Finite Group Databasebeta

← #481859

Subgroups — #481859

Order 512.

989 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2#20688C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#11812C214yesmaximal, complemented
5#11808C214yesmaximal
6#11717C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×#117C2211yescharacteristic
9#2920—21no—
10#2628—21no—
11#2628—21no—
12#2628—21no—
13#2628—21no—
14#2628C412yescomplemented
15#2628C412yescomplemented
16#2628C2212yes—
17#2628C2212yes—
18C2×#400—21no—
19C2×#399—21no—
20C2×#395C412yescomplemented
21C2×#393C2212yes—
22C2×#389—21no—
23C2×#382C411yescharacteristic, complemented
24#2280—21no—
25C2×#380—21no—
26#1820—21no—
27#1820C2212yes—
28#1819—21no—
29#1819C2212yes—
30#1740—21no—
31#1740C2212yes—
32#1464—21no—
33#1464C2212yes—
34#1463—21no—
35#1462C2212yes—
36#1458—21no—
37#1458C2212yes—
38#1351C22116yes—
39C23×Q8C2×C412yes—
40C24×C4D411yescharacteristic
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×#117—22no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117—41no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73#432—22no—
74#432—41no—
75#431—22no—
76#431—41no—
77#423—41no—
78#407—22no—
79C4×M16—41no—
80#400—22no—
81#400—22no—
82#400—22no—
83#400—22no—
84#399—22no—
85#399—22no—
86#399—22no—
87#399—22no—
88#395—22no—
89#395—22no—
90#395C2×C414yes—
91#395C2×C414yes—
92#393—22no—
93#393—22no—
94#393C2×C414yes—
95#393C2314yes—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#382—21no—
101#382—21no—
102#382C2×C412yes—
103#382C2×C412yes—
104#381—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—41no—
110#380—41no—
111#379—22no—
112C4×C22⋊C4—41no—
113C4×C22⋊C4—41no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95C2×C412yes—
118#342—22no—
119#342—22no—
120#342—22no—
121#342C2314yes—
122#324—22no—
123#324—22no—
124#323—22no—
125#323C2314yes—
126C25C2×D411yescharacteristic
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—84no—
131C22×Q8C22×C414yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—42no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—41no—
172#117—41no—
173#117—41no—
174#117—41no—
175#117—44no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—84no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—84no—
216C2×C22⋊C4C22⋊C411yescharacteristic
217C2×C22⋊C4C22⋊C411yescharacteristic
218C2×C22⋊C4C22×C411yescharacteristic
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C22⋊C4C2×D411yescharacteristic
223C2×C22⋊C4C2411yescharacteristic
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—42no—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C4⋊C8—42no—
242C4⋊C8—42no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245#97—42no—
246C4×C8—42no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—24no—
252#95—22no—
253#95—24no—
254#95—42no—
255#95—42no—
256#95C22×C414yes—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—41no—
261C24—42no—
262C24—42no—
263C24C2×C22⋊C411yescharacteristic
264C24C2×C22⋊C411yescharacteristic
265C24C22×D411yescharacteristic
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—88no—
271C2×Q8—88no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4#9911yescharacteristic
320C22×C4#9911yescharacteristic
321C22×C4#9911yescharacteristic
322C22×C4#9911yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4#13714yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336M16—84no—
337C2×C8—44no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C22⋊C4—84no—
370C22⋊C4—84no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—84no—
388C23—22no—
389C23—21no—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—42no—
398C23—42no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—48no—
404C23—42no—
405C23—42no—
406C23—81no—
407C23—81no—
408C23—84no—
409C23C2×#9911yescharacteristic
410C23C2×#9911yescharacteristic
411C23C22×C22⋊C411yescharacteristic
412C23C2×#13712yes—
413Q8—88no—
414Q8—88no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
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—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—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—88no—
467C2×C4—84no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4C2×#9911yescharacteristic
473C2×C4C2×#9911yescharacteristic
474C2×C4C2×#9911yescharacteristic
475C2×C4C2×#9911yescharacteristic
476C2×C4#41912yes—
477C2×C4#41912yes—
478C2×C4#42112yes—
479C8—84no—
480C22—24no—
481C22—24no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—81no—
497C22—84no—
498C22—84no—
499C22—81no—
500C22#212212yes—
501C22#212912yes—
502C22C22×#9911yescharacteristic
503C22C2×#41911yescharacteristic
504C22C2×#42111yescharacteristic
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—84no—
518C4—84no—
519C4—164no—
520C4#151112yes—
521C4#151112yes—
522C2—21no—
523C2—21no—
524C2—22no—
525C2—22no—
526C2—42no—
527C2—81no—
528C2C2×#151111yescharacteristic
529C2C2×#212211yescharacteristic
530C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11812C2semidirect
6. #11717C2semidirect
7. C22×C4⋊Q8C4semidirect
14. #2628C4semidirect
15. #2628C4semidirect
20. C2×#395C4semidirect
23. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
#206882
#173572
#118122
#118082
#117172