Finite Group Databasebeta

← #192390

Subgroups — #192390

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34070C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4C2×#1517C211yesmaximal, characteristic
5#14277C211yesmaximal, characteristic, complemented
6#8873C214yesmaximal, complemented
7#8870C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#7906C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×C4×#41—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2328—21no—
15#2327—21no—
16#2326—21no—
17#2326—21no—
18#2325—21no—
19#2325—21no—
20#2325C2212yes—
21#2325C2212yes—
22#2297—21no—
23C2×#389—21no—
24#2290—22no—
25#2290—22no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1990—21no—
31#1957—21no—
32#1948—22no—
33#1874C2214yes—
34#1853C2214yes—
35#1844—21no—
36#1834—21no—
37#1802—22no—
38#1517C418yescomplemented
39C2×#327C2214yes—
40#1444—21no—
41#1444C2212yes—
42#1442—21no—
43#1442C2212yes—
44C4⋊#95—21no—
45#1294C2214yes—
46#1280C2212yes—
47#1280C2212yes—
48C22×#41—21no—
49#532—22no—
50#532—22no—
51#532C2×C414yes—
52#532Q814yes—
53C2×C4⋊Q8D411yescharacteristic
54C2×#127D411yescharacteristic
55C2×#124—21no—
56C2×#121—21no—
57C2×#121C2312yes—
58C4×#41—44no—
59C4×#41—44no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4D412yes—
68C2×#117—22no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C42—22no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79#431—22no—
80#431—22no—
81#431—22no—
82#431C2×C414yescomplemented
83C2×#106—24no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×#102—24no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#391—21no—
92#391—21no—
93#390—21no—
94#390—21no—
95#389—44no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384D411yescharacteristic
119#384D411yescharacteristic
120#382—21no—
121#382—22no—
122#382C2×C414yescomplemented
123#380—21no—
124#380C2312yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#376—21no—
130#376—21no—
131#376—21no—
132#376—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142#340—24no—
143#340—24no—
144#327C2×C4116yescomplemented
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—42no—
151C22×Q8—21no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4C2×D412yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C4⋊Q8—22no—
171C4⋊Q8C4⋊C414yescomplemented
172#127—22no—
173#127C4⋊C414yescomplemented
174#124—24no—
175#124—24no—
176#121—24no—
177#121—24no—
178#121—22no—
179#121—22no—
180#121C22×C414yes—
181#121C2×Q814yes—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—44no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—44no—
193C4×D4C4⋊C418yescomplemented
194#117—48no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—48no—
243C2×C42SD1614yes—
244C2×C42C22×C412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×Q812yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252#106—84no—
253C4⋊C8—24no—
254C4⋊C8—42no—
255#102—84no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8C22×C414yes—
260C4×C8—24no—
261C4×C8—42no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—22no—
277#95—22no—
278#95—44no—
279#95—82no—
280#95—82no—
281#95C22×C414yes—
282C24—22no—
283C24C22×D411yescharacteristic
284#41—48no—
285#41—48no—
286#41—48no—
287#41—48no—
288C2×Q8—24no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—81no—
305C2×D4—81no—
306C2×D4C2×C4⋊C414yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
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—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4#12011yescharacteristic
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12011yescharacteristic
361C22×C4C22⋊Q812yes—
362C22×C4C22⋊Q812yes—
363C22×C4C2×SD1612yes—
364C22×C4C2×SD1612yes—
365C22×C4C2×SD1612yes—
366C22×C4#13512yes—
367C22×C4AGL(1,ℤ/8)12yes—
368C22×C4AGL(1,ℤ/8)12yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C2×C8—24no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—81no—
393C4⋊C4—81no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—88no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4C2×C4⋊C414yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—28no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—24no—
412C42—24no—
413C42—44no—
414C42—44no—
415C42—42no—
416C42—42no—
417C42—44no—
418C42C2×C4⋊C412yes—
419C42C2×C4⋊C412yes—
420C42C4×D412yes—
421C42C4×D412yes—
422C42C22⋊Q812yes—
423C42C22⋊Q812yes—
424C42#12312yes—
425C42#12312yes—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C2×C4×D411yescharacteristic
438C23C2×#12011yescharacteristic
439C23C2×C22⋊Q811yescharacteristic
440C23C22×SD1611yescharacteristic
441C23C2×#13511yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×AGL(1,ℤ/8)11yescharacteristic
444Q8—48no—
445D4—44no—
446D4—44no—
447D4—44no—
448D4—44no—
449D4—44no—
450D4—48no—
451D4—82no—
452D4—84no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—416no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—88no—
499C2×C4—88no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#38512yes—
514C2×C4#38512yes—
515C2×C4#38512yes—
516C2×C4#38512yes—
517C2×C4C4×SD1614yes—
518C2×C4#44214yes—
519C2×C4#44912yes—
520C2×C4#44912yes—
521C2×C4#45012yes—
522C2×C4#45012yes—
523C2×C4#47614yes—
524C2×C4#47814yes—
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C8—44no—
530C8—82no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22C2×#38511yescharacteristic
551C22C2×C4×SD1611yescharacteristic
552C22C2×#44211yescharacteristic
553C22C2×#45011yescharacteristic
554C22C2×#44911yescharacteristic
555C22#302111yescharacteristic
556C22#302211yescharacteristic
557C22#302711yescharacteristic
558C22#302811yescharacteristic
559C22C2×#47611yescharacteristic
560C22C2×#47811yescharacteristic
561C4—22no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—42no—
568C4—42no—
569C4—44no—
570C4—84no—
571C4—82no—
572C4—81no—
573C4—82no—
574C4—81no—
575C4—162no—
576C4#186414yes—
577C4#186414yes—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—84no—
583C2#1830111yescharacteristic
584C2#1830411yescharacteristic
585C2#1830911yescharacteristic
586C2#1831211yescharacteristic
587C2#2086511yescharacteristic
588C2C2×#186411yescharacteristic
589C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21981C2semidirect
5. #14277C2semidirect
6. #8873C2semidirect
7. #8870C2semidirect
8. #8026C2semidirect
9. #7906C2semidirect
38. #1517C4semidirect
82. #431C2×C4semidirect
122. #382C2×C4semidirect
144. #327C2×C4semidirect
171. C4⋊Q8C4⋊C4semidirect
173. #127C4⋊C4semidirect
193. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340702
#219812
#218592
C2×#15172
#142772
#88732
#88702
#80262
#79062