Finite Group Databasebeta

← #192376

Subgroups — #192376

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#21942C211yesmaximal, characteristic
3#21765C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic
5#13966C211yesmaximal, characteristic
6#8872C214yesmaximal, complemented
7#8867C214yesmaximal, complemented
8#8019C211yesmaximal, characteristic
9#7909C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2329—22no—
14#2329—22no—
15#2329—22no—
16#2329C2214yes—
17C2×#391—21no—
18C2×#389—21no—
19C2×#387—21no—
20C2×#387—21no—
21C2×#385—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1894—22no—
28#1872C2212yes—
29#1872C2212yes—
30#1853—21no—
31#1853C2212yes—
32#1852—21no—
33#1852C2212yes—
34#1823—21no—
35#1809—22no—
36#1787—21no—
37#1508C418yescomplemented
38C2×#328C2214yes—
39#1442C2212yes—
40#1442C2212yes—
41#1434—22no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44#1294C2214yes—
45#1280C2214yes—
46C23×Q8D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C22×M16—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8C2×C414yes—
57C2×C22⋊Q8D412yes—
58C2×C22⋊Q8Q814yes—
59C2×C22⋊Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—21no—
75C2×#107—22no—
76C2×#106—22no—
77C2×#103—22no—
78C2×#103—22no—
79#407—22no—
80#407—22no—
81#407—22no—
82#407C2×C414yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87#391—24no—
88#391—24no—
89#389—24no—
90#389—24no—
91#387—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#385—24no—
96#385—24no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380D412yes—
104#380C2312yes—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4C2×C414yes—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#328C2×C4116yescomplemented
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—81no—
135C22×Q8—81no—
136C22×Q8C4⋊C414yes—
137C22×Q8C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×Q812yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—81no—
172C2×M16—81no—
173C2×M16—81no—
174C2×M16—81no—
175C22×C8—22no—
176C22×C8—22no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8C4⋊C418yes—
185C22⋊Q8C22×C414yes—
186C22⋊Q8C2×Q814yes—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C4⋊C414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—81no—
228C2×C42—81no—
229C2×C42—82no—
230C2×C42—82no—
231#107—82no—
232#106—82no—
233#103—82no—
234#103—82no—
235C22⋊C8—24no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—42no—
240C22⋊C8C22×C414yes—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—44no—
253#95—41no—
254#95—41no—
255#95—44no—
256#95—82no—
257#95—82no—
258#95C22×C414yes—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24C4×D412yes—
266C24#12011yescharacteristic
267C24#12011yescharacteristic
268C24C22⋊Q812yes—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—81no—
275C2×Q8—81no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8C2×C4⋊C414yes—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
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—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—88no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—88no—
349C22×C4C2×C4⋊C412yes—
350C22×C4C2×C4⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4C4×D412yes—
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4#12312yes—
360C22×C4#12312yes—
361C22×C4#13714yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×Q811yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368M16—84no—
369M16—84no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4C2×C4⋊C414yes—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—84no—
401C42—84no—
402C42—84no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23#38512yes—
428C23#38512yes—
429C23#45412yes—
430C23#45512yes—
431C23C2×C4×D411yescharacteristic
432C23C2×#12011yescharacteristic
433C23C2×C22⋊Q811yescharacteristic
434C23C2×#13712yes—
435C23#57512yes—
436Q8—44no—
437Q8—82no—
438Q8—84no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—48no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
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—42no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4#38512yes—
507C2×C4#38512yes—
508C2×C4C22×C4⋊C411yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4C2×#12311yescharacteristic
512C8—44no—
513C8—82no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—21no—
526C22—22no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—48no—
532C22—48no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22#186014yes—
538C22C2×#38511yescharacteristic
539C22C4×#13712yes—
540C22#300411yescharacteristic
541C22#300511yescharacteristic
542C22C2×#45511yescharacteristic
543C22C2×#45411yescharacteristic
544C22#302611yescharacteristic
545C22#302611yescharacteristic
546C22#307812yes—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—22no—
553C4—44no—
554C4—44no—
555C4—44no—
556C4—84no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—161no—
561C4—161no—
562C2—22no—
563C2—22no—
564C2—24no—
565C2—42no—
566C2#1831911yescharacteristic
567C2#1832011yescharacteristic
568C2#1832611yescharacteristic
569C2#1832911yescharacteristic
570C2C2×#186011yescharacteristic
571C2#2087411yescharacteristic
572C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
6. #8872C2semidirect
7. #8867C2semidirect
37. #1508C4semidirect
130. #328C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#219422
#217652
C2×#15082
#139662
#88722
#88672
#80192
#79092