Finite Group Databasebeta

← #164747

Subgroups — #164747

Order 512.

1075 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36410C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic, complemented
3#20684C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7684C211yesmaximal, characteristic, complemented
6#7664C211yesmaximal, characteristic, complemented
7#7654C212yesmaximal, complemented
8#7645C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7578C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×D4—21no—
15C2×#408—21no—
16#2856C2211yescharacteristic
17#2854C2212yes—
18#2582C411yescharacteristic
19#2548C411yescharacteristic
20#2476—22no—
21#2464—22no—
22#2445—21no—
23#2431—21no—
24#2414C412yes—
25#2407C412yes—
26#2367—21no—
27#2363—21no—
28#2354C411yescharacteristic
29C4×C22⋊Q8C2211yescharacteristic
30#2297C2212yes—
31#1984—21no—
32#1869—22no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1728C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C412yes—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57C23×C8—22no—
58#550—44no—
59C2×#127—21no—
60C2×#127C2×C411yescharacteristic
61C2×#124—21no—
62C2×#123—22no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—22no—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C4×#41—44no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C412yes—
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D412yes—
78C2×#117C2×C412yes—
79C2×#117D412yes—
80C2×#117D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2312yes—
83C8×D4—44no—
84C2×C4⋊C8—21no—
85#408—44no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92#398—22no—
93#394—22no—
94#394—22no—
95#394—22no—
96#392—22no—
97#392—22no—
98#391—21no—
99#391C2×C412yes—
100#391C2311yescharacteristic
101#390—21no—
102#390—21no—
103#390—22no—
104#390—21no—
105#390—21no—
106#390C2×C412yes—
107#390C2×C412yes—
108#390C2312yes—
109#389—21no—
110#389C2×C412yes—
111#389C2311yescharacteristic
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#387—21no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#385—21no—
122#385—22no—
123#384C2×C411yescharacteristic
124#380C2×C412yes—
125#380C2×C411yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—24no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—44no—
151C2×#41—82no—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22×C411yescharacteristic
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22⋊C412yes—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22×C412yes—
162C23×C4—22no—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—42no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—44no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—44no—
182C22×C8—42no—
183C22×C8—82no—
184#127—42no—
185#127—44no—
186#124—44no—
187#123—82no—
188#123—82no—
189C22⋊Q8—22no—
190C22⋊Q8—41no—
191#121—48no—
192#121—44no—
193#121—42no—
194#121—42no—
195#121—81no—
196#121—81no—
197#121—82no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—44no—
206#117—44no—
207#117—42no—
208#117—82no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
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—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C22⋊C412yes—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2411yescharacteristic
250C4⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—81no—
256C4×C8—42no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—21no—
266C24—22no—
267C24C2×C22⋊C412yes—
268C24C2×C22⋊C411yescharacteristic
269#41—48no—
270#41—48no—
271#41—88no—
272#41—88no—
273C2×Q8—21no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8#13112yes—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4#9914yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—82no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4#13512yes—
342C22×C4#13512yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346M16—84no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—88no—
366C2×C8—84no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—81no—
382C4⋊C4—82no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—81no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—22no—
401C42—22no—
402C42—41no—
403C42—42no—
404C42—41no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—82no—
410C23—21no—
411C23—22no—
412C23—21no—
413C23—22no—
414C23—21no—
415C23—22no—
416C23—22no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—21no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23C2×#9914yes—
430C23#41612yes—
431C23C22×C22⋊C411yescharacteristic
432Q8—42no—
433Q8—42no—
434Q8—44no—
435D4—28no—
436D4—48no—
437D4—82no—
438D4—82no—
439D4—88no—
440D4—84no—
441D4—162no—
442D4—164no—
443D4—164no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—88no—
493C2×C4C2×#9912yes—
494C2×C4#41612yes—
495C2×C4#42112yes—
496C2×C4C2×#13111yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4C2×#13511yescharacteristic
499C2×C4C2×#13511yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57711yescharacteristic
502C8—42no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—82no—
507C8—82no—
508C22—24no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—41no—
515C22—42no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—42no—
521C22—41no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22#288011yescharacteristic
530C22C22×#9911yescharacteristic
531C22#288811yescharacteristic
532C22C2×#41611yescharacteristic
533C22#289711yescharacteristic
534C22C2×#42111yescharacteristic
535C22#290111yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—41no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—81no—
549C4—82no—
550C4—81no—
551C4—81no—
552C4#151614yes—
553C2—44no—
554C2—42no—
555C2—162no—
556C2#1741111yescharacteristic
557C2#1751011yescharacteristic
558C2#1752611yescharacteristic
559C2C2×#151611yescharacteristic
560C2#1760311yescharacteristic
561C2#1768411yescharacteristic
562C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36410C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22469C2semidirect
3. #20684C2semidirect
4. #7718C2semidirect
5. #7684C2semidirect
6. #7664C2semidirect
7. #7654C2semidirect
8. #7645C2semidirect
9. #7617C2semidirect
10. #7578C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364102
#224692
#206842
#77182
#76842
#76642
#76542
#76452
#76172
#75782
#75662