Finite Group Databasebeta

← #497581

Subgroups — #497581

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36547C211yesmaximal, characteristic
2#22515C211yesmaximal, characteristic
3#22513C211yesmaximal, characteristic
4#13297C212yesmaximal
5#13296C211yesmaximal, characteristic
6#13294C212yesmaximal
7#13293C211yesmaximal, characteristic
8#12794C211yesmaximal, characteristic
9#12794C212yesmaximal
10C4×#352C212yesmaximal
11C4×#352C211yesmaximal, characteristic
12C2×#554C2211yescharacteristic
13C22×C4×Q8C411yescharacteristic, complemented
14Q8×M16—21no—
15#2855C412yescomplemented
16#2855C411yescharacteristic, complemented
17#2583C412yescomplemented
18#2579C2212yes—
19#2465—22no—
20#2462—22no—
21#2451—21no—
22#2434C411yescharacteristic, complemented
23#2429—21no—
24#2409C2211yescharacteristic
25#2365—21no—
26#2360—21no—
27#2345—21no—
28#2345C411yescharacteristic, complemented
29C4×#123C2212yes—
30C4×#123C2211yescharacteristic
31C2×#352C2212yes—
32C2×#352C2211yescharacteristic
33C2×#352C2211yescharacteristic
34#1998—22no—
35#1997—21no—
36#1996—21no—
37#1996—22no—
38#1800C2211yescharacteristic
39#1800C2212yes—
40#1799C2211yescharacteristic
41#1799C2212yes—
42C4×#100C2211yescharacteristic
43C4×#100C2212yes—
44C4×#99C2211yescharacteristic
45C4×#99C2212yes—
46#1390C2212yes—
47#1390C2211yescharacteristic
48#1390C2212yes—
49#1390C2211yescharacteristic
50#1390C2212yes—
51#1390C2212yes—
52#1390C2212yes—
53C23×Q8C2×C411yescharacteristic
54#554—41no—
55#554—42no—
56#554—41no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8D411yescharacteristic
59C2×#125D412yes—
60C2×#123C2311yescharacteristic
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8D411yescharacteristic
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—41no—
70C2×C4×Q8—41no—
71C2×C4×Q8—42no—
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4C2312yes—
74C2×C4×D4C2311yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4D411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42C2×C411yescharacteristic
80#446—42no—
81#446—41no—
82C8×Q8—41no—
83#423—21no—
84#423—22no—
85C2×#100C2312yes—
86C2×#100C2311yescharacteristic
87C2×#100C2311yescharacteristic
88C2×#99C2312yes—
89C2×#99C2311yescharacteristic
90C2×#99C2311yescharacteristic
91C4×M16—21no—
92C4×M16—22no—
93#398—22no—
94#398—22no—
95#396—22no—
96#395—22no—
97#395—22no—
98#395—21no—
99#393—21no—
100#391—21no—
101#391—22no—
102#391—21no—
103#391—21no—
104#389—22no—
105#389—22no—
106#389—21no—
107#389—22no—
108#389—21no—
109#389—41no—
110#388—22no—
111#388—21no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—21no—
118#387—22no—
119#387—42no—
120#386C2311yescharacteristic
121#386C2312yes—
122#385—22no—
123#385C2×C412yes—
124#385C2×C412yes—
125#385C2×C411yescharacteristic
126#385C2×C412yes—
127#385C2×C412yes—
128#381C2×C411yescharacteristic
129#381C2×C412yes—
130#380—22no—
131#380—21no—
132#380C2×C412yes—
133#380C2×C411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—41no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4C2×C412yes—
143C4×C22⋊C4C2×C411yescharacteristic
144#352—24no—
145#352—22no—
146#352Q814yescomplemented
147#343—22no—
148#343—21no—
149#343—22no—
150#343—21no—
151#342—21no—
152#342—22no—
153#342—21no—
154#342—22no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—44no—
160C22×Q8C22⋊C411yescharacteristic
161C22×Q8C22⋊C412yes—
162C22×D4C2411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—44no—
176#125—42no—
177#125—42no—
178#123—21no—
179#123—21no—
180#123C2×Q812yes—
181#123#4112yes—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—44no—
191C4×Q8—21no—
192C4×Q8—22no—
193C4×Q8—21no—
194C4×Q8—44no—
195C4×Q8—44no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×D4—42no—
202C4×D4—41no—
203C4×D4—42no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D412yes—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4C22×C411yescharacteristic
242C2×C22⋊C4C22×C412yes—
243C2×C22⋊C4C22×C411yescharacteristic
244C2×C22⋊C4C22×C411yescharacteristic
245C2×C22⋊C4C22×C412yes—
246C2×C22⋊C4C2×D411yescharacteristic
247C2×C22⋊C4C2×D412yes—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—41no—
259C2×C42—42no—
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C4⋊C8—42no—
263C4⋊C8—41no—
264C4⋊C8—42no—
265C4⋊C8—41no—
266#100—21no—
267#100—22no—
268#100—21no—
269#100—22no—
270#100—21no—
271#100C2×Q812yes—
272#99—21no—
273#99—22no—
274#99—21no—
275#99—22no—
276#99—21no—
277#99C2×Q812yes—
278#97—42no—
279#97—41no—
280C4×C8—42no—
281C4×C8—41no—
282#95—21no—
283#95—22no—
284#95—21no—
285#95—22no—
286#95—21no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—21no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—22no—
297#95—41no—
298#95—42no—
299#95—42no—
300#95—41no—
301C24C23×C411yescharacteristic
302C24C22×D411yescharacteristic
303C2×Q8—24no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—44no—
312C2×Q8—48no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—82no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4C22×Q811yescharacteristic
320C2×D4C2×#4111yescharacteristic
321C2×D42−1+411yescharacteristic
322C2×D42−1+411yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4—81no—
350C22×C4—82no—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C4×Q812yes—
360C22×C4C22×D411yescharacteristic
361M16—22no—
362M16—41no—
363M16—41no—
364M16—42no—
365M16—41no—
366M16—42no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—41no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—81no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—162no—
398C4⋊C4#9912yes—
399C4⋊C4#9911yescharacteristic
400C4⋊C4#9911yescharacteristic
401C22⋊C4—21no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C22⋊C4—41no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—42no—
415C22⋊C4—41no—
416C22⋊C4—42no—
417C22⋊C4—81no—
418C22⋊C4C4×Q812yes—
419C42—22no—
420C42—21no—
421C42—22no—
422C42—21no—
423C42—44no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—41no—
429C42—41no—
430C42—82no—
431C42—82no—
432C42—162no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23C22×C22⋊C411yescharacteristic
442C23C2×C4×Q811yescharacteristic
443C23#51911yescharacteristic
444C23#54811yescharacteristic
445C23D4×Q811yescharacteristic
446Q8—44no—
447Q8—48no—
448Q8—42no—
449Q8—44no—
450Q8—81no—
451Q8—162no—
452D4—41no—
453D4—41no—
454D4—41no—
455D4—41no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4C2×#9912yes—
513C2×C4C2×#9912yes—
514C2×C4C2×#9911yescharacteristic
515C2×C4C2×#9911yescharacteristic
516C2×C4#54811yescharacteristic
517C2×C4D4×Q811yescharacteristic
518C8—42no—
519C8—82no—
520C8—81no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—81no—
537C22Q8×C22⋊C411yescharacteristic
538C22C22×#9911yescharacteristic
539C22#288411yescharacteristic
540C4—21no—
541C4—22no—
542C4—22no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—44no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—42no—
554C4—81no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4—161no—
559C4—162no—
560C4#212012yes—
561C2—21no—
562C2—21no—
563C2—41no—
564C2—41no—
565C2—41no—
566C2—41no—
567C2Q8×#9911yescharacteristic
568C2C2×#212011yescharacteristic
569C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, prime radical, cosocleC22×D4C24
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. C22×C4×Q8C4semidirect
15. #2855C4semidirect
16. #2855C4semidirect
17. #2583C4semidirect
22. #2434C4semidirect
28. #2345C4semidirect
146. #352Q8semidirect

Maximal subgroups

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

subgroupindex
#365472
#225152
#225132
#132972
#132962
#132942
#132932
#127942
C4×#3522