Finite Group Databasebeta

← #201405

Subgroups — #201405

Order 512.

1695 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic, complemented
2#35000C211yesmaximal, characteristic
3#21758C212yesmaximal, complemented
4C2×#1799C211yesmaximal, characteristic, complemented
5#9470C214yesmaximal, complemented
6#9468C214yesmaximal, complemented
7C2×#1289C212yesmaximal, complemented
8C2×#534—21no—
9C22×#121C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2972—24no—
12#2972—22no—
13#2972C2214yes—
14C2×C8×D4—24no—
15C2×#431—21no—
16C22×#99—21no—
17C22×C22⋊C8—21no—
18C22×C22⋊C8C2212yes—
19#2854—21no—
20#2847—21no—
21#2618C412yescomplemented
22#2617C412yescomplemented
23#2543C414yescomplemented
24#2534—22no—
25#2486C2212yes—
26#2416—22no—
27#2408C2212yes—
28#2308—21no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31#1893—24no—
32#1874—24no—
33#1823C2214yescomplemented
34#1799—22no—
35#1799C2214yes—
36#1786C2214yescomplemented
37C2×#342—21no—
38#1289C2218yescomplemented
39C23×D4—21no—
40C23×D4D411yescharacteristic
41C23×D4C2311yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8D414yescomplemented
46#534—44no—
47#534—44no—
48C2×#123—22no—
49C2×#123—22no—
50C2×#121—24no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121C2×C412yes—
59C2×#121C2×C412yes—
60C2×#120—22no—
61C2×#120—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4D412yes—
68C2×C4×D4D412yes—
69C2×C4×D4C2312yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42D411yescharacteristic
80C8×D4—216no—
81C8×D4—216no—
82#431—24no—
83#431—24no—
84C2×C4⋊C8—22no—
85C2×#102—28no—
86C2×#99—24no—
87C2×#99—44no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8D414yescomplemented
96C2×C22⋊C8C2314yes—
97C2×C4×C8—22no—
98#398C2×C414yescomplemented
99#397C2×C414yescomplemented
100#396—24no—
101#394—24no—
102#394C2×C414yescomplemented
103#392C2×C414yescomplemented
104#390—22no—
105#389—22no—
106#388—24no—
107#386—22no—
108#386—22no—
109#386—24no—
110#386—24no—
111#381—22no—
112#380—21no—
113#380—22no—
114#380—21no—
115#380—22no—
116#380—22no—
117#380—22no—
118#380C2×C412yes—
119#380C2314yes—
120#380C2312yes—
121#379—22no—
122C2×#95—22no—
123C2×#95C2×C412yes—
124#342—24no—
125#340—28no—
126C25—21no—
127C25C22⋊C411yescharacteristic
128C25C2×D411yescharacteristic
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—44no—
143C22×D4—82no—
144C22×D4C22⋊C412yes—
145C22×D4D814yes—
146C22×D4SD1614yes—
147C22×D4C22×C412yes—
148C22×D4C2×D412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—82no—
157C23×C4—82no—
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C22×C8—24no—
174C22×C8—28no—
175C22×C8—24no—
176C22×C8—84no—
177C22×C8—84no—
178C22×C8C2×D418yes—
179#123—44no—
180#123—44no—
181#123—48no—
182#121—24no—
183#121—24no—
184#121—24no—
185#121—24no—
186#121—416no—
187#121—41no—
188#121—42no—
189#121—41no—
190#121—41no—
191#121—41no—
192#121—42no—
193#121—84no—
194#120—48no—
195#120—44no—
196#120—44no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22×C412yes—
243C2×C22⋊C4C2×D412yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22⋊C412yes—
249C2×C42C2×D412yes—
250C4⋊C8—28no—
251#102—88no—
252#99—48no—
253C22⋊C8—28no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C4×C8—28no—
259#95—22no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—42no—
264#95C22×C414yes—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—42no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—81no—
274C24—81no—
275C24—81no—
276C24—81no—
277C24—82no—
278C24—82no—
279C24#10214yescomplemented
280C24C2×C22⋊C412yes—
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C411yescharacteristic
283C24#12112yes—
284C24#12112yes—
285C24C2×D812yes—
286C24C2×SD1612yes—
287C24#13514yes—
288C24C22×D411yescharacteristic
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—24no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—48no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—82no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4#10218yescomplemented
319C2×D4C2×D814yes—
320C2×D4C2×SD1614yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—82no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—81no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4#10214yescomplemented
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4C4×D412yes—
373C22×C4#12011yescharacteristic
374C22×C4#12012yes—
375C22×C4#12011yescharacteristic
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12311yescharacteristic
379C22×C4#12311yescharacteristic
380C22×C4#12312yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4112yes—
385C2×C8—28no—
386C2×C8—48no—
387C2×C8—48no—
388C2×C8—84no—
389C2×C8—88no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—48no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—81no—
445C23—81no—
446C23—84no—
447C23—84no—
448C23—82no—
449C23—82no—
450C23—84no—
451C23—84no—
452C23—84no—
453C23—82no—
454C23—82no—
455C23—162no—
456C23—162no—
457C23—164no—
458C23#38612yes—
459C23#38611yescharacteristic
460C23#38611yescharacteristic
461C23C2×#10214yes—
462C23C2×#10212yes—
463C23C2×#10212yes—
464C23#41612yes—
465C23#41612yes—
466C23#46514yes—
467C23#46514yes—
468C23#46614yes—
469C23#46614yes—
470C23C22×C22⋊C411yescharacteristic
471C23C2×#12112yes—
472C23C22×D811yescharacteristic
473C23C22×SD1611yescharacteristic
474C23C2×#13512yes—
475D4—216no—
476D4—28no—
477D4—28no—
478D4—48no—
479D4—82no—
480D4—84no—
481D4—84no—
482D4—88no—
483D4—82no—
484D4—168no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—164no—
531C2×C4—168no—
532C2×C4#38611yescharacteristic
533C2×C4#38611yescharacteristic
534C2×C4#38612yes—
535C2×C4C2×#10214yes—
536C2×C4C2×C4×D412yes—
537C2×C4C2×#12011yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C8—48no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—21no—
544C22—22no—
545C22—21no—
546C22—21no—
547C22—22no—
548C22—21no—
549C22—22no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—48no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—164no—
574C22—164no—
575C22#189312yes—
576C22#189312yes—
577C22#189314yes—
578C22C2×#38611yescharacteristic
579C22C22×#10211yescharacteristic
580C22C2×#41611yescharacteristic
581C22C2×#46512yes—
582C22C2×#46612yes—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—22no—
587C4—22no—
588C4—24no—
589C4—42no—
590C4—42no—
591C4—84no—
592C4—82no—
593C4—164no—
594C4#302612yes—
595C2—21no—
596C2—22no—
597C2—22no—
598C2—21no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—44no—
603C2—162no—
604C2#2096011yescharacteristic
605C2C2×#189311yescharacteristic
606C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35000C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2972C2semidirect
3. #21758C2semidirect
4. C2×#1799C2semidirect
5. #9470C2semidirect
6. #9468C2semidirect
7. C2×#1289C2semidirect
21. #2618C4semidirect
22. #2617C4semidirect
23. #2543C4semidirect
33. #1823C22semidirect
36. #1786C22semidirect
38. #1289C22semidirect
45. C23×C8D4semidirect
95. C2×C22⋊C8D4semidirect
98. #398C2×C4semidirect
99. #397C2×C4semidirect
102. #394C2×C4semidirect
103. #392C2×C4semidirect
279. C24#102semidirect
318. C2×D4#102semidirect
363. C22×C4#102semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#350002
#217582
C2×#17992
#94702
#94682
C2×#12892