Finite Group Databasebeta

← #495185

Subgroups — #495185

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2133C214yesmaximal, complemented
3C2×#1801C211yesmaximal, characteristic, complemented
4C2×C4×#101C211yesmaximal, characteristic, complemented
5#13228C214yesmaximal, complemented
6#13227C212yesmaximal, complemented
7#13226C212yesmaximal, complemented
8C2×#536C2212yes—
9C2×#532C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#423—21no—
12C22×#101C2212yes—
13C22×#101C2212yes—
14C2×C4×M16—21no—
15#2854—21no—
16#2854—21no—
17#2618—21no—
18#2618—21no—
19#2548C411yescharacteristic
20#2548C2211yescharacteristic
21#2546C411yescharacteristic
22#2546C2211yescharacteristic
23#2545C412yes—
24#2545C2212yes—
25#2538—21no—
26#2538—21no—
27#2536—21no—
28#2536—21no—
29#2311C412yes—
30#2285C411yescharacteristic
31#2133—216no—
32C2×#356C2218yes—
33C2×#355C2218yes—
34#1801—21no—
35#1801—21no—
36#1801C2212yes—
37#1801C2212yes—
38C4×#101—21no—
39C4×#101—21no—
40C4×#101C2212yes—
41C4×#101C2212yes—
42C2×#343—21no—
43C2×#343—21no—
44C22×#41C2×C412yes—
45C23×Q8C2311yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48#536—28no—
49#536—28no—
50#532—41no—
51#532—41no—
52#532—41no—
53#532—41no—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×#127C2×C411yescharacteristic
58C2×#124C2×C412yes—
59C2×#124C2×C412yes—
60C2×#124C2312yes—
61C2×#123—22no—
62C2×#123—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×Q8—41no—
68C2×C4×Q8—41no—
69C2×C4×Q8—41no—
70C2×C4×Q8—41no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2311yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×#117C2×C412yes—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88#423—22no—
89#423—22no—
90#423—41no—
91#423—41no—
92C2×C4⋊C8—41no—
93C2×C4⋊C8—41no—
94C2×#101—22no—
95C2×#101—24no—
96C2×#101—28no—
97C2×#101—22no—
98C2×#101D418yescomplemented
99C2×#101C2314yes—
100C2×#101C2314yes—
101C4×M16—22no—
102C4×M16—22no—
103C4×M16—41no—
104C4×M16—41no—
105C2×#97—41no—
106C2×C4×C8—41no—
107#398—21no—
108#398—21no—
109#398—21no—
110#398—21no—
111#396—22no—
112#396—22no—
113#396—22no—
114#396—22no—
115#394—21no—
116#394—21no—
117#394—21no—
118#394—21no—
119#391C2×C412yes—
120#391C2×C412yes—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390C2×C412yes—
126#390C2×C412yes—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#388—22no—
132#388—22no—
133#388—22no—
134#388—22no—
135#386—22no—
136#386—22no—
137#384—21no—
138#382—21no—
139#380—21no—
140#380—21no—
141#380—22no—
142#380—22no—
143C4×C4⋊C4C2×C412yes—
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146#356—216no—
147#355—216no—
148#343—22no—
149#343—22no—
150#343—22no—
151#343—22no—
152C2×#41—28no—
153C2×#41—42no—
154C2×#41—42no—
155C2×#41—44no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—42no—
159C22×Q8C22×C412yes—
160C22×Q8C22×C411yescharacteristic
161C22×Q8C2×D411yescharacteristic
162C22×Q8C2×D411yescharacteristic
163C22×Q8C2×D414yes—
164C22×Q8#4114yes—
165C22×Q8C2411yescharacteristic
166C22×D4—22no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C412yes—
170C23×C4—22no—
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×M16—28no—
179C2×M16—24no—
180C2×M16—24no—
181C2×M16—44no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—44no—
185C2×M16—44no—
186C22×C8—42no—
187C22×C8—42no—
188C4⋊Q8—28no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193#127—41no—
194#127—41no—
195#124—28no—
196#124—24no—
197#124—24no—
198#124—42no—
199#124—42no—
200#123—48no—
201#123—44no—
202#123—44no—
203C22⋊Q8—48no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206#121—42no—
207#121—42no—
208#121—41no—
209#121—41no—
210#121—41no—
211#121—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4—42no—
232#117—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22×C412yes—
291C2×C42C22×C412yes—
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302#101—28no—
303#101—24no—
304#101—28no—
305#101—44no—
306#101C2×D418yes—
307#97—42no—
308#97—42no—
309C4×C8—42no—
310C4×C8—42no—
311#95—22no—
312#95—21no—
313#95—21no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318C24—22no—
319C24C2×C22⋊C411yescharacteristic
320#41—88no—
321#41—88no—
322C2×Q8—24no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8C4×D414yes—
338C2×Q8#12114yes—
339C2×Q8#12412yes—
340C2×Q8#12712yes—
341C2×Q8C22×D412yes—
342C2×Q8C2×#4112yes—
343C2×D4—24no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—82no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4#9911yescharacteristic
382C22×C4#9911yescharacteristic
383C22×C4#9911yescharacteristic
384C22×C4#9911yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C4×D414yes—
397C22×C4#12114yes—
398C22×C4#12412yes—
399C22×C4#12712yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403M16—28no—
404M16—44no—
405M16—48no—
406M16—44no—
407M16—44no—
408M16—84no—
409C2×C8—44no—
410C2×C8—48no—
411C2×C8—44no—
412C2×C8—84no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—82no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—24no—
475C42—24no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23#39012yes—
486C23C2×#9911yescharacteristic
487C23C2×#9911yescharacteristic
488C23#41012yes—
489C23C22×C22⋊C411yescharacteristic
490Q8—44no—
491Q8—44no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—84no—
497D4—84no—
498D4—82no—
499D4—84no—
500D4—84no—
501D4—84no—
502D4—82no—
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—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—88no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4#39012yes—
558C2×C4#39012yes—
559C2×C4#39012yes—
560C2×C4C2×#9911yescharacteristic
561C2×C4C2×#9911yescharacteristic
562C2×C4C2×#9911yescharacteristic
563C2×C4C2×#9911yescharacteristic
564C2×C4C2×C4×D412yes—
565C2×C4C2×#12112yes—
566C2×C4C2×#12411yescharacteristic
567C2×C4C2×#12711yescharacteristic
568C8—48no—
569C8—84no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—42no—
577C22—42no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22#196412yes—
586C22#196412yes—
587C22C2×#39011yescharacteristic
588C22C22×#9911yescharacteristic
589C22C2×#41011yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—21no—
595C4—22no—
596C4—22no—
597C4—21no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#213312yescomplemented
610C2—21no—
611C2—21no—
612C2—22no—
613C2—22no—
614C2—162no—
615C2C2×#196411yescharacteristic
616C2C2×#213311yescharacteristic
617C2#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. C2×#2133C2semidirect
3. C2×#1801C2semidirect
4. C2×C4×#101C2semidirect
5. #13228C2semidirect
6. #13227C2semidirect
7. #13226C2semidirect
98. C2×#101D4semidirect
609. C4#2133semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#21332
C2×#18012
C2×C4×#1012
#132282
#132272
#132262