Finite Group Databasebeta

← #495426

Subgroups — #495426

Order 512.

1033 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#13303C214yesmaximal, complemented
5#13302C212yesmaximal, complemented
6#13301C212yesmaximal, complemented
7#12585C214yesmaximal
8C2×#533C411yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C2×#423—21no—
11C2×#413C2212yes—
12C2×#413C2212yes—
13C2×C4×M16—21no—
14#2568C2211yescharacteristic
15#2567C2211yescharacteristic
16#2566C412yes—
17#2565C2212yes—
18#2564—21no—
19#2564—21no—
20#2563C411yescharacteristic
21#2542C411yescharacteristic
22#2417—21no—
23#2417—21no—
24#2396—21no—
25#2396—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28#2297C2212yes—
29#2285C412yes—
30C2×C4×C4⋊C4C411yescharacteristic
31#1801C2212yes—
32#1801C2212yes—
33#1800—21no—
34#1800—21no—
35C4×#101C2212yes—
36C4×#101C2212yes—
37C4×#100—21no—
38C4×#100—21no—
39#1742—21no—
40#1742—21no—
41#1394C2218yes—
42#1394C2218yes—
43C22×#41C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46#533—41no—
47#533—41no—
48#533—41no—
49#533—41no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125C2×C412yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×#117C2×C412yes—
72C2×#117C2×C411yescharacteristic
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C412yes—
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80#423—22no—
81#423—22no—
82#423—41no—
83#423—41no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86#413—24no—
87#413—28no—
88#413Q818yescomplemented
89C2×#101C2314yes—
90C2×#101C2314yes—
91C2×#100—22no—
92C2×#100—22no—
93C4×M16—22no—
94C4×M16—22no—
95C4×M16—41no—
96C4×M16—41no—
97C2×#97—41no—
98C2×C4×C8—41no—
99#400—21no—
100#400—21no—
101#399—21no—
102#399—21no—
103#398—22no—
104#398—22no—
105#396—22no—
106#396—22no—
107#395—21no—
108#395—21no—
109#393—21no—
110#393—21no—
111#391—21no—
112#391—21no—
113#391C2312yes—
114#391C2312yes—
115#390C2×C412yes—
116#390C2×C412yes—
117#389—21no—
118#389—21no—
119#387—22no—
120#387—22no—
121#386—22no—
122#386—22no—
123#385—22no—
124#385—22no—
125#384C2×C412yes—
126#384C2×C412yes—
127#382C2×C412yes—
128#382C2×C412yes—
129#380—21no—
130#380—21no—
131#380—22no—
132#380—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C412yes—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C2×#95—21no—
143#343—22no—
144#343—22no—
145#343—22no—
146#343—22no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C2×#41C2×Q814yes—
151C2×#41#4114yes—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C2411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—22no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—28no—
169C2×M16—24no—
170C2×M16—24no—
171C2×M16—44no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16—44no—
175C2×M16—44no—
176C22×C8—42no—
177C22×C8—42no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182#125—42no—
183#125—42no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212#117—41no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—41no—
217#117—41no—
218#117—42no—
219#117—42no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
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—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—44no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C412yes—
273C2×C42C22×C412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282#101—24no—
283#101—28no—
284#101C2×Q818yes—
285#100—28no—
286#100—44no—
287#97—42no—
288#97—42no—
289C4×C8—42no—
290C4×C8—42no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24C2×C22⋊C411yescharacteristic
305#41—44no—
306#41—44no—
307#41—44no—
308#41—44no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—82no—
317C2×Q8C22×Q812yes—
318C2×Q8C2×#4112yes—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4C4×Q814yes—
326C2×D4C22⋊Q814yes—
327C2×D4#12412yes—
328C2×D4C4⋊Q812yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—82no—
362C22×C4—81no—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C4×Q814yes—
373C22×C4C22⋊Q814yes—
374C22×C4#12412yes—
375C22×C4C4⋊Q812yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379M16—28no—
380M16—44no—
381M16—48no—
382M16—44no—
383M16—44no—
384M16—84no—
385C2×C8—44no—
386C2×C8—48no—
387C2×C8—44no—
388C2×C8—84no—
389C4⋊C4—24no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—82no—
409C4⋊C4—82no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
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—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—44no—
444C42#9911yescharacteristic
445C42#9911yescharacteristic
446C42#9911yescharacteristic
447C42#9911yescharacteristic
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23#39112yes—
461C23#39112yes—
462C23#39112yes—
463C23C22×C22⋊C411yescharacteristic
464Q8—42no—
465Q8—42no—
466Q8—82no—
467Q8—84no—
468Q8—82no—
469D4—44no—
470D4—44no—
471D4—42no—
472D4—42no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—88no—
526C2×C4#39112yes—
527C2×C4C2×#9911yescharacteristic
528C2×C4C2×#9911yescharacteristic
529C2×C4C2×#9911yescharacteristic
530C2×C4C2×#9911yescharacteristic
531C2×C4C2×#9911yescharacteristic
532C2×C4C2×#9911yescharacteristic
533C2×C4#41012yes—
534C2×C4C2×C4×Q812yes—
535C2×C4C2×C22⋊Q812yes—
536C2×C4C2×#12411yescharacteristic
537C2×C4C2×C4⋊Q811yescharacteristic
538C8—48no—
539C8—84no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22C2×#39111yescharacteristic
558C22C22×#9911yescharacteristic
559C22C2×#41011yescharacteristic
560C4—22no—
561C4—21no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—44no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4#199612yes—
578C4#199612yes—
579C4#213212yes—
580C2—21no—
581C2—21no—
582C2—42no—
583C2—42no—
584C2—42no—
585C2C2×#199611yescharacteristic
586C2C2×#213211yescharacteristic
587C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13303C2semidirect
5. #13302C2semidirect
6. #13301C2semidirect
88. #413Q8semidirect

Maximal subgroups

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

subgroupindex
#349472
#206372
C4×#4132
#133032
#133022
#133012
#125852