Finite Group Databasebeta

← #222924

Subgroups — #222924

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2489C211yesmaximal, characteristic
2#22603C211yesmaximal, characteristic, complemented
3#22599C211yesmaximal, characteristic, complemented
4#9878C211yesmaximal, characteristic, complemented
5#9878C211yesmaximal, characteristic, complemented
6#9084C212yesmaximal
7#9080C212yesmaximal
8#8703C212yesmaximal, complemented
9#8557C211yesmaximal, characteristic
10C4×#323C211yesmaximal, characteristic
11#7914C212yesmaximal, complemented
12C22×#125C411yescharacteristic
13C22×C22⋊Q8C411yescharacteristic
14C2×#432C2212yes—
15C2×#431C2212yes—
16#2489—22no—
17#2489—22no—
18#2489—21no—
19#2489—21no—
20#2489—21no—
21#2489—21no—
22C2×#400C412yes—
23C2×#397C412yes—
24C2×#391C411yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#388C2212yes—
27C2×#382C2212yes—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1916—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1770—21no—
40C8×C4⋊C4—21no—
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43C2×#323C2211yescharacteristic
44C2×#323C2211yescharacteristic
45C2×#323C2212yes—
46#1280C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2212yes—
50#1280C2212yes—
51#1280C2211yescharacteristic
52#1280C2211yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C24×C4C2×C411yescharacteristic
55C2×#125—22no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—44no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—44no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2311yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#432—24no—
77#432—42no—
78#431—24no—
79#431—42no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2311yescharacteristic
90C2×#97—22no—
91C2×C4×C8—22no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#400—22no—
95#400—22no—
96#400—22no—
97#400—22no—
98#397—22no—
99#397—22no—
100#397—22no—
101#397—22no—
102#391—22no—
103#391—21no—
104#391—21no—
105#391—21no—
106#391—21no—
107#391—21no—
108#391—21no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#382—22no—
114#382—22no—
115#382—22no—
116#382—22no—
117#380—22no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138#336—21no—
139#336—21no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—21no—
145#336—21no—
146#323—22no—
147#323—22no—
148#323—42no—
149C25C22×C411yescharacteristic
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178#125—48no—
179#125—44no—
180#125—44no—
181C22⋊Q8—44no—
182C22⋊Q8—48no—
183C22⋊Q8—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—44no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C4⋊C8—44no—
239C4⋊C8—44no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8#4112yes—
253C22⋊C8#4112yes—
254#97—44no—
255C4×C8—44no—
256C4×C8—42no—
257C4×C8—42no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C22⋊C412yes—
275#95C22⋊C412yes—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24#13112yes—
287C24C23×C411yescharacteristic
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—84no—
295C2×Q8—84no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4#13112yes—
356C22×C4#13112yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4#10014yes—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—44no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—41no—
416C42—41no—
417C42—42no—
418C42—42no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23C22×C22⋊C411yescharacteristic
447C23C4×#4111yescharacteristic
448C23#52011yescharacteristic
449C23C2×#13111yescharacteristic
450C23C2×#13111yescharacteristic
451C23C2×#13111yescharacteristic
452C23#56811yescharacteristic
453Q8—84no—
454Q8—84no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—84no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—84no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#40812yes—
520C2×C4#40812yes—
521C2×C4#41012yes—
522C2×C4C2×#10014yes—
523C2×C4C2×#10012yes—
524C2×C4#54611yescharacteristic
525C2×C4#54611yescharacteristic
526C2×C4#54711yescharacteristic
527C2×C4#54711yescharacteristic
528C8—42no—
529C8—42no—
530C8—82no—
531C8—82no—
532C8—81no—
533C8—81no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#233911yescharacteristic
558C22C2×#40811yescharacteristic
559C22#288011yescharacteristic
560C22C2×#41011yescharacteristic
561C22#288411yescharacteristic
562C22C22×#10011yescharacteristic
563C22#288911yescharacteristic
564C22#296611yescharacteristic
565C22#296911yescharacteristic
566C22#299311yescharacteristic
567C22#299411yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—24no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—81no—
580C4—82no—
581C4—81no—
582C4—81no—
583C4—82no—
584C4#146914yes—
585C2—21no—
586C2—21no—
587C2—21no—
588C2—21no—
589C2—82no—
590C2C2×#146911yescharacteristic
591C2#1739111yescharacteristic
592C2#1741311yescharacteristic
593C2#1742511yescharacteristic
594C2#2247411yescharacteristic
595C2#2247811yescharacteristic
596C2#2247911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22603C2semidirect
3. #22599C2semidirect
4. #9878C2semidirect
5. #9878C2semidirect
8. #8703C2semidirect
11. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24892
#226032
#225992
#98782
#90842
#90802
#87032
#85572
C4×#3232
#79142