Finite Group Databasebeta

← #215079

Subgroups — #215079

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36030C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#9493C212yesmaximal, complemented
5#9389C212yesmaximal
6#9188C212yesmaximal, complemented
7#9030C212yesmaximal, complemented
8#8774C212yesmaximal, complemented
9#8368C212yesmaximal, complemented
10C2×#556C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2983C2212yes—
13#2921C2212yes—
14#2632C411yescharacteristic
15#2632C2211yescharacteristic
16#2628C411yescharacteristic
17#2610C411yescharacteristic
18#2610C2211yescharacteristic
19#2599C411yescharacteristic
20#2599C2211yescharacteristic
21#2567C411yescharacteristic
22#2536C411yescharacteristic
23#2479—21no—
24#2479—21no—
25#2469—21no—
26#2469—21no—
27#2423—21no—
28#2409—21no—
29#2399C411yescharacteristic
30#2399C2211yescharacteristic
31#2392C411yescharacteristic
32#2362—21no—
33#2356C2211yescharacteristic
34#2354—21no—
35#1822—21no—
36#1801—21no—
37#1801—21no—
38#1784—21no—
39C4×#101C2212yes—
40C4×#101C2212yes—
41#1527C2214yes—
42#1491C2214yes—
43#1467—21no—
44#1467C2212yes—
45#1464—21no—
46#1464C2212yes—
47#1461—21no—
48#1458—21no—
49#1300C2214yes—
50#1288C2214yes—
51C23×Q8D411yescharacteristic
52#556—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#126—21no—
55C2×#126—21no—
56C2×#124C2×C411yescharacteristic
57C2×#124C2311yescharacteristic
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60#519—44no—
61C2×C4×Q8C2×C411yescharacteristic
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×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×#117—21no—
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117C2311yescharacteristic
80#446—44no—
81#433—44no—
82#432—22no—
83#432C2314yes—
84#431—22no—
85#423—22no—
86C2×#101—22no—
87C2×#101—22no—
88#407—22no—
89C4×M16C2314yes—
90#400—21no—
91#399—21no—
92#399—21no—
93#399—21no—
94#398—21no—
95#398—21no—
96#398C2×C411yescharacteristic
97#398C2×C411yescharacteristic
98#398C2×C411yescharacteristic
99#397C2×C411yescharacteristic
100#396—21no—
101#395—21no—
102#395—21no—
103#395C2×C411yescharacteristic
104#395C2×C411yescharacteristic
105#395C2×C411yescharacteristic
106#395C2×C411yescharacteristic
107#395C2×C411yescharacteristic
108#395C2×C411yescharacteristic
109#394—21no—
110#394C2×C411yescharacteristic
111#394C2×C411yescharacteristic
112#394C2×C411yescharacteristic
113#393—21no—
114#393—21no—
115#393C2×C411yescharacteristic
116#393C2×C411yescharacteristic
117#392C2×C411yescharacteristic
118#391—21no—
119#391—21no—
120#390—21no—
121#390—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#386—21no—
144#386—21no—
145#385—21no—
146#385—21no—
147#384—21no—
148#383C2×C411yescharacteristic
149#383C2311yescharacteristic
150#382C2×C411yescharacteristic
151#380—21no—
152#380—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C22⋊C4—21no—
160#376C2×C411yescharacteristic
161#376C2311yescharacteristic
162#343—22no—
163#343—22no—
164#329—44no—
165Q8⋊C8—44no—
166#324—22no—
167#324—22no—
168#324—22no—
169#323—22no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C22×Q8C2×D411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C2×M16—24no—
186C4⋊Q8—42no—
187#126—44no—
188#126—44no—
189#124—24no—
190#124—42no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×D4—42no—
204C4×D4—44no—
205#117—24no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
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⋊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×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22×C411yescharacteristic
258C2×C22⋊C4C2×D411yescharacteristic
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—24no—
277C4⋊C8—44no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280#101—44no—
281#101—44no—
282C22⋊C8—24no—
283C22⋊C8—44no—
284#97—24no—
285#97—24no—
286#97—42no—
287C4×C8—24no—
288C4×C8—42no—
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#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310#95C22×C411yescharacteristic
311#95C22×C411yescharacteristic
312C24—21no—
313C24C2×C22⋊C411yescharacteristic
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—416no—
329C2×Q8—81no—
330C2×Q8—81no—
331C2×D4—22no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—81no—
335C2×D4—81no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—44no—
370M16—44no—
371C2×C8—44no—
372C2×C8—44no—
373C4⋊C4—22no—
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—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—81no—
416C22⋊C4—81no—
417C22⋊C4—81no—
418C22⋊C4—81no—
419C22⋊C4—81no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42H(5,𝔽2)14yes—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23C22×C22⋊C411yescharacteristic
450Q8—44no—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—82no—
456D4—82no—
457D4—84no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
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—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
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×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—84no—
512C2×C4#42112yes—
513C2×C4#42112yes—
514C2×C4#51812yes—
515C2×C4#53412yes—
516C2×C4#53412yes—
517C8—84no—
518C8—84no—
519C22—21no—
520C22—21no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#230811yescharacteristic
534C22C2×#42111yescharacteristic
535C22C2×#42111yescharacteristic
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C2—21no—
558C2—21no—
559C2—81no—
560C2—161no—
561C2#2177411yescharacteristic
562C2#2178111yescharacteristic
563C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#392C2×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
2. #20730C2semidirect
3. #17371C2semidirect
4. #9493C2semidirect
6. #9188C2semidirect
7. #9030C2semidirect
8. #8774C2semidirect
9. #8368C2semidirect

Maximal subgroups

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

subgroupindex
#360302
#207302
#173712
#94932
#93892
#91882
#90302
#87742
#83682