Finite Group Databasebeta

← #189828

Subgroups — #189828

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21820C211yesmaximal, characteristic
3#13786C211yesmaximal, characteristic
4#8717C212yesmaximal
5#8716C212yesmaximal
6#8714C212yesmaximal
7#8714C212yesmaximal
8#8022C211yesmaximal, characteristic
9#8021C211yesmaximal, characteristic
10#8014C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2355C411yescharacteristic, complemented
16#2355C411yescharacteristic, complemented
17#2355C412yescomplemented
18#2355C412yescomplemented
19Q8×C4⋊C4C411yescharacteristic, complemented
20Q8×C4⋊C4C411yescharacteristic, complemented
21#2344—21no—
22#2344—22no—
23Q8×C22⋊C4—21no—
24#2320—21no—
25#2320—22no—
26C2×#391—21no—
27#2285C2212yes—
28#2285C2211yescharacteristic
29#2283C2212yes—
30#2283C2211yescharacteristic
31#2276—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1985—22no—
35#1984—21no—
36#1984—21no—
37#1817C2212yes—
38#1817C2212yes—
39#1436—22no—
40C4⋊#95—21no—
41#1434C2212yes—
42#1434C2212yes—
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2211yescharacteristic
53C23×Q8D411yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4×Q8D411yescharacteristic
66C2×#117—22no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#431—22no—
78#431—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8C2311yescharacteristic
82C2×C22⋊C8C2311yescharacteristic
83C2×C22⋊C8C2312yes—
84C2×#97—21no—
85C2×C4×C8—21no—
86#391—28no—
87#391—21no—
88#391—21no—
89#391—22no—
90#391—22no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#384C2×C412yes—
102#384C2×C412yes—
103#384C2×C411yescharacteristic
104#384C2×C411yescharacteristic
105#383—22no—
106#383—21no—
107#382—22no—
108#382—21no—
109#382C2×C412yes—
110#382C2×C412yes—
111#382C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#381—22no—
114#381—21no—
115#380—22no—
116#380—21no—
117#380C2311yescharacteristic
118#380C2312yes—
119#380C2311yescharacteristic
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
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×C412yes—
134C4×C4⋊C4C2×C412yes—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137#376—22no—
138#376—21no—
139#376—22no—
140#376—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—42no—
157C22×Q8C22⋊C411yescharacteristic
158C22×Q8C22⋊C412yes—
159C22×Q8C22⋊C411yescharacteristic
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—24no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—24no—
183C4×Q8—44no—
184C4×Q8—44no—
185#117—81no—
186#117—82no—
187#117—81no—
188#117—82no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—81no—
236C2×C42—81no—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257#97—42no—
258C4×C8—42no—
259#95—28no—
260#95—28no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—22no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95—42no—
286#95—82no—
287#95—82no—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8#10414yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
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—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4#12312yes—
354C22×C4#12312yes—
355C22×C4#12312yes—
356C22×C4#12312yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—42no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—48no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—81no—
391C4⋊C4—81no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C22⋊C4—42no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—48no—
409C42—44no—
410C42—44no—
411C42—42no—
412C42—42no—
413C42—82no—
414C42—82no—
415C42—82no—
416C42—84no—
417C42#11712yes—
418C42#11712yes—
419C42#11712yes—
420C42#11712yes—
421C42#12312yes—
422C42#12312yes—
423C42#12312yes—
424C42#12312yes—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23C22×C22⋊C411yescharacteristic
430C23C2×#12311yescharacteristic
431C23C2×#12311yescharacteristic
432C23#57611yescharacteristic
433C23#57711yescharacteristic
434C23#57711yescharacteristic
435C23#57811yescharacteristic
436Q8—28no—
437Q8—48no—
438Q8—44no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
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—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4#38112yes—
516C2×C4#38112yes—
517C2×C4#38112yes—
518C2×C4#38112yes—
519C2×C4C2×#10414yes—
520C2×C4C2×#10412yes—
521C2×C4#42112yes—
522C2×C4C2×#11711yescharacteristic
523C2×C4C2×#11711yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C8—81no—
527C8—81no—
528C8—82no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22C2×#38111yescharacteristic
542C22#288411yescharacteristic
543C22#288411yescharacteristic
544C22#289711yescharacteristic
545C22#289811yescharacteristic
546C22C22×#10411yescharacteristic
547C22C2×#42111yescharacteristic
548C22#309611yescharacteristic
549C22#309711yescharacteristic
550C22#309811yescharacteristic
551C22#309911yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—24no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—84no—
564C4—82no—
565C4—82no—
566C4—81no—
567C4—82no—
568C4—81no—
569C4—81no—
570C4—82no—
571C4—81no—
572C4#182614yes—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2#1754911yescharacteristic
578C2#1755011yescharacteristic
579C2#1760111yescharacteristic
580C2#1760211yescharacteristic
581C2#2069111yescharacteristic
582C2#2070311yescharacteristic
583C2C2×#182611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2355C4semidirect
16. #2355C4semidirect
17. #2355C4semidirect
18. #2355C4semidirect
19. Q8×C4⋊C4C4semidirect
20. Q8×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#341962
#218202
#137862
#87172
#87162
#87142
#80222
#80212
#80142
#80132