Finite Group Databasebeta

← #106163

Subgroups — #106163

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2C2×#1520C212yesmaximal
3#10846C211yesmaximal, characteristic
4#10730C212yesmaximal
5#7907C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C22×C4×Q8C411yescharacteristic, complemented
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2360—24no—
11#2360—24no—
12C2×#391C411yescharacteristic, complemented
13C2×#389—21no—
14C2×#389C411yescharacteristic, complemented
15C2×#387—21no—
16C2×#385—21no—
17C2×#380C2212yes—
18C2×C4×C4⋊C4C411yescharacteristic, complemented
19C2×C4×C22⋊C4—21no—
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×#342—21no—
22C2×#342—21no—
23C2×#340—24no—
24#1520—24no—
25#1520C418yescomplemented
26C2×#328—24no—
27#1280—21no—
28#1280—21no—
29C23×Q8C2×C411yescharacteristic
30C24×C4—21no—
31C24×C4C2311yescharacteristic
32C23×C8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—21no—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8C2×C414yes—
38C2×C22⋊Q8D414yes—
39C2×C22⋊Q8D412yes—
40C2×C22⋊Q8Q812yes—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—44no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—44no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D414yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2×C414yes—
69#391—22no—
70#391—22no—
71#389—28no—
72#389—22no—
73#389—22no—
74#387—28no—
75#385—28no—
76#380—22no—
77#380—22no—
78#380—22no—
79#380C2×C414yes—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C22⋊C4—28no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95D411yescharacteristic
91C2×#95D411yescharacteristic
92#342—42no—
93#342—42no—
94#340—48no—
95#328—216no—
96C25C2×D411yescharacteristic
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—42no—
101C22×Q8—42no—
102C22×Q8C4212yes—
103C22×Q8C22⋊C412yes—
104C22×Q8C4⋊C412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—24no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—44no—
113C23×C4—42no—
114C23×C4—41no—
115C23×C4—41no—
116C23×C4—42no—
117C23×C4—82no—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×Q811yescharacteristic
129C23×C4#4111yescharacteristic
130C23×C4#4111yescharacteristic
131C23×C4#4112yes—
132C22×C8—24no—
133C22×C8—44no—
134C22×C8—44no—
135C22×C8—42no—
136C22×C8—42no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8C22⋊C418yes—
144C22⋊Q8C4⋊C418yes—
145C4×Q8—48no—
146C4×Q8—44no—
147C4×Q8—44no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C4212yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C4⋊C412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4SD1612yes—
194C2×C22⋊C4Q1612yes—
195C2×C42—24no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—24no—
201C2×C42—44no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—82no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—42no—
208C22⋊C8—42no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—22no—
215#95—24no—
216#95—42no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24C2×C22⋊C411yescharacteristic
222C24C4×D412yes—
223C24#12012yes—
224C24#12111yescharacteristic
225C24C22⋊Q811yescharacteristic
226C24C2×SD1611yescharacteristic
227C24C2×Q1611yescharacteristic
228C24#13512yes—
229C24AGL(1,ℤ/8)11yescharacteristic
230C24#13711yescharacteristic
231C24#13712yes—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8#9514yes—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—84no—
301C22×C4—88no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4#10312yes—
311C22×C4#10312yes—
312C22×C4C2×C4211yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C4⋊C411yescharacteristic
317C22×C4C2×C4⋊C412yes—
318C22×C4#11711yescharacteristic
319C22×C4#11711yescharacteristic
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12012yes—
324C22×C4#12112yes—
325C22×C4#12111yescharacteristic
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q811yescharacteristic
328C22×C4#12312yes—
329C22×C4#12312yes—
330C2×C8—48no—
331C2×C8—44no—
332C2×C8—82no—
333C2×C8—82no—
334C2×C8—84no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—48no—
344C4⋊C4—42no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C4⋊C4#9514yes—
350C4⋊C4#9914yes—
351C22⋊C4—28no—
352C22⋊C4—28no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—42no—
362C42—42no—
363C42—48no—
364C42—42no—
365C42—42no—
366C42—84no—
367C23—21no—
368C23—22no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—22no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23C4×C22⋊C411yescharacteristic
394C23#37911yescharacteristic
395C23#38011yescharacteristic
396C23#38512yes—
397C23#38612yes—
398C23C2×#10311yescharacteristic
399C23#41611yescharacteristic
400C23#41711yescharacteristic
401C23#41911yescharacteristic
402C23C4×SD1612yes—
403C23C4×Q1612yes—
404C23#44012yes—
405C23#44112yes—
406C23#44812yes—
407C23#44912yes—
408C23#45112yes—
409C23#45212yes—
410C23#46011yescharacteristic
411C23#46211yescharacteristic
412C23#46412yes—
413C23#47511yescharacteristic
414C23#47711yescharacteristic
415C23#47912yes—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419Q8—44no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—88no—
474C2×C4—88no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—84no—
484C2×C4—162no—
485C2×C4—162no—
486C2×C4—162no—
487C2×C4—162no—
488C2×C4C2×#9511yescharacteristic
489C2×C4C4×C22⋊C411yescharacteristic
490C2×C4#38011yescharacteristic
491C2×C4#38111yescharacteristic
492C2×C4#38512yes—
493C2×C4#38612yes—
494C2×C4C2×#9912yes—
495C2×C4#41012yes—
496C8—84no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—22no—
509C22—21no—
510C22—21no—
511C22—22no—
512C22—22no—
513C22—21no—
514C22—21no—
515C22—22no—
516C22—22no—
517C22—21no—
518C22—21no—
519C22—22no—
520C22—22no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22C22⋊#9511yescharacteristic
532C22C4×#10311yescharacteristic
533C22#176311yescharacteristic
534C22#178111yescharacteristic
535C22#178211yescharacteristic
536C22#178711yescharacteristic
537C22#178811yescharacteristic
538C22#179511yescharacteristic
539C22#179611yescharacteristic
540C22#180311yescharacteristic
541C22#180511yescharacteristic
542C22#186312yes—
543C22#186512yes—
544C22#187812yes—
545C22#188012yes—
546C4—22no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—44no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—81no—
556C4—82no—
557C4—81no—
558C4—164no—
559C4C4×#9912yes—
560C4#179912yes—
561C2—22no—
562C2—21no—
563C2—21no—
564C2—22no—
565C2—22no—
566C2—22no—
567C2—21no—
568C2—21no—
569C2#837911yescharacteristic
570C2#838011yescharacteristic
571C2#838111yescharacteristic
572C2#838211yescharacteristic
573C2#1076111yescharacteristic
574C2#1077511yescharacteristic
575C2#1078911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C22×C4×Q8C4semidirect
12. C2×#391C4semidirect
14. C2×#389C4semidirect
18. C2×C4×C4⋊C4C4semidirect
25. #1520C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#15202
#108462
#107302
#79072