Finite Group Databasebeta

← #224499

Subgroups — #224499

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2421C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3Q8×C22⋊C8C211yesmaximal, characteristic, complemented
4#9652C212yesmaximal
5#9651C212yesmaximal
6#9650C211yesmaximal, characteristic, complemented
7#9650C211yesmaximal, characteristic, complemented
8#8562C212yesmaximal, complemented
9#8557C211yesmaximal, characteristic
10#8154C212yesmaximal, complemented
11C4×#323C211yesmaximal, characteristic
12C22×C22⋊Q8C411yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#446—21no—
15C2×C8×Q8—21no—
16#2421—21no—
17#2421—21no—
18#2421—21no—
19#2421—21no—
20#2421—22no—
21#2421—22no—
22C2×#400C412yes—
23C2×#397C412yes—
24C2×#391C411yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2212yes—
27C2×#387C2212yes—
28C2×#382C411yescharacteristic
29C2×#380C411yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1985—22no—
32#1984—21no—
33#1984—22no—
34#1984—21no—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2212yes—
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2212yes—
43C2×#324C2212yes—
44C2×#324C2212yes—
45C2×#324C2212yes—
46C2×#324C2212yes—
47C2×#323C2211yescharacteristic
48C2×#323C2211yescharacteristic
49C2×#323C2212yes—
50#1280C2212yes—
51#1280C2211yescharacteristic
52#1280C2211yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—44no—
59C2×C4×Q8—44no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4×Q8D412yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79#446—44no—
80C8×Q8—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2311yescharacteristic
90C2×C22⋊C8C2311yescharacteristic
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95#400—22no—
96#400—22no—
97#400—22no—
98#400—22no—
99#397—22no—
100#397—22no—
101#397—22no—
102#397—22no—
103#391—21no—
104#391—22no—
105#391—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#382—21no—
119#382—21no—
120#382—22no—
121#380—21no—
122#380—21no—
123#380—22no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2311yescharacteristic
136#336—21no—
137#336—22no—
138#336—22no—
139#336—21no—
140#336—21no—
141#336—22no—
142#336—22no—
143#336—21no—
144#324—24no—
145#324—24no—
146#324—42no—
147#324—42no—
148#323—22no—
149#323—22no—
150#323—42no—
151C25C22×C411yescharacteristic
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22⋊C412yes—
158C22×Q8C22⋊C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C412yes—
174C23×C4C2411yescharacteristic
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22⋊Q8—44no—
182C22⋊Q8—48no—
183C22⋊Q8—44no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—44no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—44no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C4⋊C8—44no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—44no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8#4112yes—
260C22⋊C8#4112yes—
261#97—44no—
262C4×C8—44no—
263C4×C8—42no—
264C4×C8—42no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24#13112yes—
292C24C23×C411yescharacteristic
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—42no—
301C2×Q8—48no—
302C2×Q8—48no—
303C2×Q8#9914yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4#13112yes—
364C22×C4#13112yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+411yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—44no—
416C42—24no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—41no—
421C42—41no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23C22×C22⋊C411yescharacteristic
454C23C4×#4111yescharacteristic
455C23#52011yescharacteristic
456C23C2×#13111yescharacteristic
457C23C2×#13111yescharacteristic
458C23C2×#13111yescharacteristic
459C23#56811yescharacteristic
460Q8—28no—
461Q8—48no—
462Q8—44no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—84no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4#40812yes—
528C2×C4#40812yes—
529C2×C4C2×#9914yes—
530C2×C4C2×#9912yes—
531C2×C4#41312yes—
532C2×C4#54811yescharacteristic
533C2×C4#54811yescharacteristic
534C2×C4#55011yescharacteristic
535C2×C4#55011yescharacteristic
536C8—42no—
537C8—42no—
538C8—81no—
539C8—82no—
540C8—81no—
541C8—82no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22#234111yescharacteristic
566C22C2×#40811yescharacteristic
567C22#288011yescharacteristic
568C22C22×#9911yescharacteristic
569C22#288411yescharacteristic
570C22C2×#41311yescharacteristic
571C22#288911yescharacteristic
572C22#296611yescharacteristic
573C22#296911yescharacteristic
574C22#299311yescharacteristic
575C22#299411yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—24no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4#146814yes—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2C2×#146811yescharacteristic
599C2#1739111yescharacteristic
600C2#1741411yescharacteristic
601C2#1742311yescharacteristic
602C2#2253111yescharacteristic
603C2#2253411yescharacteristic
604C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. Q8×C22⋊C8C2semidirect
6. #9650C2semidirect
7. #9650C2semidirect
8. #8562C2semidirect
10. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24212
#225102
Q8×C22⋊C82
#96522
#96512
#96502
#85622
#85572
#81542
C4×#3232