Finite Group Databasebeta

← #215033

Subgroups — #215033

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20399C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#9491C212yesmaximal
5#9388C212yesmaximal
6#9190C212yesmaximal, complemented
7#9030C212yesmaximal, complemented
8#8775C212yesmaximal, complemented
9#8374C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13C22×#101—21no—
14#2873C2212yes—
15#2634C411yescharacteristic
16#2634C2211yescharacteristic
17#2621C411yescharacteristic
18#2611C411yescharacteristic
19#2611C2211yescharacteristic
20#2599C411yescharacteristic
21#2599C2211yescharacteristic
22#2581C411yescharacteristic
23#2546C411yescharacteristic
24#2475—21no—
25#2475—21no—
26#2462—21no—
27#2462—21no—
28#2446—21no—
29#2434—21no—
30#2399C411yescharacteristic
31#2399C2211yescharacteristic
32#2396C411yescharacteristic
33#2365—21no—
34#2362—21no—
35#2356C2211yescharacteristic
36C4×#101—21no—
37C4×#101—21no—
38C4×#101C2212yes—
39C4×#101C2212yes—
40#1742—21no—
41#1528C2214yes—
42#1491C2214yes—
43#1467—21no—
44#1467C2212yes—
45#1464—21no—
46#1464C2212yes—
47#1461—21no—
48#1458—21no—
49#1300C2214yes—
50#1288C2214yes—
51C23×Q8D411yescharacteristic
52C22×M16—22no—
53#557—44no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×#125—21no—
59C2×#125C2×C411yescharacteristic
60C2×#125C2311yescharacteristic
61C2×C22⋊Q8—21no—
62#519—44no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C2×#117C2311yescharacteristic
81#446—44no—
82#432—22no—
83#432C2314yes—
84#431—22no—
85C2×#101—28no—
86C2×#101—22no—
87C2×#101—22no—
88#405—44no—
89C4×M16—22no—
90C4×M16C2314yes—
91#400—21no—
92#400—21no—
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#399C2×C411yescharacteristic
97#398—21no—
98#398—21no—
99#398—21no—
100#398—21no—
101#398C2×C411yescharacteristic
102#398C2×C411yescharacteristic
103#398C2×C411yescharacteristic
104#398C2×C411yescharacteristic
105#398C2×C411yescharacteristic
106#398C2×C411yescharacteristic
107#397—21no—
108#397C2×C411yescharacteristic
109#397C2×C411yescharacteristic
110#396—21no—
111#395—21no—
112#395—21no—
113#395—21no—
114#395—21no—
115#395C2×C411yescharacteristic
116#395C2×C411yescharacteristic
117#395C2×C411yescharacteristic
118#393C2×C411yescharacteristic
119#391—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#383C2×C411yescharacteristic
151#383C2311yescharacteristic
152#382C2×C411yescharacteristic
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2×C411yescharacteristic
160C4×C22⋊C4—21no—
161#376C2×C411yescharacteristic
162#376C2311yescharacteristic
163#343—22no—
164#343—22no—
165#330—44no—
166Q8⋊C8—44no—
167#324—22no—
168#324—22no—
169#324—22no—
170#323—22no—
171C22×Q8—24no—
172C22×Q8—21no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8C22⋊C411yescharacteristic
176C22×Q8C22⋊C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×Q8C2×D411yescharacteristic
179C22×Q8C2×D411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—28no—
184C2×M16—24no—
185C2×M16—44no—
186C2×M16—42no—
187C22×C8—42no—
188C4⋊Q8—42no—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191#125—24no—
192#125—42no—
193#125—44no—
194#125—44no—
195C22⋊Q8—44no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×Q8—44no—
208#117—24no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22×C411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2411yescharacteristic
275C4⋊C8—24no—
276C4⋊C8—42no—
277C4⋊C8—42no—
278#101—28no—
279#101—28no—
280C22⋊C8—24no—
281#97—24no—
282#97—24no—
283#97—42no—
284#97—42no—
285C4×C8—24no—
286C4×C8—42no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—42no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4#10414yes—
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—28no—
369M16—48no—
370M16—44no—
371C2×C8—48no—
372C2×C8—44no—
373C2×C8—44no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42H(5,𝔽2)14yes—
435C23—21no—
436C23—21no—
437C23—41no—
438C23—41no—
439C23C2×#10412yes—
440C23C22×C22⋊C411yescharacteristic
441Q8—24no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—42no—
447Q8—84no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
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—41no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4C2×#10412yes—
505C2×C4C2×#10412yes—
506C2×C4#42112yes—
507C2×C4#51812yes—
508C2×C4#53412yes—
509C2×C4#53412yes—
510C8—48no—
511C8—84no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—42no—
516C22—41no—
517C22—41no—
518C22—81no—
519C22#230811yescharacteristic
520C22C22×#10411yescharacteristic
521C22C2×#42111yescharacteristic
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4—81no—
544C2—21no—
545C2—21no—
546C2—42no—
547C2#2177311yescharacteristic
548C2#2178011yescharacteristic
549C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9190C2semidirect
7. #9030C2semidirect
8. #8775C2semidirect
9. #8374C2semidirect

Maximal subgroups

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

subgroupindex
#357492
#203992
#173712
#94912
#93882
#91902
#90302
#87752
#83742