Finite Group Databasebeta

← #192800

Subgroups — #192800

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34373C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#21982C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic
5C2×#1497C211yesmaximal, characteristic
6#8892C212yesmaximal, complemented
7#8891C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic
11#7894C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2378—22no—
17#2362—21no—
18#2362—21no—
19#2355—21no—
20#2355—21no—
21#2355C2211yescharacteristic
22#2355C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25#2336—22no—
26#2331—21no—
27#2327—21no—
28#2314—21no—
29#2314—21no—
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2283C2211yescharacteristic
33#1990—21no—
34#1956—21no—
35#1948C2212yes—
36#1947—22no—
37#1947C2212yes—
38#1872—21no—
39#1872—21no—
40#1853—21no—
41#1852—22no—
42#1852—21no—
43#1847—21no—
44#1845C2212yes—
45#1836—21no—
46#1834C2212yes—
47#1803C2212yes—
48#1803C2212yes—
49#1497C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D411yescharacteristic
65C2×C22⋊Q8D411yescharacteristic
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432C2×C414yes—
79#431—22no—
80#431—22no—
81#431—22no—
82#431C2×C414yes—
83C2×#107—24no—
84C2×#107—22no—
85C2×#106—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#391—22no—
98#391—22no—
99#391—21no—
100#391—21no—
101#389—21no—
102#389—21no—
103#389—22no—
104#389—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2312yes—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#383—21no—
128#382—21no—
129#382—21no—
130#382—21no—
131#382D411yescharacteristic
132#382D411yescharacteristic
133#381—21no—
134#380—21no—
135#380D411yescharacteristic
136#380D411yescharacteristic
137#380D412yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4D412yes—
148C4×C4⋊C4C2311yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C22⋊C4—21no—
151#376—21no—
152#340—22no—
153#340—22no—
154#340—24no—
155#340—22no—
156#340—22no—
157Q8⋊C8C2×C418yescomplemented
158Q8⋊C8C2×C418yescomplemented
159C22×Q8—21no—
160C22×Q8C2×D411yescharacteristic
161C22×Q8C2×D411yescharacteristic
162C23×C4—41no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8C4⋊C414yes—
172C4⋊Q8C4⋊C414yes—
173#125—24no—
174#125—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8C4⋊C414yes—
180C22⋊Q8C4⋊C414yes—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—82no—
188C4×Q8—82no—
189C4×Q8C22×C412yes—
190C4×Q8C22×C412yes—
191C4×Q8C2×Q812yes—
192C4×Q8C2×Q812yes—
193#117—24no—
194#117—24no—
195#117—82no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C22×C412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250#107—82no—
251#107—84no—
252#106—82no—
253C4⋊C8—24no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8C22×C414yes—
258#103—82no—
259#103—82no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—42no—
263#97—24no—
264C4×C8—22no—
265C4×C8—42no—
266C4×C8C22×C414yes—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—21no—
285C24#12011yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×Q8C2×C4⋊C412yes—
296C2×Q8C2×C4⋊C412yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4C2×C4⋊C412yes—
318C22×C4C4×D412yes—
319C22×C4C4×D412yes—
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4#12312yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4C2×C4⋊C412yes—
372C4⋊C4C2×C4⋊C412yes—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—81no—
376C22⋊C4—81no—
377C22⋊C4—82no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—42no—
384C42—82no—
385C42—82no—
386C42—82no—
387C42—82no—
388C42—82no—
389C42C2×C4⋊C412yes—
390C42C4×D412yes—
391C42C4×D412yes—
392C42C22⋊Q812yes—
393C42C22⋊Q812yes—
394C42#12312yes—
395C42C23×C411yescharacteristic
396C42C22×Q811yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23#38512yes—
406C23C2×#12011yescharacteristic
407Q8—24no—
408Q8—24no—
409Q8—42no—
410Q8—42no—
411Q8—84no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—164no—
460C2×C4#38512yes—
461C2×C4#38512yes—
462C2×C4#38512yes—
463C2×C4C22×C4⋊C411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C22⋊Q811yescharacteristic
467C2×C4C2×C22⋊Q811yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4#57612yes—
470C2×C4#57712yes—
471C2×C4#57812yes—
472C2×C4#57812yes—
473C8—44no—
474C8—44no—
475C8—82no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22C2×#38511yescharacteristic
484C22#300811yescharacteristic
485C22#301011yescharacteristic
486C22#301111yescharacteristic
487C22#301111yescharacteristic
488C22#302111yescharacteristic
489C22#302211yescharacteristic
490C4—22no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—22no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—81no—
500C4—82no—
501C4—81no—
502C4—81no—
503C4—82no—
504C4—81no—
505C4—162no—
506C4#291312yes—
507C4#294712yes—
508C4#308212yes—
509C4#309812yes—
510C2—42no—
511C2#1895811yescharacteristic
512C2#1896111yescharacteristic
513C2#1897011yescharacteristic
514C2#1897311yescharacteristic
515C2#2084711yescharacteristic
516C2#2086511yescharacteristic
517C2#2087711yescharacteristic
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. #8892C2semidirect
7. #8891C2semidirect
8. #8884C2semidirect
9. #8883C2semidirect
49. #1497C4semidirect
157. Q8⋊C8C2×C4semidirect
158. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343732
#219872
#219822
#218602
C2×#14972
#88922
#88912
#88842
#88832
#80462
#78942