C22#261711yescharacteristic
507C22#285411yescharacteristic
508C22#305711yescharacteristic
509C22#305711yescharacteristic
510C22#306011yescharacteristic
511C22#306111yescharacteristic
512C22#308111yescharacteristic
513C22#308111yescharacteristic
514C22#308211yescharacteristic
515C22#308211yescharacteristic
516C22#308412yes—
517C22#308412yes—
518C22#317312yes—
519C22#317412yes—
520C22#317812yes—
521C22#317912yes—
522C22#319812yes—
523C22#320012yes—
524C22#320112yes—
525C22#320412yes—
526C4—24no—
527C4—82no—
528C4—82no—
529C4—84no—
530C4—84no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4—161no—
535C4#248211yescharacteristic
536C4#248311yescharacteristic
537C4#250211yescharacteristic
538C4#250512yes—
539C4#250512yes—
540C4#258311yescharacteristic
541C2—44no—
542C2#1501411yescharacteristic
543C2#2366811yescharacteristic
544C2#2366911yescharacteristic
545C2#2916812yes—
546C2#2917012yes—
547C2#2917212yes—
548C2#2917412yes—
549C2#2974011yescharacteristic
550C2#2974111yescharacteristic
551C2#3313111yescharacteristic
552C2#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. #11327C2semidirect
7. #11293C2semidirect
8. #11292C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18072
C2×#17292
#142642
#113782
#113272
#112932
#112922
#107222
#107212