Finite Group Databasebeta

← #496540

Subgroups — #496540

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34955C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3C2×#1801C211yesmaximal, characteristic, complemented
4#13301C212yesmaximal, complemented
5#13226C212yesmaximal, complemented
6#13092C214yesmaximal, complemented
7#12801C214yesmaximal, complemented
8C2×#532C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C2×#423—21no—
12C22×#101C2212yes—
13#2592—21no—
14#2592—21no—
15#2591—21no—
16#2591—21no—
17#2568C411yescharacteristic
18#2568C2211yescharacteristic
19#2548C411yescharacteristic
20#2548C2211yescharacteristic
21#2504C412yes—
22#2436C2212yes—
23#2398—21no—
24#2398—21no—
25C4×#124C412yes—
26C4×#124C2212yes—
27#2297—21no—
28#2297—21no—
29C4×#117C411yescharacteristic
30C2×#355C2218yes—
31#1821—21no—
32#1801—21no—
33#1801—21no—
34#1801—21no—
35#1801C2212yes—
36#1801C2212yes—
37#1801C2212yes—
38#1784C2212yes—
39C4×#101—21no—
40C4×#101C2212yes—
41C2×#343—21no—
42#1394C2218yes—
43C22×#41—21no—
44C23×Q8C2311yescharacteristic
45C22×M16—22no—
46#532—41no—
47#532—41no—
48#532—41no—
49#532—41no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124—21no—
53C2×#124C2×C412yes—
54C2×#124C2312yes—
55C2×#121—21no—
56C2×#121C2×C412yes—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2×C412yes—
71C2×#117—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78#432—41no—
79#432—41no—
80#431—41no—
81#431—41no—
82#423—22no—
83#423—22no—
84#423—22no—
85#423—41no—
86#423—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8—41no—
89C2×#101—22no—
90C2×#101—22no—
91C2×#101—44no—
92C2×#101C2314yes—
93C2×#101C2314yes—
94#407—22no—
95C4×M16—22no—
96#400—21no—
97#400—21no—
98#400—21no—
99#400—21no—
100#397—22no—
101#397—22no—
102#394—21no—
103#394—21no—
104#394—21no—
105#394—21no—
106#393—22no—
107#393—22no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—21no—
112#391C2×C412yes—
113#391C2312yes—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390C2×C412yes—
119#390C2×C412yes—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#382C2×C412yes—
129#382C2×C412yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4C2×C412yes—
139#376—21no—
140C43C2×C412yes—
141#355—48no—
142#343—22no—
143#343—22no—
144#343—22no—
145#343—22no—
146C2×#41—44no—
147C2×#41—44no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—41no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—21no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8C22×C411yescharacteristic
159C22×Q8C2×D411yescharacteristic
160C22×Q8C2×D411yescharacteristic
161C22×Q8C2411yescharacteristic
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4C22×C412yes—
166C23×C4—22no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—42no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8—42no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182#127—41no—
183#127—41no—
184#124—24no—
185#124—24no—
186#124—42no—
187#124—42no—
188#124—44no—
189#121—42no—
190#121—42no—
191#121—42no—
192#121—42no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22×C412yes—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281#101—24no—
282#101—28no—
283#101—24no—
284#101—44no—
285#101—44no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289C4×C8—42no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301C24—22no—
302C24C2×C22⋊C411yescharacteristic
303#41—84no—
304#41—84no—
305#41—84no—
306#41—84no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8H(5,𝔽2)12yes—
328C2×Q82−1+412yes—
329C2×D4—24no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—82no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4#9911yescharacteristic
373C22×C4#9911yescharacteristic
374C22×C4#9911yescharacteristic
375C22×C4#9911yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390M16—44no—
391M16—44no—
392M16—44no—
393M16—44no—
394M16—84no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—84no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—24no—
451C42—42no—
452C42—44no—
453C42—82no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23C2×#9911yescharacteristic
466C23C2×#9911yescharacteristic
467C23C22×C22⋊C411yescharacteristic
468Q8—42no—
469Q8—42no—
470Q8—44no—
471Q8—44no—
472Q8—42no—
473Q8—42no—
474Q8—82no—
475Q8—82no—
476D4—84no—
477D4—82no—
478D4—84no—
479D4—84no—
480D4—84no—
481D4—82no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—84no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—84no—
538C2×C4#35514yescomplemented
539C2×C4C2×#9911yescharacteristic
540C2×C4C2×#9911yescharacteristic
541C2×C4C2×#9911yescharacteristic
542C2×C4C2×#9911yescharacteristic
543C2×C4#41012yes—
544C2×C4#52012yes—
545C2×C4#53511yescharacteristic
546C2×C4#53611yescharacteristic
547C2×C4#53712yes—
548C8—84no—
549C8—84no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—42no—
558C22—42no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22C2×#35512yes—
567C22#231111yescharacteristic
568C22C22×#9911yescharacteristic
569C22C2×#41011yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—21no—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4C2×#35512yes—
589C4C2×#35512yes—
590C2—21no—
591C2—21no—
592C2—22no—
593C2—41no—
594C2—162no—
595C2#2187411yescharacteristic
596C2C22×#35511yescharacteristic
597C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. C2×#1801C2semidirect
4. #13301C2semidirect
5. #13226C2semidirect
6. #13092C2semidirect
7. #12801C2semidirect
538. C2×C4#355semidirect

Maximal subgroups

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

subgroupindex
#349552
#207272
C2×#18012
#133012
#132262
#130922
#128012