Finite Group Databasebeta

← #164754

Subgroups — #164754

Order 512.

1007 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36456C211yesmaximal, characteristic
2#22469C211yesmaximal, characteristic
3#20630C211yesmaximal, characteristic
4#7725C212yesmaximal, complemented
5#7688C211yesmaximal, characteristic, complemented
6#7668C212yesmaximal, complemented
7#7665C211yesmaximal, characteristic, complemented
8#7648C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7589C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×Q82C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×D4—21no—
15C2×#424—21no—
16C2×#408—21no—
17#2857C2211yescharacteristic
18#2857C2212yes—
19#2563C412yes—
20#2563C411yescharacteristic
21#2538C412yes—
22#2538C411yescharacteristic
23#2452—21no—
24#2452—22no—
25#2439—21no—
26#2439—22no—
27#2351—21no—
28Q8×C4⋊C4C411yescharacteristic
29#2341—21no—
30C4×C22⋊Q8C2211yescharacteristic
31C4×C22⋊Q8C2212yes—
32#1984—21no—
33#1869—22no—
34#1817C2212yes—
35#1798—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1728C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56C23×C8—22no—
57Q82—44no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C412yes—
66C4×#41—44no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D412yes—
77C2×#117D411yescharacteristic
78C2×#117D412yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C8×D4—44no—
82#424—44no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86#408—44no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C4×C8—21no—
93#396—22no—
94#396—22no—
95#396—22no—
96#393—22no—
97#393—22no—
98#393—22no—
99#391—22no—
100#391—21no—
101#391C2311yescharacteristic
102#390—21no—
103#390—22no—
104#390—21no—
105#390—22no—
106#389—22no—
107#389—21no—
108#389—21no—
109#389—22no—
110#389—21no—
111#389—22no—
112#389C2312yes—
113#389C2311yescharacteristic
114#387—22no—
115#387—21no—
116#385—22no—
117#385—21no—
118#384C2×C411yescharacteristic
119#384C2×C412yes—
120#382—21no—
121#382—22no—
122#380—21no—
123#380—22no—
124#380C2×C411yescharacteristic
125#380C2×C412yes—
126#380C2×C412yes—
127#380C2×C412yes—
128#380C2×C412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C412yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135#336—21no—
136#336—21no—
137#336—22no—
138#336—22no—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2312yes—
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145C2×#41—42no—
146C2×#41—42no—
147C2×#41SD1614yes—
148C2×#41Q1614yes—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C411yescharacteristic
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22⋊C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—42no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—44no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—44no—
174C22×C8—42no—
175C22×C8—82no—
176C4⋊Q8—44no—
177C4⋊Q8—42no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180C22⋊Q8—22no—
181C22⋊Q8—24no—
182C22⋊Q8—41no—
183C22⋊Q8—42no—
184C22⋊Q8—82no—
185C22⋊Q8—81no—
186C22⋊Q8—82no—
187C22⋊Q8—81no—
188C4×Q8—24no—
189C4×Q8—42no—
190C4×Q8—44no—
191C4×Q8—42no—
192C4×Q8—44no—
193C4×Q8—44no—
194C4×D4—24no—
195C4×D4—44no—
196C4×D4—42no—
197C4×D4—44no—
198#117—24no—
199#117—44no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
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×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2411yescharacteristic
244C4⋊C8—44no—
245C4⋊C8—44no—
246C4⋊C8—42no—
247C22⋊C8—42no—
248C22⋊C8—44no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—81no—
252C4×C8—42no—
253#95—21no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—21no—
264C24C2×C22⋊C411yescharacteristic
265C24C2×C22⋊C412yes—
266#41—48no—
267#41—48no—
268C2×Q8—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—82no—
276C2×Q8—81no—
277C2×Q8—82no—
278C2×Q8—81no—
279C2×Q8#10314yes—
280C2×Q8#13112yes—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4#10314yes—
286C2×D4#10318yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×SD1612yes—
320C22×C4C2×SD1614yes—
321C22×C4C2×Q1612yes—
322C22×C4C2×Q1614yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—84no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—88no—
346C2×C8—84no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—81no—
371C4⋊C4#9914yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C22⋊C4—81no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—22no—
383C42—24no—
384C42—24no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—42no—
389C42—44no—
390C42—44no—
391C42—42no—
392C42—42no—
393C23—21no—
394C23—22no—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—22no—
399C23—22no—
400C23—21no—
401C23—21no—
402C23—22no—
403C23—22no—
404C23—21no—
405C23C2×#10312yes—
406C23C2×#10314yes—
407C23C22×C22⋊C411yescharacteristic
408Q8—28no—
409Q8—42no—
410Q8—42no—
411Q8—164no—
412Q8—162no—
413D4—28no—
414D4—48no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4C2×#9914yes—
465C2×C4C2×#9912yes—
466C2×C4C2×#10314yes—
467C2×C4C2×#10312yes—
468C2×C4#42112yes—
469C2×C4C2×#13111yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4C22×Q1611yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57711yescharacteristic
475C8—42no—
476C8—81no—
477C8—81no—
478C8—81no—
479C8—82no—
480C8—82no—
481C22—22no—
482C22—24no—
483C22—22no—
484C22—24no—
485C22—22no—
486C22—24no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—42no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22#288011yescharacteristic
496C22C22×#9911yescharacteristic
497C22#288911yescharacteristic
498C22C22×#10311yescharacteristic
499C22#289711yescharacteristic
500C22C2×#42111yescharacteristic
501C22#290111yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—24no—
507C4—21no—
508C4—21no—
509C4—24no—
510C4—41no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—81no—
515C4—82no—
516C4—81no—
517C4—81no—
518C4—82no—
519C4#152014yes—
520C2—42no—
521C2—44no—
522C2#1741411yescharacteristic
523C2#1750911yescharacteristic
524C2#1752611yescharacteristic
525C2C2×#152011yescharacteristic
526C2#1760311yescharacteristic
527C2#1768311yescharacteristic
528C2#1772311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
4. #7725C2semidirect
5. #7688C2semidirect
6. #7668C2semidirect
7. #7665C2semidirect
8. #7648C2semidirect
9. #7617C2semidirect
10. #7589C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364562
#224692
#206302
#77252
#76882
#76682
#76652
#76482
#76172
#75892
#75662