Finite Group Databasebeta

← #238216

Subgroups — #238216

Order 512.

880 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5#10356C212yesmaximal, complemented
6#10355C212yesmaximal, complemented
7#10353C212yesmaximal, complemented
8#10352C212yesmaximal, complemented
9C2×#1296C211yesmaximal, characteristic
10#7893C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2859—21no—
17#2859—21no—
18#2647—22no—
19#2571—21no—
20#2571C2212yes—
21#2552—21no—
22#2552C2212yes—
23#2442—22no—
24#2442—22no—
25#2317—21no—
26#2316—21no—
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1990C2212yes—
31#1989—21no—
32#1989—21no—
33#1988C2211yescharacteristic
34#1988C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2212yes—
38#1981C2212yes—
39#1956C2212yes—
40#1892—22no—
41#1872—22no—
42#1803—22no—
43#1803—21no—
44#1803—21no—
45#1735—21no—
46#1735—21no—
47#1296C2212yes—
48#1296C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C23×C8—21no—
54C23×C8—21no—
55#533—22no—
56#533—22no—
57#533—22no—
58#533Q814yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D412yes—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—22no—
91C2×C22⋊C8—21no—
92C2×C4×C8C2312yes—
93#400—24no—
94#400—24no—
95#398—24no—
96#398—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#385—24no—
108#384—22no—
109#384—21no—
110#384—21no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#384D412yes—
114#384C2312yes—
115#382—22no—
116#382—22no—
117#382—22no—
118#382—21no—
119#380—24no—
120#380—21no—
121#380D412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2312yes—
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130#376—21no—
131C43—21no—
132#340—24no—
133#340—22no—
134#340—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—81no—
158C22×C8—81no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C2×Q812yes—
166C4⋊Q8C2×Q812yes—
167#125—24no—
168#125—42no—
169#125—82no—
170#125—82no—
171#125—84no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—42no—
179C4×Q8—42no—
180#117—24no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—42no—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—41no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—41no—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×Q812yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248#103—82no—
249#103—82no—
250#103—84no—
251C22⋊C8—42no—
252C4×C8—22no—
253C4×C8—22no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264C24#12011yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—44no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q812yes—
298C22×C4C22⋊Q812yes—
299C22×C4#12412yes—
300C22×C4#12412yes—
301C22×C4#12412yes—
302C22×C4#13512yes—
303C22×C4#13512yes—
304C22×C4#13712yes—
305C22×C4#13712yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—84no—
322C2×C8—84no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—82no—
357C4⋊C4—84no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—44no—
374C42—42no—
375C42—42no—
376C42C22⋊Q812yes—
377C42C22⋊Q812yes—
378C42C22⋊Q812yes—
379C42#12412yes—
380C42#12412yes—
381C42#12412yes—
382C42C22×Q811yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23#39312yes—
391C23#45212yes—
392C23#45212yes—
393C23C2×#12011yescharacteristic
394Q8—44no—
395Q8—42no—
396Q8—42no—
397Q8—82no—
398Q8—81no—
399Q8—81no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—164no—
440C2×C4#39312yes—
441C2×C4#39312yes—
442C2×C4#39312yes—
443C2×C4#45212yes—
444C2×C4#45212yes—
445C2×C4#47914yes—
446C2×C4#47914yes—
447C2×C4#48814yes—
448C2×C4#49114yes—
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×#12411yescharacteristic
453C2×C4C2×#12411yescharacteristic
454C2×C4C2×#12411yescharacteristic
455C2×C4C2×#13511yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13711yescharacteristic
458C2×C4C2×#13711yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57811yescharacteristic
463C8—82no—
464C8—81no—
465C8—81no—
466C8—82no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22#202614yes—
474C22C2×#39311yescharacteristic
475C22C2×#45211yescharacteristic
476C22C2×#45211yescharacteristic
477C22#301211yescharacteristic
478C22#301311yescharacteristic
479C22#301911yescharacteristic
480C22#302011yescharacteristic
481C4—22no—
482C4—21no—
483C4—21no—
484C4—22no—
485C4—82no—
486C4—82no—
487C4—81no—
488C4—84no—
489C4—81no—
490C4—82no—
491C4—81no—
492C4—81no—
493C4—82no—
494C4#202614yes—
495C4C2×#47911yescharacteristic
496C4C2×#47911yescharacteristic
497C4#308311yescharacteristic
498C4#308411yescharacteristic
499C4C2×#48811yescharacteristic
500C4C2×#49111yescharacteristic
501C4#313611yescharacteristic
502C4#313711yescharacteristic
503C2—24no—
504C2#1887511yescharacteristic
505C2#1887711yescharacteristic
506C2#1888711yescharacteristic
507C2#1888911yescharacteristic
508C2C2×#202611yescharacteristic
509C2#2345811yescharacteristic
510C2#2346211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10356C2semidirect
6. #10355C2semidirect
7. #10353C2semidirect
8. #10352C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#218852
#218602
#213112
#103562
#103552
#103532
#103522
C2×#12962
#78932
#78912