Finite Group Databasebeta

← #164164

Subgroups — #164164

Order 512.

1113 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36410C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#20378C211yesmaximal, characteristic, complemented
4#7684C211yesmaximal, characteristic, complemented
5#7674C212yesmaximal, complemented
6#7664C211yesmaximal, characteristic, complemented
7#7658C212yesmaximal, complemented
8#7617C211yesmaximal, characteristic, complemented
9#7612C212yesmaximal, complemented
10#7566C211yesmaximal, characteristic, complemented
11#7563C212yesmaximal, complemented
12C2×#550C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×Q8—21no—
15C2×#408—21no—
16C2×#405—21no—
17#2856C2211yescharacteristic
18#2854C2212yes—
19#2582C411yescharacteristic
20#2548C411yescharacteristic
21#2476—22no—
22#2464—22no—
23#2445—21no—
24#2431—21no—
25#2414C412yes—
26#2407C412yes—
27#2367—21no—
28#2363—21no—
29#2354C411yescharacteristic
30C4×C22⋊Q8C2211yescharacteristic
31#2297C2212yes—
32#1954—21no—
33#1954—22no—
34#1797C2212yes—
35#1797C2211yescharacteristic
36#1750—22no—
37#1749C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C412yes—
54C22×#41D411yescharacteristic
55C22×M16—21no—
56C23×C8—21no—
57#550—44no—
58C2×#127—21no—
59C2×#127C2×C411yescharacteristic
60C2×#124—21no—
61C2×#123—22no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—22no—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C4×#41—44no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D412yes—
77C2×#117C2×C412yes—
78C2×#117D411yescharacteristic
79C2×#117D412yes—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C8×Q8—44no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85#408—28no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—22no—
92#405—44no—
93C2×#97—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97#398—22no—
98#394—22no—
99#394—22no—
100#394—22no—
101#392—22no—
102#392—22no—
103#391—21no—
104#391C2×C412yes—
105#391C2311yescharacteristic
106#390—21no—
107#390—21no—
108#390—22no—
109#390—21no—
110#390—21no—
111#390C2×C412yes—
112#390C2×C412yes—
113#390C2312yes—
114#389—21no—
115#389C2×C412yes—
116#389C2311yescharacteristic
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#387—21no—
122#386—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#385—21no—
127#385—22no—
128#384C2×C411yescharacteristic
129#380C2×C412yes—
130#380C2×C411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137#336—21no—
138#336—21no—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—24no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—44no—
152C2×#41—82no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22⋊C412yes—
162C22×D4C22×C412yes—
163C23×C4—22no—
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C2×M16—24no—
171C2×M16—42no—
172C2×M16—42no—
173C22×C8—24no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—42no—
181C22×C8—42no—
182#127—42no—
183#127—44no—
184#124—44no—
185#123—82no—
186#123—82no—
187C22⋊Q8—22no—
188C22⋊Q8—41no—
189#121—48no—
190#121—42no—
191#121—42no—
192#121—44no—
193#121—81no—
194#121—81no—
195#121—82no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×D4—24no—
200C4×D4—44no—
201C4×D4—42no—
202C4×D4—44no—
203C4×D4—44no—
204#117—24no—
205#117—44no—
206#117—82no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C22⋊C412yes—
235C2×C22⋊C4C22⋊C412yes—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2411yescharacteristic
248C4⋊C8—44no—
249C4⋊C8—42no—
250C22⋊C8—28no—
251C22⋊C8—28no—
252C22⋊C8—44no—
253C22⋊C8—42no—
254C22⋊C8—81no—
255C22⋊C8—82no—
256#97—44no—
257C4×C8—44no—
258C4×C8—44no—
259C4×C8—42no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—22no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24—22no—
270C24C2×C22⋊C412yes—
271C24C2×C22⋊C411yescharacteristic
272#41—48no—
273#41—48no—
274#41—88no—
275#41—88no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8#10414yes—
282C2×Q8#13112yes—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—81no—
302C2×D4—81no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—82no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—82no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4#13512yes—
345C22×C4#13512yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349M16—48no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—48no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4#9914yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—81no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C42—24no—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—41no—
407C42—42no—
408C42—41no—
409C42—42no—
410C42—42no—
411C42—82no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23#41612yes—
433C23C22×C22⋊C411yescharacteristic
434Q8—28no—
435Q8—42no—
436Q8—42no—
437D4—48no—
438D4—44no—
439D4—82no—
440D4—82no—
441D4—88no—
442D4—84no—
443D4—162no—
444D4—164no—
445D4—164no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—88no—
493C2×C4C2×#9914yes—
494C2×C4C2×#9912yes—
495C2×C4#41612yes—
496C2×C4C2×#10414yes—
497C2×C4C2×#10412yes—
498C2×C4C2×#13111yescharacteristic
499C2×C4#56811yescharacteristic
500C2×C4C2×#13511yescharacteristic
501C2×C4C2×#13511yescharacteristic
502C2×C4#57611yescharacteristic
503C2×C4#57711yescharacteristic
504C8—42no—
505C8—42no—
506C8—42no—
507C8—44no—
508C8—82no—
509C8—81no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22#288011yescharacteristic
532C22C22×#9911yescharacteristic
533C22#288811yescharacteristic
534C22C2×#41611yescharacteristic
535C22#289711yescharacteristic
536C22C22×#10411yescharacteristic
537C22#290111yescharacteristic
538C4—21no—
539C4—21no—
540C4—24no—
541C4—21no—
542C4—21no—
543C4—24no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—42no—
548C4—42no—
549C4—41no—
550C4—82no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4#150114yes—
555C2—42no—
556C2—82no—
557C2—162no—
558C2#1741111yescharacteristic
559C2#1751011yescharacteristic
560C2#1752011yescharacteristic
561C2C2×#150111yescharacteristic
562C2#1761111yescharacteristic
563C2#1765511yescharacteristic
564C2#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. #22530C2semidirect
3. #20378C2semidirect
4. #7684C2semidirect
5. #7674C2semidirect
6. #7664C2semidirect
7. #7658C2semidirect
8. #7617C2semidirect
9. #7612C2semidirect
10. #7566C2semidirect
11. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#364102
#225302
#203782
#76842
#76742
#76642
#76582
#76172
#76122
#75662
#75632