Finite Group Databasebeta

← #215601

Subgroups — #215601

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4#9880C214yesmaximal, complemented
5#9879C212yesmaximal, complemented
6#9878C212yesmaximal, complemented
7#8702C214yesmaximal
8C22×C22⋊Q8C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#431—21no—
11C2×#431—21no—
12#2855—21no—
13#2855—21no—
14#2855—21no—
15#2855—21no—
16#2855—21no—
17#2855—21no—
18#2855—21no—
19#2855—21no—
20C2×#391C411yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387C412yes—
25C2×#387C2212yes—
26C2×#385C412yes—
27C2×#381C2212yes—
28C2×#380C411yescharacteristic
29#1817—21no—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1779C2212yes—
36#1779C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1280C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—44no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C42—44no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#431—44no—
74#431—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8C2314yes—
82C2×C22⋊C8C2314yes—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85#391—21no—
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391—21no—
92#391—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#381—22no—
114#381—22no—
115#381—22no—
116#381—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132C25C22×C411yescharacteristic
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—24no—
136C22×Q8—42no—
137C22×Q8—42no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—24no—
170C22×C8—24no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C4⋊C8—44no—
228C4⋊C8—44no—
229C22⋊C8—28no—
230C22⋊C8—28no—
231C22⋊C8—44no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C22⋊C8—44no—
235C4×C8—44no—
236C4×C8—44no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—41no—
257C24—41no—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C411yescharacteristic
260C24C23×C411yescharacteristic
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—88no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—416no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4#9914yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4#13114yes—
349C22×C4#13114yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C42−1+414yes—
353C2×C8—44no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—44no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—28no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23C2×#9912yes—
415C23#41012yes—
416C23C22×C22⋊C411yescharacteristic
417C23#51912yes—
418C23C2×#13112yes—
419C23C2×#13112yes—
420Q8—84no—
421Q8—84no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4#40812yes—
483C2×C4#40812yes—
484C2×C4#40812yes—
485C2×C4#40812yes—
486C2×C4C2×#9912yes—
487C2×C4C2×#9912yes—
488C2×C4#41312yes—
489C2×C4#41312yes—
490C2×C4#53612yes—
491C2×C4#53612yes—
492C8—84no—
493C8—84no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—416no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22#231011yescharacteristic
516C22C2×#40811yescharacteristic
517C22C2×#40811yescharacteristic
518C22C22×#9911yescharacteristic
519C22C2×#41011yescharacteristic
520C22C2×#41311yescharacteristic
521C22C2×#41311yescharacteristic
522C22#298614yes—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4#146814yes—
539C2—24no—
540C2—42no—
541C2—42no—
542C2C2×#146811yescharacteristic
543C2#1737611yescharacteristic
544C2#1738111yescharacteristic
545C2#1738311yescharacteristic
546C2#2181911yescharacteristic
547C2#2182511yescharacteristic
548C2#2183111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9880C2semidirect
5. #9879C2semidirect
6. #9878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#207142
#98802
#98792
#98782
#87022