Finite Group Databasebeta

← #192649

Subgroups — #192649

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic
3#20797C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C22×Q8⋊C8C211yesmaximal, characteristic
6#8888C212yesmaximal, complemented
7#8887C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic
11#7898C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#427—22no—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2355—22no—
18#2355C2212yes—
19Q8×C4⋊C4C2212yes—
20#2344—22no—
21#2344—21no—
22Q8×C22⋊C4—21no—
23#2320—22no—
24#2320—21no—
25#2285—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2283—21no—
29#2283C2211yescharacteristic
30#2276—21no—
31#1949—22no—
32#1948C2212yes—
33#1947C2212yes—
34#1846—22no—
35#1845C2212yes—
36#1844C2212yes—
37#1823—21no—
38#1808—21no—
39#1807—21no—
40#1803C2212yes—
41#1803C2212yes—
42#1787—21no—
43#1763—21no—
44#1763—21no—
45C2×Q8⋊C8C418yescomplemented
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2211yescharacteristic
49C2×Q8⋊C8C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8D411yescharacteristic
55C23×C8—21no—
56C23×C8—21no—
57C2×C4×Q8—22no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8Q814yes—
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×#117—22no—
69C2×#117—21no—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#427—48no—
78C2×#107—24no—
79C2×#107—22no—
80C2×#106—24no—
81C2×#106—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yes—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—22no—
89C2×#103—22no—
90C2×C22⋊C8—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8C2×C414yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—24no—
98#391—22no—
99#389—22no—
100#389—24no—
101#387—24no—
102#387—24no—
103#387—24no—
104#384—22no—
105#384C2312yes—
106#384C2312yes—
107#383—22no—
108#383—21no—
109#382—22no—
110#382—22no—
111#382—21no—
112#382D412yes—
113#382D412yes—
114#381—22no—
115#381—21no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380D411yescharacteristic
122#380D411yescharacteristic
123C4×C4⋊C4—22no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135#376—22no—
136#376—21no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141Q8⋊C8C2×C414yes—
142Q8⋊C8C2×C414yes—
143Q8⋊C8C2×C414yes—
144Q8⋊C8C2×C414yes—
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8C4⋊C414yes—
150C22×Q8C2×D411yescharacteristic
151C22×Q8C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C22×C8—24no—
158C22×C8—24no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—24no—
165C22×C8—22no—
166C22×C8—82no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×Q8C4⋊C414yes—
180C4×Q8C4⋊C414yes—
181C4×Q8C22×C412yes—
182C4×Q8C22×C412yes—
183C4×Q8C2×Q812yes—
184C4×Q8C2×Q812yes—
185#117—44no—
186#117—44no—
187#117—84no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4C4⋊C414yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—41no—
213C2×C42—24no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—24no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—41no—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42C2411yescharacteristic
232#107—48no—
233#107—82no—
234#106—48no—
235#106—82no—
236C4⋊C8—24no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243#103—44no—
244#103—44no—
245C22⋊C8—42no—
246#97—42no—
247C4×C8—24no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C22×C412yes—
251C4×C8C22×C412yes—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—41no—
264#95—41no—
265C24#12011yescharacteristic
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—28no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—48no—
276C2×Q8C2×C4⋊C412yes—
277C2×Q8C2×C4⋊C412yes—
278C2×Q8#13514yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×C4⋊C412yes—
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4#12312yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—216no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—84no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4C2×C4⋊C412yes—
351C4⋊C4C2×C4⋊C412yes—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—82no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—24no—
362C42—48no—
363C42—42no—
364C42—42no—
365C42—82no—
366C42—82no—
367C42—84no—
368C42C2×C4⋊C412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C42#12312yes—
374C42C23×C411yescharacteristic
375C42C22×Q811yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C23—22no—
379C23—22no—
380C23#38512yes—
381C23#45312yes—
382C23#45612yes—
383C23C2×#12011yescharacteristic
384Q8—24no—
385Q8—22no—
386Q8—22no—
387Q8—22no—
388Q8—48no—
389Q8—42no—
390Q8#42714yescomplemented
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4#38512yes—
430C2×C4#38512yes—
431C2×C4#38512yes—
432C2×C4C22×C4⋊C411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4C2×#12311yescharacteristic
438C2×C4C2×#13512yes—
439C2×C4#57512yes—
440C2×C4#57712yes—
441C2×C4#57812yes—
442C8—24no—
443C8—24no—
444C8—42no—
445C8—42no—
446C8—82no—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—24no—
451C22#185914yes—
452C22C2×#38511yescharacteristic
453C22#300212yes—
454C22#301312yes—
455C22C2×#45311yescharacteristic
456C22C2×#45611yescharacteristic
457C4—22no—
458C4—24no—
459C4—21no—
460C4—21no—
461C4—22no—
462C4—22no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4—162no—
470C4C2×#42711yescharacteristic
471C4#291111yescharacteristic
472C4#294911yescharacteristic
473C4#295011yescharacteristic
474C4#308212yes—
475C4#309112yes—
476C2—24no—
477C2#1861912yes—
478C2#1864912yes—
479C2#2084311yescharacteristic
480C2C2×#185911yescharacteristic
481C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ 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
6. #8888C2semidirect
7. #8887C2semidirect
8. #8884C2semidirect
9. #8883C2semidirect
45. C2×Q8⋊C8C4semidirect
390. Q8#427semidirect

Maximal subgroups

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

subgroupindex
#341962
#219622
#207972
#207402
C22×Q8⋊C82
#88882
#88872
#88842
#88832
#80442
#78982