Finite Group Databasebeta

← #219836

Subgroups — #219836

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34932C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4C2×#1494C212yesmaximal, complemented
5#9673C211yesmaximal, characteristic, complemented
6#9637C211yesmaximal, characteristic, complemented
7#9071C212yesmaximal, complemented
8#8941C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8315C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#529C2212yes—
13C2×C8×D4—21no—
14C2×#432C2212yes—
15C22×C4×C8C2212yes—
16#2553—22no—
17#2552C411yescharacteristic
18#2547C411yescharacteristic
19#2542C411yescharacteristic
20#2537C411yescharacteristic
21#2496—21no—
22#2489C412yes—
23#2427C412yes—
24#2422—21no—
25#2395—22no—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C4×#121—21no—
29#2285—21no—
30C42×D4C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1924—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1770—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1494—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2211yescharacteristic
51#1279C2211yescharacteristic
52#1279C2212yes—
53#1279C2211yescharacteristic
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56#529—24no—
57#529—24no—
58#529—42no—
59#529—42no—
60C2×#126—22no—
61C2×#125C2×C412yes—
62C2×#124C2×C412yes—
63C2×#123—22no—
64C2×C22⋊Q8C2×C412yes—
65C2×C22⋊Q8C2×C412yes—
66C2×#121—21no—
67C2×#121C2×C412yes—
68C2×C4×Q8C2312yes—
69C2×C4×D4—22no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—22no—
77C22×C42—21no—
78C22×C42C2312yes—
79C8×D4—28no—
80#432—24no—
81#432—42no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—24no—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—42no—
98C2×C4×C8C2312yes—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#400—22no—
102#399—22no—
103#398—22no—
104#397—22no—
105#397—22no—
106#397—22no—
107#394—22no—
108#394—22no—
109#391C2×C412yes—
110#391C2×C412yes—
111#390—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389C2×C412yes—
116#388—22no—
117#387—22no—
118#386—22no—
119#385—22no—
120#384—21no—
121#384—21no—
122#384C2×C411yescharacteristic
123#382—22no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380C2×C412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4D412yes—
145C43C2311yescharacteristic
146#336—22no—
147#336—21no—
148#336—22no—
149#336—21no—
150Q8⋊C8—24no—
151Q8⋊C8—24no—
152Q8⋊C8—42no—
153Q8⋊C8—42no—
154D4⋊C8—24no—
155D4⋊C8—22no—
156D4⋊C8D414yescomplemented
157C22×Q8C22×C412yes—
158C22×D4—22no—
159C22×D4C22⋊C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C22×C8—21no—
168C22×C8—28no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—44no—
175C22×C8—44no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178#126—44no—
179#126—44no—
180#125—24no—
181#125—42no—
182#124—24no—
183#124—42no—
184#123—44no—
185#123—44no—
186C22⋊Q8—24no—
187C22⋊Q8—24no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190#121—24no—
191#121—42no—
192#121—44no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×D4—24no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—41no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4C2×D412yes—
211C4×D4#4112yes—
212#117—44no—
213#117—44no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C412yes—
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2411yescharacteristic
263C4⋊C8—24no—
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—22no—
269C4⋊C8—42no—
270C4⋊C8—81no—
271C4⋊C8C2×D412yes—
272C22⋊C8—28no—
273C22⋊C8—44no—
274C22⋊C8—82no—
275#97—44no—
276C4×C8—24no—
277C4×C8—22no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—22no—
282C4×C8—44no—
283C4×C8—42no—
284C4×C8—42no—
285C4×C8C2×D412yes—
286#95—21no—
287#95—21no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—21no—
295#95C22⋊C412yes—
296#95C22⋊C412yes—
297#95C22⋊C412yes—
298C24—21no—
299C24C2×C22⋊C412yes—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—82no—
316C2×D4#10414yes—
317C2×D4C4×D412yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—81no—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—416no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—42no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—81no—
399C4⋊C4—82no—
400C4⋊C4C4×D412yes—
401C4⋊C4#13112yes—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—22no—
426C42—41no—
427C42—42no—
428C42—42no—
429C42—41no—
430C42—41no—
431C42—42no—
432C42—42no—
433C42#13112yes—
434C42#13112yes—
435C42#13512yes—
436C42#13512yes—
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C42H(5,𝔽2)11yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23C2×#10414yes—
455C23C22×C22⋊C411yescharacteristic
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460D4—22no—
461D4—22no—
462D4—28no—
463D4—41no—
464D4—41no—
465D4—88no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
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—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4#40812yes—
516C2×C4#40812yes—
517C2×C4#41612yes—
518C2×C4#41612yes—
519C2×C4C2×#10412yes—
520C2×C4C2×C4×D411yescharacteristic
521C2×C4#51811yescharacteristic
522C2×C4D4211yescharacteristic
523C2×C4D4211yescharacteristic
524C2×C4#54611yescharacteristic
525C2×C4#54611yescharacteristic
526C2×C4C2×#13111yescharacteristic
527C2×C4C2×#13111yescharacteristic
528C2×C4C2×#13111yescharacteristic
529C2×C4#56811yescharacteristic
530C2×C4C2×#13511yescharacteristic
531C2×C4C2×#13511yescharacteristic
532C2×C4#57711yescharacteristic
533C2×C4#57811yescharacteristic
534C8—24no—
535C8—42no—
536C8—44no—
537C8—42no—
538C8—82no—
539C8—81no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22D4×C22⋊C411yescharacteristic
555C22C2×#40811yescharacteristic
556C22#288011yescharacteristic
557C22C2×#41611yescharacteristic
558C22#289811yescharacteristic
559C22C22×#10411yescharacteristic
560C22#290111yescharacteristic
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—81no—
572C4—82no—
573C4—82no—
574C4#149414yescomplemented
575C4#293111yescharacteristic
576C4#293511yescharacteristic
577C4#297911yescharacteristic
578C4#298111yescharacteristic
579C4#328311yescharacteristic
580C4#328411yescharacteristic
581C4#330211yescharacteristic
582C4#330311yescharacteristic
583C2—44no—
584C2—82no—
585C2C2×#149411yescharacteristic
586C2#1748411yescharacteristic
587C2#1750811yescharacteristic
588C2#1751911yescharacteristic
589C2#2240211yescharacteristic
590C2#2241711yescharacteristic
591C2D4×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22799C2semidirect
3. C4×C8×D4C2semidirect
4. C2×#1494C2semidirect
5. #9673C2semidirect
6. #9637C2semidirect
7. #9071C2semidirect
8. #8941C2semidirect
9. #8563C2semidirect
10. #8315C2semidirect
11. C4×D4⋊C8C2semidirect
156. D4⋊C8D4semidirect
574. C4#1494semidirect

Maximal subgroups

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

subgroupindex
#349322
#227992
C4×C8×D42
C2×#14942
#96732
#96372
#90712
#89412
#85632
#83152
C4×D4⋊C82