Finite Group Databasebeta

← #118558

Subgroups — #118558

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21982C211yesmaximal, characteristic
2#20647C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4C2×#1461C211yesmaximal, characteristic
5C2×#1301C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7893C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×#419—21no—
14C22×#103—21no—
15#2285C2211yescharacteristic
16C2×#384C2211yescharacteristic
17#1947—22no—
18#1852—22no—
19#1845—21no—
20#1834—21no—
21#1805—22no—
22#1803—22no—
23#1803—22no—
24#1803—21no—
25#1803—21no—
26#1736—21no—
27C2×#340—21no—
28#1461—22no—
29#1461C414yescomplemented
30C2×#324—22no—
31#1301—22no—
32#1296—22no—
33#1279—22no—
34C23×Q8D411yescharacteristic
35C22×M16—21no—
36C23×C8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8Q812yes—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431C2×C412yes—
64#431D412yes—
65C2×#107—24no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D412yes—
79C2×C4⋊C8D412yes—
80#419—42no—
81#419—42no—
82C2×#103—24no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—42no—
86C2×#103—42no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8D412yes—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384D412yes—
97#382—21no—
98#380—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102#340—24no—
103#340—22no—
104#340—22no—
105#340—24no—
106#324—28no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8C4⋊C412yes—
119C22×Q8D812yes—
120C22×Q8Q1612yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—41no—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4#4111yescharacteristic
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—41no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C4⋊C412yes—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×Q811yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213#107—84no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226#103—48no—
227#103—44no—
228#103—44no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—41no—
232C22⋊C8C22⋊C414yes—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—41no—
237#95—41no—
238#95—41no—
239C24#12111yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8#10714yescomplemented
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4#11711yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4#12411yescharacteristic
306C22×C4#12611yescharacteristic
307C22×C4#12611yescharacteristic
308C22×C4#12711yescharacteristic
309C22×C4C2×D811yescharacteristic
310C22×C4C2×Q1611yescharacteristic
311C22×C4#13511yescharacteristic
312C22×C4#13511yescharacteristic
313C22×C4AGL(1,ℤ/8)11yescharacteristic
314C22×C4AGL(1,ℤ/8)11yescharacteristic
315C22×C4#13711yescharacteristic
316C22×C4#13711yescharacteristic
317M16—81no—
318M16—81no—
319M16—81no—
320M16—81no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—81no—
332C2×C8—81no—
333C2×C8—81no—
334C2×C8—81no—
335C2×C8—81no—
336C2×C8—81no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—82no—
361C22⋊C4—81no—
362C22⋊C4—81no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—81no—
368C42—81no—
369C42C4×D411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42C22⋊Q811yescharacteristic
380C42C22⋊Q811yescharacteristic
381C42#12311yescharacteristic
382C42#12311yescharacteristic
383C42#12411yescharacteristic
384C42#12411yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23#38011yescharacteristic
390C23#39211yescharacteristic
391C23#39711yescharacteristic
392C23#46611yescharacteristic
393C23#46711yescharacteristic
394C23#46911yescharacteristic
395C23#47011yescharacteristic
396Q8—24no—
397Q8—24no—
398Q8—42no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—162no—
440C2×C4—162no—
441C2×C4#38511yescharacteristic
442C2×C4#38511yescharacteristic
443C2×C4#38611yescharacteristic
444C2×C4#38811yescharacteristic
445C2×C4#38811yescharacteristic
446C2×C4#39011yescharacteristic
447C2×C4#39311yescharacteristic
448C2×C4#39411yescharacteristic
449C2×C4#39411yescharacteristic
450C2×C4#39611yescharacteristic
451C2×C4#39611yescharacteristic
452C2×C4#39911yescharacteristic
453C2×C4C2×#10711yescharacteristic
454C2×C4#42711yescharacteristic
455C2×C4#42811yescharacteristic
456C2×C4#42811yescharacteristic
457C2×C4#44711yescharacteristic
458C2×C4#44711yescharacteristic
459C2×C4#44911yescharacteristic
460C2×C4#44911yescharacteristic
461C2×C4#45111yescharacteristic
462C2×C4#45111yescharacteristic
463C2×C4#45211yescharacteristic
464C2×C4#45211yescharacteristic
465C2×C4#46611yescharacteristic
466C2×C4#46711yescharacteristic
467C2×C4#46811yescharacteristic
468C2×C4#46811yescharacteristic
469C2×C4#47112yes—
470C2×C4#47312yes—
471C2×C4#48011yescharacteristic
472C2×C4#48011yescharacteristic
473C2×C4#48211yescharacteristic
474C2×C4#48211yescharacteristic
475C2×C4#48411yescharacteristic
476C2×C4#48411yescharacteristic
477C2×C4#48511yescharacteristic
478C2×C4#48511yescharacteristic
479C8—44no—
480C8—82no—
481C8—82no—
482C8—82no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22#143911yescharacteristic
496C22#180811yescharacteristic
497C22#180911yescharacteristic
498C22#199911yescharacteristic
499C22#200111yescharacteristic
500C22#200811yescharacteristic
501C22#200911yescharacteristic
502C22#201911yescharacteristic
503C22#202011yescharacteristic
504C22#207511yescharacteristic
505C22#207611yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—44no—
513C4—82no—
514C4—161no—
515C4—162no—
516C4—161no—
517C4#185211yescharacteristic
518C4#185211yescharacteristic
519C4#185411yescharacteristic
520C4#185411yescharacteristic
521C4#193012yes—
522C4#203311yescharacteristic
523C4#203311yescharacteristic
524C4#203611yescharacteristic
525C4#203611yescharacteristic
526C4#206412yes—
527C2—21no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2#1017711yescharacteristic
532C2#1018311yescharacteristic
533C2#1018511yescharacteristic
534C2#1019111yescharacteristic
535C2#1109311yescharacteristic
536C2#1109611yescharacteristic
537C2#1114511yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1461C4semidirect
267. C2×Q8#107semidirect

Maximal subgroups

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

subgroupindex
#219822
#206472
C2×#18032
C2×#14612
C2×#13012
#80432
#78932