Finite Group Databasebeta

← #215044

Subgroups — #215044

Order 512.

927 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
1#36021C211yesmaximal, characteristic
2#20397C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#9491C212yesmaximal, complemented
5#9388C212yesmaximal, complemented
6#9190C212yesmaximal, complemented
7#9030C212yesmaximal, complemented
8#8772C212yesmaximal, complemented
9#8371C212yesmaximal, complemented
10C2×#552C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13C2×#413—21no—
14#2873C2212yes—
15#2638C411yescharacteristic
16#2634C2211yescharacteristic
17#2627C411yescharacteristic
18#2611C2211yescharacteristic
19#2599C2211yescharacteristic
20#2583C411yescharacteristic
21#2564C411yescharacteristic
22#2486C411yescharacteristic
23#2484C411yescharacteristic
24#2478—21no—
25#2473—21no—
26#2468—21no—
27#2462—21no—
28#2419—21no—
29#2406—21no—
30#2399C2211yescharacteristic
31#2395C411yescharacteristic
32C2×#398C411yescharacteristic
33#2359—21no—
34#2353C2211yescharacteristic
35#2344—21no—
36C4×#101—21no—
37C4×#101C2212yes—
38C4×#100—21no—
39C4×#100C2212yes—
40C2×#343—21no—
41#1528C2214yes—
42#1491C2214yes—
43#1466—21no—
44#1466C2212yes—
45#1464—21no—
46#1464C2212yes—
47#1460—21no—
48#1458—21no—
49#1300C2214yes—
50#1288C2214yes—
51C22×#41D411yescharacteristic
52C22×M16—22no—
53#552—44no—
54C2×#126—21no—
55C2×#125—21no—
56C2×#125C2311yescharacteristic
57C2×#123—21no—
58C2×#123—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic
61#520—44no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117C2×C411yescharacteristic
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117C2311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81#446—44no—
82#432—22no—
83#432C2314yes—
84#431—22no—
85#413—28no—
86C2×#101—22no—
87C2×#100—22no—
88#405—44no—
89C4×M16—22no—
90C4×M16C2314yes—
91#400—21no—
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#399C2×C411yescharacteristic
96#398—21no—
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#397—21no—
109#397C2×C411yescharacteristic
110#397C2×C411yescharacteristic
111#395—21no—
112#395—21no—
113#395C2×C411yescharacteristic
114#395C2×C411yescharacteristic
115#395C2×C411yescharacteristic
116#394—21no—
117#394—21no—
118#393C2×C411yescharacteristic
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#386—21no—
137#386—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#383C2311yescharacteristic
147#381—21no—
148#381—21no—
149#381—21no—
150#381C2×C411yescharacteristic
151#380—21no—
152#380—21no—
153#380C2×C411yescharacteristic
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4C2×C411yescharacteristic
160#376C2311yescharacteristic
161C2×#95—21no—
162C2×#95C2×C411yescharacteristic
163#343—22no—
164#343—22no—
165#330—44no—
166Q8⋊C8—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C2×#41—24no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8C22×C411yescharacteristic
175C22×Q8C2×D411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×M16—28no—
191C2×M16—24no—
192C2×M16—44no—
193C2×M16—42no—
194C22×C8—42no—
195#126—44no—
196#125—24no—
197#125—44no—
198#123—44no—
199#123—44no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—42no—
205C4×D4—24no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211#117—24no—
212#117—42no—
213#117—42no—
214#117—44no—
215#117—42no—
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—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
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⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22×C411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C22⋊C4C2×D411yescharacteristic
266C2×C22⋊C4C2×D411yescharacteristic
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2411yescharacteristic
277C4⋊C8—24no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280#101—28no—
281#100—28no—
282C22⋊C8—24no—
283#97—24no—
284#97—24no—
285#97—42no—
286#97—42no—
287C4×C8—24no—
288C4×C8—42no—
289C4×C8—42no—
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
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320#41—48no—
321#41—48no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—81no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4#10414yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383M16—28no—
384M16—48no—
385M16—44no—
386C2×C8—48no—
387C2×C8—44no—
388C2×C8—44no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—41no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—81no—
444C42H(5,𝔽2)14yes—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23C2×#10412yes—
459C23#42112yes—
460C23C22×C22⋊C411yescharacteristic
461Q8—44no—
462Q8—44no—
463D4—44no—
464D4—44no—
465D4—44no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—44no—
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×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—161no—
522C2×C4—161no—
523C2×C4—161no—
524C2×C4—161no—
525C2×C4C2×#10412yes—
526C2×C4C2×#10412yes—
527C2×C4#51812yes—
528C2×C4#53412yes—
529C2×C4#53412yes—
530C8—48no—
531C8—84no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22#230811yescharacteristic
550C22C22×#10411yescharacteristic
551C22C2×#42111yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—81no—
577C2#2177311yescharacteristic
578C2#2177911yescharacteristic
579C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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
4. #9491C2semidirect
5. #9388C2semidirect
6. #9190C2semidirect
7. #9030C2semidirect
8. #8772C2semidirect
9. #8371C2semidirect

Maximal subgroups

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

subgroupindex
#360212
#203972
#173722
#94912
#93882
#91902
#90302
#87722
#83712