Finite Group Databasebeta

← #188468

Subgroups — #188468

Order 512.

1084 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34235C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic
6#8578C212yesmaximal
7#8576C212yesmaximal
8#8574C212yesmaximal
9#8572C212yesmaximal
10#8021C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2365C411yescharacteristic, complemented
16#2365C411yescharacteristic, complemented
17#2365C411yescharacteristic, complemented
18#2365C411yescharacteristic, complemented
19#2360—21no—
20#2360—21no—
21#2356C412yescomplemented
22#2351C412yescomplemented
23#2343—22no—
24#2336—22no—
25#2320—21no—
26C4×C22⋊Q8C2212yes—
27C4×C22⋊Q8C2212yes—
28C2×#389—21no—
29#2283C2211yescharacteristic
30#2276—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1940—22no—
35#1939—21no—
36#1939—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1506—24no—
40#1503—24no—
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2212yes—
44C2×#328C2212yes—
45#1436—22no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48#1434C2212yes—
49#1294C2211yescharacteristic
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54#1294C2212yes—
55#533—22no—
56#533—22no—
57#533D414yes—
58#533Q814yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2312yes—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C412yes—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#431—22no—
82#431—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2311yescharacteristic
89C2×C22⋊C8C2312yes—
90C2×#97—21no—
91C2×C4×C8—21no—
92#391—21no—
93#391—21no—
94#391C2×C412yes—
95#391C2×C412yes—
96#389—28no—
97#389—21no—
98#389—21no—
99#389—22no—
100#389—22no—
101#389C2×C412yes—
102#389C2×C412yes—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—22no—
112#385—22no—
113#383—21no—
114#383—22no—
115#382—22no—
116#382C2×C411yescharacteristic
117#382C2×C411yescharacteristic
118#381—21no—
119#380C2311yescharacteristic
120#380C2311yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D412yes—
137C4×C4⋊C4D411yescharacteristic
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141#376—21no—
142#376—22no—
143#376—22no—
144#376—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95D412yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153#336—22no—
154#336—22no—
155#328—24no—
156#328—24no—
157#328—24no—
158#328—24no—
159C22×Q8—21no—
160C22×Q8C22×C412yes—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8C4⋊C414yescomplemented
175C4⋊Q8C4⋊C414yescomplemented
176#125—24no—
177#125—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8C2×D414yes—
183C22⋊Q8C2×Q814yes—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—82no—
190C4×Q8—82no—
191C4×Q8C4⋊C418yescomplemented
192#117—24no—
193#117—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—82no—
237C2×C42—81no—
238C2×C42—81no—
239C2×C42C22×C412yes—
240C2×C42C22×C412yes—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8—82no—
254C4⋊C8—82no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—22no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—22no—
261#97—42no—
262C4×C8—42no—
263#95—28no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—21no—
276#95—22no—
277#95—22no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—22no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—42no—
287#95—82no—
288#95—82no—
289#95C22⋊C412yes—
290#95C22⋊C412yes—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293C24C22×D411yescharacteristic
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8C2×C4⋊C414yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×Q811yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—48no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—88no—
393C4⋊C4#10414yes—
394C4⋊C4C2×C4⋊C414yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C42—28no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—48no—
407C42—44no—
408C42—44no—
409C42—42no—
410C42—42no—
411C42—82no—
412C42—82no—
413C42—84no—
414C42C2×C4⋊C412yes—
415C42C2×C4⋊C412yes—
416C42#11712yes—
417C42#11712yes—
418C42#12112yes—
419C42#12112yes—
420C42C22⋊Q812yes—
421C42C22⋊Q812yes—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23C22×C22⋊C411yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×C22⋊Q811yescharacteristic
429C23#57711yescharacteristic
430C23#57711yescharacteristic
431C23#57811yescharacteristic
432C23#57811yescharacteristic
433Q8—44no—
434Q8—44no—
435Q8—48no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4#38012yes—
511C2×C4#38012yes—
512C2×C4#38012yes—
513C2×C4#38012yes—
514C2×C4C2×#10414yes—
515C2×C4C2×#10412yes—
516C2×C4#42112yes—
517C2×C4C22×C4⋊C411yescharacteristic
518C2×C4C2×#11711yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C8—81no—
522C8—81no—
523C8—82no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22C2×#38011yescharacteristic
537C22#288311yescharacteristic
538C22#288411yescharacteristic
539C22#289811yescharacteristic
540C22#289811yescharacteristic
541C22C22×#10411yescharacteristic
542C22C2×#42111yescharacteristic
543C22#304311yescharacteristic
544C22#304411yescharacteristic
545C22#308211yescharacteristic
546C22#308211yescharacteristic
547C4—24no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—44no—
557C4—44no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—84no—
562C4—82no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4#180614yes—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2#1754811yescharacteristic
573C2#1754911yescharacteristic
574C2#1760011yescharacteristic
575C2#1760111yescharacteristic
576C2#2063411yescharacteristic
577C2#2065111yescharacteristic
578C2C2×#180611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2365C4semidirect
16. #2365C4semidirect
17. #2365C4semidirect
18. #2365C4semidirect
21. #2356C4semidirect
22. #2351C4semidirect
174. C4⋊Q8C4⋊C4semidirect
175. C4⋊Q8C4⋊C4semidirect
191. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342352
#219532
C2×#15062
C2×#15032
#137832
#85782
#85762
#85742
#85722
#80212
#80132