Finite Group Databasebeta

← #493749

Subgroups — #493749

Order 512.

1155 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2C2×#2133C212yesmaximal
3C2×#1801C212yesmaximal, complemented
4#12804C212yesmaximal, complemented
5#12803C212yesmaximal, complemented
6#12802C212yesmaximal, complemented
7#12801C212yesmaximal, complemented
8#12586C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×C4×#41C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423—22no—
13C22×#101C2214yes—
14#2327C411yescharacteristic
15#2327C2211yescharacteristic
16#2326C411yescharacteristic
17#2326C2211yescharacteristic
18#2323—22no—
19#2322—22no—
20#2317C411yescharacteristic
21#2311C411yescharacteristic
22#2310C2211yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C2211yescharacteristic
27C4×C22⋊Q8—22no—
28C4×#123—22no—
29C4×#117C411yescharacteristic
30#2133—28no—
31C2×#356C2214yes—
32C2×#355C2214yes—
33#1801—22no—
34#1801—22no—
35#1801C2214yes—
36#1801C2214yes—
37C2×#343—22no—
38#1394C2214yes—
39#1394C2214yes—
40C22×#41C2×C411yescharacteristic
41C23×Q8C2311yescharacteristic
42C22×M16—24no—
43#536—24no—
44#536D414yes—
45#536Q814yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#124C2×C411yescharacteristic
49C2×#124C2311yescharacteristic
50C2×#123—22no—
51C2×C22⋊Q8—22no—
52C4×#41—42no—
53C4×#41—42no—
54C2×C4×Q8—42no—
55C2×C4×Q8—42no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2311yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×#117—21no—
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2×C411yescharacteristic
76C22×C42D412yes—
77#423—24no—
78#423—24no—
79#423—42no—
80#423—42no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8—42no—
83C2×#101—28no—
84C2×#101—28no—
85C2×#101—24no—
86C2×#101C2318yes—
87#391—22no—
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#391C2×C411yescharacteristic
92#391C2311yescharacteristic
93#391C2311yescharacteristic
94#390C2×C411yescharacteristic
95#390C2×C411yescharacteristic
96#389—22no—
97#389—22no—
98#389—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382C2×C411yescharacteristic
114#382C2×C411yescharacteristic
115#381—22no—
116#380—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125#376C2×C411yescharacteristic
126#376C2×C411yescharacteristic
127C43C2×C411yescharacteristic
128C43C2×C411yescharacteristic
129#356—28no—
130#355—28no—
131#343—24no—
132#343—24no—
133C2×#41—22no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41C4⋊C414yes—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C22×C411yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×Q8C2×D412yes—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q812yes—
144C22×Q8#4114yes—
145C22×Q8C2411yescharacteristic
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C22⋊C412yes—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C2×M16—28no—
157C2×M16—44no—
158C2×M16—48no—
159C2×M16—48no—
160C22×C8—44no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8C4⋊C414yes—
164C4⋊Q8C2×D412yes—
165C4⋊Q8C2×Q812yes—
166#124—22no—
167#124—22no—
168#124C4⋊C414yes—
169#124C2×D412yes—
170#124C2×Q812yes—
171#123—28no—
172#123—28no—
173C22⋊Q8—28no—
174C22⋊Q8—28no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—42no—
191C4×D4—44no—
192C4×D4—42no—
193C4×D4—42no—
194#117—22no—
195#117—44no—
196#117—42no—
197#117—42no—
198#117C4⋊C414yes—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C412yes—
245C2×C42C22⋊C412yes—
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C4⋊C8—44no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258C4⋊C8—44no—
259#101—28no—
260#101—28no—
261#101—48no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24—21no—
275C24C2×C22⋊C411yescharacteristic
276#41—48no—
277#41—48no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—44no—
284C2×Q8—42no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C2×C4⋊C412yes—
292C2×Q8C2×C4⋊C412yes—
293C2×Q8#11712yes—
294C2×Q8#12114yes—
295C2×Q8C22⋊Q814yes—
296C2×Q8C22×D411yescharacteristic
297C2×Q8C22×Q811yescharacteristic
298C2×Q8C2×#4112yes—
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4C2×C4⋊C412yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—81no—
332C22×C4#9912yes—
333C22×C4#9912yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C4⋊C412yes—
343C22×C4#11712yes—
344C22×C4#12114yes—
345C22×C4C22⋊Q814yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D412yes—
348M16—48no—
349M16—48no—
350M16—88no—
351C2×C8—48no—
352C2×C8—88no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—28no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C42—21no—
382C42—21no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—21no—
396C42—21no—
397C42—41no—
398C42—41no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—41no—
404C42—41no—
405C42—81no—
406C42—81no—
407C42C2×C4⋊C412yes—
408C42C2×C4⋊C412yes—
409C23—22no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23#38012yes—
417C23C2×#9912yes—
418C23#41012yes—
419C23C22×C22⋊C411yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—44no—
423Q8—44no—
424Q8—41no—
425Q8—41no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429D4—42no—
430D4—42no—
431D4—48no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—48no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4#38014yes—
485C2×C4#38012yes—
486C2×C4C2×#9911yescharacteristic
487C2×C4C2×#9912yes—
488C2×C4C2×#9911yescharacteristic
489C2×C4C22×C4⋊C411yescharacteristic
490C2×C4C2×#11711yescharacteristic
491C2×C4C2×#12112yes—
492C2×C4C2×C22⋊Q812yes—
493C8—88no—
494C22—21no—
495C22—22no—
496C22—21no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—42no—
502C22—82no—
503C22—81no—
504C22—81no—
505C22#179914yes—
506C22C2×#38011yescharacteristic
507C22C22×#9911yescharacteristic
508C22C2×#41011yescharacteristic
509C4—21no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—24no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4#213312yescomplemented
529C2—21no—
530C2—21no—
531C2—24no—
532C2—82no—
533C2C2×#179911yescharacteristic
534C2C2×#213311yescharacteristic
535C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1801C2semidirect
4. #12804C2semidirect
5. #12803C2semidirect
6. #12802C2semidirect
7. #12801C2semidirect
8. #12586C2semidirect
528. C4#2133semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
C2×#21332
C2×#18012
#128042
#128032
#128022
#128012
#125862