Finite Group Databasebeta

← #495612

Subgroups — #495612

Order 512.

982 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#34953C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×C4×#101C211yesmaximal, characteristic
4#13302C212yesmaximal, complemented
5#13227C212yesmaximal, complemented
6#13094C214yesmaximal
7C4×#356C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C22×#101C2212yes—
12C2×C4×M16—21no—
13#2584—21no—
14#2584—21no—
15#2583—21no—
16#2583—21no—
17#2567C411yescharacteristic
18#2567C2211yescharacteristic
19#2546C411yescharacteristic
20#2546C2211yescharacteristic
21#2502C412yes—
22#2434C2212yes—
23#2399—21no—
24#2399—21no—
25C4×C4⋊Q8C412yes—
26C4×C4⋊Q8C2212yes—
27#2296—21no—
28#2296—21no—
29C4×#117C411yescharacteristic
30C2×#356C2218yes—
31#1821—21no—
32#1801—21no—
33#1801C2212yes—
34#1784C2212yes—
35C4×#101—21no—
36C4×#101—21no—
37C4×#101—21no—
38C4×#101C2212yes—
39C4×#101C2212yes—
40C4×#101C2212yes—
41C2×#343—21no—
42#1394C2218yes—
43C23×Q8C2311yescharacteristic
44C22×M16—22no—
45#533—41no—
46#533—41no—
47#533—41no—
48#533—41no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2312yes—
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×Q8—41no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117—21no—
71C2×#117—21no—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78#432—41no—
79#432—41no—
80#431—41no—
81#431—41no—
82#423—22no—
83C2×#101—22no—
84C2×#101—28no—
85C2×#101—22no—
86C2×#101C2314yes—
87C2×#101C2314yes—
88#407—22no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16—22no—
92C4×M16—41no—
93C4×M16—41no—
94C2×#97—41no—
95C2×C4×C8—41no—
96#398—21no—
97#398—21no—
98#398—22no—
99#398—22no—
100#398—21no—
101#398—21no—
102#395—22no—
103#395—22no—
104#395—21no—
105#395—21no—
106#395—21no—
107#395—21no—
108#391C2×C412yes—
109#391C2×C412yes—
110#391C2×C412yes—
111#391C2312yes—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#385—22no—
125#385—22no—
126#385—22no—
127#385—22no—
128#384C2×C412yes—
129#384C2×C412yes—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4C2×C412yes—
139#376—21no—
140C43C2×C412yes—
141#356—216no—
142#343—22no—
143#343—22no—
144#343—22no—
145#343—22no—
146C22×Q8—24no—
147C22×Q8—21no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—42no—
151C22×Q8C22×C412yes—
152C22×Q8C22×C411yescharacteristic
153C22×Q8C2×D411yescharacteristic
154C22×Q8C2×D411yescharacteristic
155C22×Q8C2411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—28no—
164C2×M16—24no—
165C2×M16—24no—
166C2×M16—44no—
167C2×M16—42no—
168C22×C8—42no—
169C4⋊Q8—28no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—44no—
175#125—42no—
176#125—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201#117—28no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—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—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22×C412yes—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C4⋊C8—42no—
272C4⋊C8—42no—
273#101—28no—
274#101—24no—
275#101—28no—
276#101—44no—
277#101#4118yes—
278C22⋊C8—42no—
279C22⋊C8—42no—
280#97—42no—
281#97—42no—
282#97—42no—
283C4×C8—42no—
284C4×C8—42no—
285C4×C8—42no—
286#95—21no—
287#95—21no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297C24C2×C22⋊C411yescharacteristic
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—82no—
316C2×Q8C2×#4114yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—81no—
343C22×C4#9911yescharacteristic
344C22×C4#9911yescharacteristic
345C22×C4#9911yescharacteristic
346C22×C4#9911yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361M16—28no—
362M16—48no—
363M16—44no—
364M16—44no—
365M16—44no—
366C2×C8—44no—
367C2×C8—48no—
368C2×C8—44no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—44no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C22⋊C4—28no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C42—28no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—24no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—82no—
432C23—21no—
433C23—22no—
434C23—21no—
435C23—41no—
436C23C2×#9911yescharacteristic
437C23C2×#9911yescharacteristic
438C23C22×C22⋊C411yescharacteristic
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—84no—
444Q8—84no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4#35614yescomplemented
501C2×C4C2×#9911yescharacteristic
502C2×C4C2×#9911yescharacteristic
503C2×C4C2×#9911yescharacteristic
504C2×C4C2×#9911yescharacteristic
505C2×C4#41012yes—
506C2×C4C4×#4112yes—
507C2×C4#52512yes—
508C2×C4#53212yes—
509C8—48no—
510C8—84no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—42no—
519C22—42no—
520C22C2×#35612yes—
521C22#229711yescharacteristic
522C22C22×#9911yescharacteristic
523C22C2×#41011yescharacteristic
524C4—21no—
525C4—21no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—44no—
531C4—44no—
532C4—41no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—82no—
541C4—82no—
542C4C2×#35612yes—
543C4C2×#35612yes—
544C2—21no—
545C2—21no—
546C2—22no—
547C2—41no—
548C2#2142011yescharacteristic
549C2C22×#35611yescharacteristic
550C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
4. #13302C2semidirect
5. #13227C2semidirect
7. C4×#356C2semidirect
500. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
#349532
#207272
C2×C4×#1012
#133022
#132272
#130942
C4×#3562