Finite Group Databasebeta

← #216764

Subgroups — #216764

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13954C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#9884C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#9679C211yesmaximal, characteristic
8#9152C212yesmaximal
9#9144C212yesmaximal
10#8578C212yesmaximal
11#8572C212yesmaximal
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2855C2212yes—
15#2646—21no—
16#2627—21no—
17#2590—22no—
18#2588—22no—
19#2511C412yescomplemented
20#2507C412yescomplemented
21#2451C412yescomplemented
22#2446C412yescomplemented
23#2329—21no—
24#2313—21no—
25C4×C22⋊Q8C2212yes—
26C2×#389—21no—
27C2×#389—21no—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1916—22no—
32#1916—22no—
33#1817C2212yes—
34#1797C2212yes—
35C4×C22⋊C8C2211yescharacteristic
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1444—21no—
42#1444—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8C2×C412yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C22⋊C8C2312yes—
70C2×C22⋊C8C2312yes—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#400—22no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#395—22no—
81#391—21no—
82#391—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—44no—
92#389—44no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—42no—
111#381—22no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380C2312yes—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—42no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D412yes—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135#336—22no—
136#336—24no—
137#336—21no—
138#336—21no—
139#328—42no—
140#328—42no—
141#328—42no—
142#328—42no—
143C22×Q8—21no—
144C22×Q8C22×C412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—81no—
159C4⋊Q8—81no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—82no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4SD1614yes—
205C2×C4⋊C4Q1614yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—81no—
219C2×C42—81no—
220C2×C42C22×C412yes—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C4⋊C8—82no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C4×C8—81no—
234C4×C8—81no—
235#95—22no—
236#95—22no—
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—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24C22×D411yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—81no—
269C2×Q8—81no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
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×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×SD1612yes—
331C22×C4C2×SD1612yes—
332C22×C4C2×SD1612yes—
333C22×C4C2×Q1612yes—
334C22×C4C2×Q1612yes—
335C22×C4C2×Q1612yes—
336C22×C4AGL(1,ℤ/8)12yes—
337C22×C4#13712yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+412yes—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4#10318yes—
375C4⋊C4#10318yes—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C42—24no—
381C42—24no—
382C42—41no—
383C42—41no—
384C42—42no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—82no—
389C42—82no—
390C42—84no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23C22×C22⋊C411yescharacteristic
396C23#53511yescharacteristic
397C23#53611yescharacteristic
398C23C22×SD1611yescharacteristic
399C23C22×Q1611yescharacteristic
400C23C2×AGL(1,ℤ/8)11yescharacteristic
401C23C2×#13711yescharacteristic
402Q8—82no—
403Q8—82no—
404Q8—162no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4C2×#10314yes—
484C2×C4C2×#10314yes—
485C2×C4C2×#10312yes—
486C2×C4C2×#10312yes—
487C2×C4#41712yes—
488C2×C4#41712yes—
489C2×C4#52012yes—
490C2×C4#53712yes—
491C8—82no—
492C8—82no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22#231111yescharacteristic
506C22#288311yescharacteristic
507C22#288411yescharacteristic
508C22C22×#10311yescharacteristic
509C22C2×#41711yescharacteristic
510C22#290111yescharacteristic
511C22#290111yescharacteristic
512C22#320811yescharacteristic
513C22#320911yescharacteristic
514C22#322911yescharacteristic
515C22#323011yescharacteristic
516C4—24no—
517C4—22no—
518C4—22no—
519C4—24no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—42no—
525C4—42no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—161no—
533C4—161no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2#1761411yescharacteristic
539C2#1761911yescharacteristic
540C2#1762211yescharacteristic
541C2#1762811yescharacteristic
542C2#2186011yescharacteristic
543C2#2187711yescharacteristic
544C2#2190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2511C4semidirect
20. #2507C4semidirect
21. #2451C4semidirect
22. #2446C4semidirect

Maximal subgroups

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

subgroupindex
#350152
#219122
#139542
#98852
#98842
#96792
#91522
#91442
#85782
#85722