Finite Group Databasebeta

← #197510

Subgroups — #197510

Order 512.

1577 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×#1520C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9200C214yesmaximal, complemented
6#9199C214yesmaximal, complemented
7C2×#1289C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C2×#518—21no—
10C22×C4×D4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12#2855—21no—
13#2848—21no—
14#2627C2212yes—
15#2626C2212yes—
16#2579—22no—
17#2564C414yescomplemented
18#2487C412yescomplemented
19#2417—22no—
20#2409C412yescomplemented
21#2349—24no—
22D4×C4⋊C4—24no—
23#2316C2212yes—
24#2315—21no—
25#2315—21no—
26#2315C2212yes—
27#2308—21no—
28#2285—21no—
29C2×#380—21no—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32#1520—24no—
33#1520C2218yes—
34C2×#328—24no—
35#1459—22no—
36#1459—22no—
37C2×#324C2212yes—
38C2×#323C2212yes—
39#1289C2218yes—
40C23×Q8C2×C411yescharacteristic, complemented
41C23×D4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yescomplemented
53C2×C22⋊Q8C2312yes—
54#518—44no—
55#518—44no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—28no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C412yes—
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×#117—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic, complemented
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4C2311yescharacteristic
78C22×C42D411yescharacteristic
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2314yes—
85#400C2×C414yescomplemented
86#399C2×C414yescomplemented
87#398—24no—
88#396—24no—
89#395C2×C414yescomplemented
90#393C2×C414yescomplemented
91#391—22no—
92#389—22no—
93#389—28no—
94#387—24no—
95#386—24no—
96#386—22no—
97#386—22no—
98#385—28no—
99#385—24no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—28no—
105#384—22no—
106#382—22no—
107#381—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—21no—
114#380—21no—
115#380—22no—
116#380—22no—
117#380—24no—
118#380C2×C412yescomplemented
119#380D414yes—
120#380C2312yes—
121#380C2314yes—
122#379—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C2×#95—22no—
126C2×#95C2×C412yescomplemented
127C2×#95D414yes—
128#328—216no—
129#324—24no—
130#323—24no—
131C25C2×D411yescharacteristic
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—82no—
137C22×Q8C22×C412yes—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C22⋊C412yes—
143C22×D4C22⋊C412yes—
144C22×D4SD1614yes—
145C22×D4Q1614yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4C22⋊C412yes—
158C23×C4C22⋊C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—84no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—84no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—432no—
196#117—44no—
197#117—44no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—84no—
217C2×C4⋊C4—84no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22×C412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—24no—
252C2×C42C22⋊C412yes—
253C2×C42C22⋊C412yes—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95C22×C414yes—
267#95C2×D418yes—
268C24—21no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—81no—
273C24—81no—
274C24—81no—
275C24—81no—
276C24C2×C22⋊C412yes—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C411yescharacteristic
279C24#12012yes—
280C24#12312yes—
281C24C2×SD1612yes—
282C24C2×SD1612yes—
283C24C2×Q1612yes—
284C24C2×Q1612yes—
285C24AGL(1,ℤ/8)12yes—
286C24#13712yes—
287C24C22×D411yescharacteristic
288C2×Q8—24no—
289C2×Q8—44no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—84no—
293C2×Q8—82no—
294C2×Q8—84no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—28no—
303C2×D4#10318yes—
304C2×D4#10318yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—28no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—44no—
332C22×C4—48no—
333C22×C4—82no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—81no—
339C22×C4—82no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—82no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C4×D412yes—
358C22×C4C4×D412yes—
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4#12011yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12311yescharacteristic
368C22×C4#12311yescharacteristic
369C22×C4C2×SD1612yes—
370C22×C4C2×Q1612yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4112yes—
375C2×C8—44no—
376C2×C8—84no—
377C2×C8—84no—
378C2×C8—84no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—48no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—48no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—48no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—41no—
424C23—41no—
425C23—42no—
426C23—41no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23—82no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23#38612yes—
448C23#38612yes—
449C23C2×#10314yes—
450C23C2×#10312yes—
451C23C2×#10312yes—
452C23#41712yes—
453C23#41712yes—
454C23#45014yes—
455C23#45114yes—
456C23#48314yes—
457C23#48414yes—
458C23C22×C22⋊C411yescharacteristic
459C23C2×#12011yescharacteristic
460C23C2×#12311yescharacteristic
461C23C22×SD1611yescharacteristic
462C23C22×Q1611yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23C2×#13711yescharacteristic
465Q8—84no—
466Q8—82no—
467Q8—82no—
468D4—216no—
469D4—216no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—164no—
519C2×C4—168no—
520C2×C4—168no—
521C2×C4#38611yescharacteristic
522C2×C4#38611yescharacteristic
523C2×C4#38611yescharacteristic
524C2×C4#38611yescharacteristic
525C2×C4C2×#10314yes—
526C2×C4C2×C4×D412yes—
527C2×C4C2×#12112yes—
528C8—84no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—21no—
533C22—22no—
534C22—21no—
535C22—21no—
536C22—22no—
537C22—21no—
538C22—22no—
539C22—24no—
540C22—42no—
541C22—44no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22#187214yes—
561C22#187214yes—
562C22C2×#38611yescharacteristic
563C22C22×#10311yescharacteristic
564C22C2×#41711yescharacteristic
565C22C2×#45011yescharacteristic
566C22C2×#45111yescharacteristic
567C22C2×#48311yescharacteristic
568C22C2×#48411yescharacteristic
569C4—22no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—84no—
578C4—88no—
579C4—82no—
580C4—82no—
581C4#191312yes—
582C2—21no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—21no—
587C2—21no—
588C2—24no—
589C2—24no—
590C2—44no—
591C2C2×#187211yescharacteristic
592C2#2101211yescharacteristic
593C2C2×#191311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
3. C2×#1520C2semidirect
5. #9200C2semidirect
6. #9199C2semidirect
7. C2×#1289C2semidirect
17. #2564C4semidirect
18. #2487C4semidirect
20. #2409C4semidirect
40. C23×Q8C2×C4semidirect
52. C2×C22⋊Q8C2×C4semidirect
70. C22×C4⋊C4C2×C4semidirect
85. #400C2×C4semidirect
86. #399C2×C4semidirect
89. #395C2×C4semidirect
90. #393C2×C4semidirect
118. #380C2×C4semidirect
126. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
#341812
C2×#15202
C2×#14592
#92002
#91992
C2×#12892