Finite Group Databasebeta

← #164179

Subgroups — #164179

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36411C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#20378C211yesmaximal, characteristic, complemented
4#7688C211yesmaximal, characteristic, complemented
5#7674C212yesmaximal, complemented
6#7665C211yesmaximal, characteristic, complemented
7#7658C212yesmaximal, complemented
8#7616C211yesmaximal, characteristic, complemented
9#7613C212yesmaximal, complemented
10#7565C211yesmaximal, characteristic, complemented
11#7564C212yesmaximal, complemented
12C2×#555C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×Q8—21no—
15C2×#408—21no—
16C2×#405—21no—
17#2857C2211yescharacteristic
18#2854C2212yes—
19#2567C411yescharacteristic
20#2538C411yescharacteristic
21#2485C412yes—
22#2474—22no—
23#2461—22no—
24#2422—21no—
25#2414C412yes—
26#2407—21no—
27#2362—21no—
28#2354—21no—
29#2354C411yescharacteristic
30C4×#121C2211yescharacteristic
31#2296C2212yes—
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#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#555—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#124—22no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×#121—21no—
63C2×#121C2×C411yescharacteristic
64C2×#121C2×C412yes—
65C4×#41—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C412yes—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D412yes—
74C2×#117—21no—
75C2×#117C2×C412yes—
76C2×#117C2×C412yes—
77C2×#117D411yescharacteristic
78C2×#117D412yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C8×Q8—44no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84#408—28no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—22no—
91#405—44no—
92C2×#97—22no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96#397—22no—
97#396—22no—
98#395—22no—
99#394—22no—
100#394—22no—
101#392—22no—
102#391—21no—
103#391—21no—
104#390—21no—
105#390—22no—
106#390C2×C412yes—
107#390C2311yescharacteristic
108#389—21no—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2312yes—
113#389C2311yescharacteristic
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#388—21no—
120#387—22no—
121#386—22no—
122#386—21no—
123#385—22no—
124#385—22no—
125#384—21no—
126#384C2×C411yescharacteristic
127#380—21no—
128#380—21no—
129#380C2×C412yes—
130#380C2×C412yes—
131#380C2×C411yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C22⋊C4—21no—
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—42no—
149C22×Q8—21no—
150C22×Q8C22⋊C411yescharacteristic
151C22×D4—21no—
152C22×D4—22no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22⋊C412yes—
155C22×D4C22×C411yescharacteristic
156C23×C4—21no—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—24no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—42no—
174C22×C8—42no—
175C4⋊Q8—42no—
176#124—48no—
177#123—82no—
178#123—82no—
179C22⋊Q8—44no—
180C22⋊Q8—81no—
181C22⋊Q8—81no—
182#121—22no—
183#121—41no—
184#121—44no—
185#121—44no—
186C4×Q8—24no—
187C4×Q8—42no—
188C4×D4—24no—
189C4×D4—44no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—44no—
193C4×D4—42no—
194C4×D4—42no—
195#117—24no—
196#117—24no—
197#117—44no—
198#117—42no—
199#117—44no—
200#117—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—44no—
237C2×C42—42no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2411yescharacteristic
245C4⋊C8—44no—
246C4⋊C8—42no—
247C22⋊C8—28no—
248C22⋊C8—28no—
249C22⋊C8—44no—
250C22⋊C8—42no—
251C22⋊C8—81no—
252C22⋊C8—82no—
253#97—44no—
254C4×C8—44no—
255C4×C8—44no—
256C4×C8—42no—
257#95—21no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—22no—
266C24—21no—
267C24C2×C22⋊C412yes—
268C24C2×C22⋊C411yescharacteristic
269#41—48no—
270#41—48no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—42no—
274C2×Q8—81no—
275C2×Q8—81no—
276C2×Q8#10414yes—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—81no—
291C2×D4—82no—
292C2×D4—82no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4#13512yes—
330C22×C4#13512yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334M16—48no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—48no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4#9914yes—
374C4⋊C4#13112yes—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—24no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—41no—
397C42—42no—
398C42—41no—
399C42—44no—
400C42—44no—
401C42—42no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23#41612yes—
423C23C22×C22⋊C411yescharacteristic
424Q8—28no—
425Q8—162no—
426D4—48no—
427D4—44no—
428D4—81no—
429D4—81no—
430D4—84no—
431D4—164no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—84no—
479C2×C4C2×#9914yes—
480C2×C4C2×#9912yes—
481C2×C4#41612yes—
482C2×C4C2×#10414yes—
483C2×C4C2×#10412yes—
484C2×C4C2×#13111yescharacteristic
485C2×C4#56811yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4#57611yescharacteristic
489C2×C4#57711yescharacteristic
490C8—42no—
491C8—42no—
492C8—42no—
493C8—44no—
494C8—82no—
495C8—81no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#288011yescharacteristic
518C22C22×#9911yescharacteristic
519C22#288911yescharacteristic
520C22C2×#41611yescharacteristic
521C22#289711yescharacteristic
522C22C22×#10411yescharacteristic
523C22#290111yescharacteristic
524C4—21no—
525C4—21no—
526C4—24no—
527C4—21no—
528C4—21no—
529C4—24no—
530C4—21no—
531C4—21no—
532C4—41no—
533C4—42no—
534C4—42no—
535C4—41no—
536C4—81no—
537C4—82no—
538C4—81no—
539C4—82no—
540C4#150114yes—
541C2—42no—
542C2—82no—
543C2—161no—
544C2#1741611yescharacteristic
545C2#1750811yescharacteristic
546C2#1752011yescharacteristic
547C2C2×#150111yescharacteristic
548C2#1761111yescharacteristic
549C2#1765411yescharacteristic
550C2#1772711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36411C2
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. #7688C2semidirect
5. #7674C2semidirect
6. #7665C2semidirect
7. #7658C2semidirect
8. #7616C2semidirect
9. #7613C2semidirect
10. #7565C2semidirect
11. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#364112
#225302
#203782
#76882
#76742
#76652
#76582
#76162
#76132
#75652
#75642