Finite Group Databasebeta

← #205029

Subgroups — #205029

Order 512.

1259 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1506C211yesmaximal, characteristic, complemented
4C2×#1509C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1500C211yesmaximal, characteristic
7#13750C211yesmaximal, characteristic, complemented
8#9713C214yesmaximal
9#9711C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#529C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2850C2212yes—
16#2664—22no—
17#2568C411yescharacteristic, complemented
18#2565C412yescomplemented
19#2547C411yescharacteristic, complemented
20#2546C411yescharacteristic, complemented
21#2545C412yescomplemented
22#2542C411yescharacteristic, complemented
23#2498—22no—
24#2417—22no—
25#2323—22no—
26C2×C4×C4⋊C4—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1940—22no—
31#1939—22no—
32#1797C2214yes—
33#1509—24no—
34#1506—24no—
35#1503—24no—
36#1500—24no—
37C2×#328C2212yes—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41C2×#327C2212yes—
42C2×#327C2212yes—
43C4×#95—22no—
44C4×#95C2214yes—
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52#529—22no—
53#529—22no—
54#529—22no—
55#529—22no—
56C2×#126—21no—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#124C2×C412yes—
61C2×#123—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2312yes—
65C2×C22⋊Q8C2311yescharacteristic
66C2×#121C2311yescharacteristic
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—24no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432—22no—
83#431—22no—
84#431—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—24no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×#97—21no—
91C2×C4×C8—21no—
92#400—22no—
93#399—22no—
94#398—24no—
95#398—22no—
96#397—22no—
97#396—24no—
98#391—22no—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2×C412yes—
102#390C2×C412yes—
103#389—22no—
104#389—24no—
105#388—24no—
106#388—24no—
107#387—24no—
108#386—24no—
109#384C2×C412yes—
110#382C2×C412yes—
111#380—24no—
112#380—22no—
113C4×C4⋊C4—216no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4D412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D414yes—
121#376—22no—
122#376D414yes—
123C2×#95—21no—
124C2×#95—24no—
125C2×#95D412yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128#336—24no—
129#328—24no—
130#328—28no—
131#328—28no—
132#328—24no—
133#327—24no—
134#327—24no—
135C22×Q8—21no—
136C22×Q8C22×C411yescharacteristic
137C22×Q8C22×C412yes—
138C22×D4—21no—
139C22×D4C22×C411yescharacteristic
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4C2×D414yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—82no—
154#126—42no—
155#126—42no—
156#125—22no—
157#125—22no—
158#125—42no—
159#125—42no—
160#124—22no—
161#124—22no—
162#124—82no—
163#123—42no—
164#123—42no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—48no—
176#121—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—42no—
189C4×D4—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—24no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—216no—
227C2×C42—24no—
228C2×C42—82no—
229C2×C42—82no—
230C2×C42—84no—
231C2×C42—84no—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D414yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4112yes—
242C2×C42#4112yes—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—84no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250#97—42no—
251C4×C8—42no—
252#95—216no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—22no—
259#95—22no—
260#95C22⋊C414yes—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C2×D414yes—
264#95C2×D414yes—
265C24—21no—
266C24C22×D411yescharacteristic
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C414yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4#12112yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—216no—
336C4⋊C4—28no—
337C4⋊C4—24no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—48no—
356C4⋊C4—84no—
357C4⋊C4#10418yes—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—48no—
368C22⋊C4—84no—
369C42—216no—
370C42—44no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—44no—
375C42—84no—
376C42—816no—
377C42C4×D412yes—
378C42C4×D412yes—
379C42C4×D412yes—
380C42C4×D412yes—
381C42#12412yes—
382C42#12412yes—
383C42#12712yes—
384C42#12712yes—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23C22×C22⋊C411yescharacteristic
394C23C2×#12111yescharacteristic
395C23C2×#12111yescharacteristic
396C23#57712yes—
397C23#57812yes—
398Q8—81no—
399Q8—81no—
400Q8—82no—
401Q8—82no—
402Q8—84no—
403D4—81no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—82no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—84no—
456C2×C4#39012yes—
457C2×C4#39012yes—
458C2×C4#39012yes—
459C2×C4#39012yes—
460C2×C4C2×#10418yes—
461C2×C4C2×#10414yes—
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×#12411yescharacteristic
465C2×C4C2×#12711yescharacteristic
466C8—82no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—81no—
477C22—82no—
478C22—82no—
479C22—81no—
480C22—81no—
481C22C2×#39011yescharacteristic
482C22#288311yescharacteristic
483C22#288411yescharacteristic
484C22#289812yes—
485C22C22×#10412yes—
486C22#304211yescharacteristic
487C22#304411yescharacteristic
488C22#304612yes—
489C4—22no—
490C4—22no—
491C4—28no—
492C4—22no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—48no—
498C4—48no—
499C4—44no—
500C4—81no—
501C4—81no—
502C4—82no—
503C4—82no—
504C4—81no—
505C4#196214yes—
506C2—21no—
507C2—22no—
508C2—21no—
509C2—161no—
510C2#1754812yes—
511C2#1754912yes—
512C2C2×#196211yescharacteristic
513C2#2121411yescharacteristic
514C2#2126211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
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
2. #21953C2semidirect
3. C2×#1506C2semidirect
5. C2×#1503C2semidirect
7. #13750C2semidirect
9. #9711C2semidirect
17. #2568C4semidirect
18. #2565C4semidirect
19. #2547C4semidirect
20. #2546C4semidirect
21. #2545C4semidirect
22. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#352412
#219532
C2×#15062
C2×#15092
C2×#15032
C2×#15002
#137502
#97132
#97112