Finite Group Databasebeta

← #192393

Subgroups — #192393

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34358C211yesmaximal, characteristic
2#21941C211yesmaximal, characteristic, complemented
3#21764C211yesmaximal, characteristic
4C2×#1519C211yesmaximal, characteristic
5#14675C211yesmaximal, characteristic, complemented
6#8873C212yesmaximal, complemented
7#8870C212yesmaximal, complemented
8#8869C212yesmaximal, complemented
9#8866C212yesmaximal, complemented
10#8016C211yesmaximal, characteristic, complemented
11#7913C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2377—21no—
16#2377—21no—
17#2364—22no—
18#2342—22no—
19D4×C22⋊C4—22no—
20#2333—22no—
21#2325C2212yes—
22#2325C2212yes—
23#2313—21no—
24#2308—21no—
25#2280C2211yescharacteristic
26C2×#381C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#1895—22no—
30#1874C2212yes—
31#1874C2212yes—
32#1854—22no—
33#1853C2212yescomplemented
34#1852C2212yescomplemented
35#1824—21no—
36#1808—21no—
37#1807—21no—
38#1786—21no—
39#1519C418yescomplemented
40C2×#327C2212yes—
41C2×#327C2212yes—
42#1444C2212yescomplemented
43#1442C2212yescomplemented
44#1441—22no—
45#1439—22no—
46#1294C2212yescomplemented
47#1294C2212yescomplemented
48#1280C2211yescharacteristic, complemented
49#1280C2211yescharacteristic, complemented
50#1280C2211yescharacteristic, complemented
51#1280C2211yescharacteristic, complemented
52C23×D4—21no—
53C23×D4D411yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121C2×C414yes—
62C2×#121D412yes—
63C2×#121Q814yes—
64C2×#121C2312yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×#117D412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C2×#107—22no—
78C2×#106—22no—
79C2×#102—22no—
80C2×#102—22no—
81#407—22no—
82#407—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8C2×C414yescomplemented
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2311yescharacteristic
90C2×C22⋊C8C2311yescharacteristic
91#390—22no—
92#390—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#381—21no—
112#381—21no—
113#381—21no—
114#381—21no—
115#381—21no—
116#380—22no—
117#380—22no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380C2×C414yescomplemented
123#380D411yescharacteristic
124#380D411yescharacteristic
125#380C2312yes—
126#379—22no—
127#379—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95D412yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#327C2×C418yescomplemented
146#327C2×C418yescomplemented
147C25—21no—
148C25C2×D411yescharacteristic
149C22×D4—22no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—82no—
156C22×D4—82no—
157C22×D4—82no—
158C22×D4C4⋊C414yescomplemented
159C22×D4C2×D412yes—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—82no—
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×Q812yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C2×M16—42no—
179C2×M16—82no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—24no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—82no—
186#121—24no—
187#121—24no—
188#121—24no—
189#121—24no—
190#121—24no—
191#121—22no—
192#121—22no—
193#121C4⋊C418yescomplemented
194#121C22×C414yes—
195#121C2×Q814yes—
196C4×D4—82no—
197C4×D4—82no—
198C4×D4—82no—
199C4×D4—82no—
200#117—82no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C4⋊C414yescomplemented
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
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—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—84no—
237C2×C22⋊C4—82no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—81no—
240C2×C22⋊C4—81no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244#107—82no—
245#106—82no—
246#102—82no—
247#102—82no—
248C22⋊C8—24no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—42no—
256C22⋊C8C22×C412yes—
257C22⋊C8C22×C412yes—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—41no—
273#95—41no—
274#95—82no—
275#95—82no—
276#95—82no—
277#95C22×C414yes—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—41no—
283C24—42no—
284C24—41no—
285C24—44no—
286C24—82no—
287C24—81no—
288C24—81no—
289C24C4×D412yes—
290C24#12011yescharacteristic
291C24#12011yescharacteristic
292C24C22⋊Q812yes—
293C24C22×D411yescharacteristic
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—28no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—82no—
306C2×D4—84no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—84no—
310C2×D4—88no—
311C2×D4—84no—
312C2×D4—84no—
313C2×D4C2×C4⋊C414yes—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
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—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C2×C4⋊C412yes—
360C22×C4C2×C4⋊C412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4C22⋊Q812yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377M16—84no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—84no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—88no—
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—81no—
408C22⋊C4—81no—
409C22⋊C4—82no—
410C22⋊C4—84no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—82no—
420C42—82no—
421C42—82no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—48no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—84no—
453C23—84no—
454C23—164no—
455C23—164no—
456C23#38512yes—
457C23#38512yes—
458C23C2×C4×D411yescharacteristic
459C23C2×#12011yescharacteristic
460C23C2×C22⋊Q811yescharacteristic
461C23#57612yes—
462C23#57712yes—
463D4—44no—
464D4—44no—
465D4—44no—
466D4—44no—
467D4—44no—
468D4—82no—
469D4—84no—
470D4—88no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—164no—
524C2×C4—164no—
525C2×C4#38512yes—
526C2×C4#38512yes—
527C2×C4C22×C4⋊C411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C2×C4C2×#12311yescharacteristic
531C8—42no—
532C8—42no—
533C8—81no—
534C8—81no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—82no—
557C22—84no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—168no—
564C22C2×#38511yescharacteristic
565C22#294811yescharacteristic
566C22#294911yescharacteristic
567C22#301212yes—
568C22#302111yescharacteristic
569C22#302211yescharacteristic
570C22#302612yes—
571C22#308112yes—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—44no—
579C4—84no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—82no—
584C4—162no—
585C2—21no—
586C2—22no—
587C2—21no—
588C2—42no—
589C2—42no—
590C2—84no—
591C2#1844612yes—
592C2#1844712yes—
593C2#2086511yescharacteristic
594C2#2087911yescharacteristic
595C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21941C2semidirect
5. #14675C2semidirect
6. #8873C2semidirect
7. #8870C2semidirect
8. #8869C2semidirect
9. #8866C2semidirect
10. #8016C2semidirect
11. #7913C2semidirect
33. #1853C22semidirect
34. #1852C22semidirect
39. #1519C4semidirect
42. #1444C22semidirect
43. #1442C22semidirect
46. #1294C22semidirect
47. #1294C22semidirect
48. #1280C22semidirect
49. #1280C22semidirect
50. #1280C22semidirect
51. #1280C22semidirect
86. C2×C22⋊C8C2×C4semidirect
122. #380C2×C4semidirect
145. #327C2×C4semidirect
146. #327C2×C4semidirect
158. C22×D4C4⋊C4semidirect
193. #121C4⋊C4semidirect
213. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#343582
#219412
#217642
C2×#15192
#146752
#88732
#88702
#88692
#88662
#80162
#79132