Finite Group Databasebeta

← #496548

Subgroups — #496548

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3C2×#1801C211yesmaximal, characteristic
4#13301C212yesmaximal, complemented
5#13227C212yesmaximal, complemented
6#13094C214yesmaximal
7#12804C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11C2×#423—21no—
12C22×#101C2212yes—
13#2855—21no—
14#2855—21no—
15#2634—21no—
16#2634—21no—
17#2581—21no—
18#2581—21no—
19#2568C411yescharacteristic
20#2568C2211yescharacteristic
21#2546C411yescharacteristic
22#2546C2211yescharacteristic
23#2503C412yes—
24#2484—21no—
25#2484—21no—
26#2434C2212yes—
27#2327C412yes—
28#2327C2212yes—
29#2283C411yescharacteristic
30C2×#356C2218yes—
31#1821—21no—
32#1801—21no—
33#1801—21no—
34#1801—21no—
35#1801C2212yes—
36#1801C2212yes—
37#1801C2212yes—
38#1784C2212yes—
39C4×#101—21no—
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—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#432—41no—
78#432—41no—
79#431—41no—
80#431—41no—
81#423—22no—
82#423—22no—
83#423—22no—
84#423—41no—
85#423—41no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×#101—22no—
89C2×#101—22no—
90C2×#101—44no—
91C2×#101C2314yes—
92C2×#101C2314yes—
93#407—22no—
94C4×M16—22no—
95#400—21no—
96#400—21no—
97#400—22no—
98#400—22no—
99#400—21no—
100#400—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#398—21no—
105#395—22no—
106#395—22no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391C2×C412yes—
112#391C2×C412yes—
113#391C2×C412yes—
114#391C2312yes—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#387—22no—
120#387—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#387—22no—
125#387—22no—
126#387—22no—
127#385—22no—
128#385—22no—
129#383—21no—
130#382—21no—
131#382C2×C412yes—
132#382C2×C412yes—
133#381—22no—
134#381—22no—
135#381—21no—
136#380—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C22⋊C4C2×C412yes—
140#376C2×C412yes—
141#356—48no—
142#343—22no—
143#343—22no—
144#343—22no—
145#343—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—42no—
152C22×Q8C22×C412yes—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C2×D411yescharacteristic
155C22×Q8C2×D411yescharacteristic
156C22×Q8C2411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—24no—
165C2×M16—24no—
166C2×M16—42no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—42no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—44no—
177#125—42no—
178#125—42no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
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—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205#117—42no—
206#117—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
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—22no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22×C412yes—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272#101—24no—
273#101—28no—
274#101—24no—
275#101—44no—
276#101—44no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279#97—42no—
280C4×C8—42no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—21no—
285#95—21no—
286#95—22no—
287#95—22no—
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—
297#95—22no—
298C24C2×C22⋊C411yescharacteristic
299C2×Q8—22no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8H(5,𝔽2)12yes—
322C2×Q82−1+412yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4#9911yescharacteristic
351C22×C4#9911yescharacteristic
352C22×C4#9911yescharacteristic
353C22×C4#9911yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368M16—44no—
369M16—44no—
370M16—44no—
371M16—44no—
372M16—84no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—84no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
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—21no—
428C42—21no—
429C42—21no—
430C42—24no—
431C42—42no—
432C42—44no—
433C23—21no—
434C23—21no—
435C23—22no—
436C23—41no—
437C23C2×#9911yescharacteristic
438C23C2×#9911yescharacteristic
439C23C22×C22⋊C411yescharacteristic
440Q8—42no—
441Q8—42no—
442Q8—44no—
443Q8—44no—
444Q8—42no—
445Q8—42no—
446Q8—84no—
447Q8—82no—
448Q8—82no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
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—21no—
475C2×C4—21no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4C2×#9911yescharacteristic
507C2×C4C2×#9911yescharacteristic
508C2×C4C2×#9911yescharacteristic
509C2×C4C2×#9911yescharacteristic
510C2×C4#41012yes—
511C2×C4#52012yes—
512C2×C4#53511yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53712yes—
515C8—84no—
516C8—84no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—42no—
525C22—42no—
526C22#213512yes—
527C22#231111yescharacteristic
528C22C22×#9911yescharacteristic
529C22C2×#41011yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—22no—
535C4—21no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4#213312yes—
551C2—21no—
552C2—21no—
553C2—22no—
554C2—41no—
555C2#2187411yescharacteristic
556C2C2×#213311yescharacteristic
557C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊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
4. #13301C2semidirect
5. #13227C2semidirect
7. #12804C2semidirect

Maximal subgroups

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

subgroupindex
#352482
#207272
C2×#18012
#133012
#132272
#130942
#128042