Finite Group Databasebeta

← #493977

Subgroups — #493977

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#532C211yesmaximal, characteristic
2#20728C212yesmaximal, complemented
3#12861C212yesmaximal, complemented
4#12802C212yesmaximal, complemented
5#12801C212yesmaximal, complemented
6#12701C212yesmaximal
7C4×#356C212yesmaximal, complemented
8C4×#355C212yesmaximal, complemented
9C2×#532C2211yescharacteristic
10C2×C4×#41C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2920—22no—
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×#127C411yescharacteristic
16C4×#127C411yescharacteristic
17C4×#124C411yescharacteristic
18C4×#124C411yescharacteristic
19C4×#124C2211yescharacteristic
20C4×#124C2211yescharacteristic
21C4×#121—22no—
22C4×#121—22no—
23#2297C411yescharacteristic
24#2297C2211yescharacteristic
25#2293—22no—
26#2285C411yescharacteristic
27C42×D4—22no—
28C2×C43C411yescharacteristic
29C2×#356C2214yes—
30C2×#355C2214yes—
31#1822—22no—
32#1801C2214yes—
33#1800—22no—
34#1785C2214yes—
35C4×#101C2214yes—
36C4×#100—22no—
37#1394C2214yes—
38#1394C2214yes—
39C22×#41C2×C411yescharacteristic
40C22×#41C2311yescharacteristic
41#532—24no—
42#532—24no—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#121—22no—
48C4×#41—41no—
49C4×#41—41no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C4×#41—41no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D412yes—
70C2×C4×D4D412yes—
71C2×#117—21no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D412yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D412yes—
82#432—42no—
83#432—42no—
84#431—42no—
85#431—42no—
86#423—24no—
87C2×#101C2318yes—
88C2×#100—24no—
89#407—24no—
90C4×M16—24no—
91#391C2×C411yescharacteristic
92#391C2×C411yescharacteristic
93#391C2311yescharacteristic
94#391C2311yescharacteristic
95#390—22no—
96#390—22no—
97#390C2×C411yescharacteristic
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#389—22no—
102#389—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#386—22no—
108#386—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#380—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C43—21no—
124C43—21no—
125C43C2×C411yescharacteristic
126C43C2×C411yescharacteristic
127C43C2×C411yescharacteristic
128C43C2×C411yescharacteristic
129#356—28no—
130#355—28no—
131#343—24no—
132#343—24no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—41no—
138C2×#41—41no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—41no—
142C2×#41—41no—
143C22×Q8C22×C411yescharacteristic
144C22×Q8C2411yescharacteristic
145C22×D4—22no—
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C22×D4C2×D412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C2×M16—28no—
162C4⋊Q8—22no—
163C4⋊Q8#4114yes—
164#127—22no—
165#127—22no—
166#124—22no—
167#124—22no—
168#124—22no—
169#124#4114yes—
170#121—216no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×D4—216no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—48no—
189C4×D4—48no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198#117—41no—
199#117—41no—
200#117—42no—
201#117—42no—
202#117—42no—
203#117—42no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C412yes—
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C4⋊C8—44no—
271C4⋊C8—44no—
272#101—28no—
273#101—28no—
274#100—48no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277#97—44no—
278C4×C8—44no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285C24—21no—
286C24C2×C22⋊C412yes—
287C24C2×C22⋊C411yescharacteristic
288#41—44no—
289#41—44no—
290#41—44no—
291#41—44no—
292#41—44no—
293#41—44no—
294#41—44no—
295#41—44no—
296C2×Q8—21no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8#11712yes—
302C2×Q8C2×#4112yes—
303C2×Q8C2×#4112yes—
304C2×D4—28no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—24no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4#11712yes—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D412yes—
374M16—48no—
375M16—48no—
376C2×C8—48no—
377C4⋊C4—28no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—21no—
404C42—21no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—22no—
416C42—21no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42#9911yescharacteristic
433C42#9911yescharacteristic
434C42#9911yescharacteristic
435C42#9911yescharacteristic
436C42#11712yes—
437C42#11712yes—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23#41012yes—
453C23C22×C22⋊C411yescharacteristic
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458D4—48no—
459D4—42no—
460D4—416no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—48no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—48no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4C2×#9911yescharacteristic
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×#9911yescharacteristic
526C2×C4C2×#9911yescharacteristic
527C2×C4C2×#9911yescharacteristic
528C2×C4C2×#11711yescharacteristic
529C2×C4C2×#11711yescharacteristic
530C2×C4#52514yes—
531C8—88no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—44no—
538C22—44no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22—81no—
548C22—81no—
549C22#228011yescharacteristic
550C22C22×#9911yescharacteristic
551C22C2×#41011yescharacteristic
552C4—24no—
553C4—21no—
554C4—21no—
555C4—24no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—42no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4#213212yes—
571C4#213212yes—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—82no—
576C2—82no—
577C2#2072211yescharacteristic
578C2C2×#213211yescharacteristic
579C2C2×#213211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20728C2semidirect
3. #12861C2semidirect
4. #12802C2semidirect
5. #12801C2semidirect
7. C4×#356C2semidirect
8. C4×#355C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5322
#207282
#128612
#128022
#128012
#127012
C4×#3562
C4×#3552