Finite Group Databasebeta

← #116285

Subgroups — #116285

Order 512.

758 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic
5C2×#1303C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7#7892C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#382—21no—
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1990—21no—
17#1989—22no—
18#1988—21no—
19#1987—21no—
20#1956—21no—
21#1872—22no—
22#1803—22no—
23#1736—21no—
24#1736—21no—
25#1503—22no—
26#1503C414yescomplemented
27C2×#328—22no—
28#1442—22no—
29C4⋊#95—22no—
30C4⋊#95—21no—
31#1303—22no—
32#1296—22no—
33#1279—22no—
34C22×M16—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D412yes—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68#423—21no—
69#423—21no—
70#423—21no—
71#423—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75C2×C4⋊C8D412yes—
76C2×#103—24no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#382—42no—
86#382—42no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4D412yes—
95C2×#95—24no—
96C2×#95—21no—
97C2×#95—24no—
98C2×#95D412yes—
99#340—24no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#328—28no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—41no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C4⋊C412yes—
130C4⋊Q8C4⋊C412yes—
131#125—22no—
132#125—22no—
133#125—41no—
134#125—41no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151#117—22no—
152#117—22no—
153#117—41no—
154#117—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—41no—
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—82no—
187C2×C4⋊C4SD1612yes—
188C2×C4⋊C4Q1612yes—
189C2×C22⋊C4—41no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—44no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220#103—84no—
221C22⋊C8—22no—
222C22⋊C8—41no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225C4×C8—22no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229#95—22no—
230#95—48no—
231#95—41no—
232#95—41no—
233#95—84no—
234#95C22⋊C414yes—
235C24#12011yescharacteristic
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—81no—
243C2×Q8—81no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4C22⋊Q811yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12411yescharacteristic
289C22×C4#12711yescharacteristic
290C22×C4C2×SD1611yescharacteristic
291C22×C4C2×Q1611yescharacteristic
292C22×C4#13511yescharacteristic
293C22×C4#13511yescharacteristic
294C22×C4AGL(1,ℤ/8)11yescharacteristic
295C22×C4#13711yescharacteristic
296C22×C4#13711yescharacteristic
297C22×C4#13711yescharacteristic
298M16—81no—
299M16—81no—
300M16—81no—
301M16—81no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—81no—
310C2×C8—81no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—44no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—82no—
350C42—82no—
351C42C2×C4⋊C411yescharacteristic
352C42#11711yescharacteristic
353C42C4×D411yescharacteristic
354C42C4×D411yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42#12111yescharacteristic
358C42C22⋊Q811yescharacteristic
359C42C22⋊Q811yescharacteristic
360C42C22⋊Q811yescharacteristic
361C42#12311yescharacteristic
362C42#12311yescharacteristic
363C42#12311yescharacteristic
364C42#12311yescharacteristic
365C42#12611yescharacteristic
366C42#12611yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#38611yescharacteristic
372C23#39211yescharacteristic
373C23#39311yescharacteristic
374C23#45011yescharacteristic
375C23#45111yescharacteristic
376C23#45211yescharacteristic
377C23#45211yescharacteristic
378Q8—44no—
379Q8—82no—
380Q8—82no—
381Q8—82no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#38011yescharacteristic
425C2×C4#38511yescharacteristic
426C2×C4#38511yescharacteristic
427C2×C4#38811yescharacteristic
428C2×C4#38811yescharacteristic
429C2×C4#39011yescharacteristic
430C2×C4#39411yescharacteristic
431C2×C4#39411yescharacteristic
432C2×C4#39611yescharacteristic
433C2×C4#39611yescharacteristic
434C2×C4#39711yescharacteristic
435C2×C4#39911yescharacteristic
436C2×C4C4×SD1611yescharacteristic
437C2×C4C4×Q1611yescharacteristic
438C2×C4#44011yescharacteristic
439C2×C4#44111yescharacteristic
440C2×C4#44811yescharacteristic
441C2×C4#44911yescharacteristic
442C2×C4#45111yescharacteristic
443C2×C4#45211yescharacteristic
444C2×C4#45412yes—
445C2×C4#45512yes—
446C2×C4#46111yescharacteristic
447C2×C4#46211yescharacteristic
448C2×C4#46311yescharacteristic
449C2×C4#46411yescharacteristic
450C2×C4#46511yescharacteristic
451C2×C4#46711yescharacteristic
452C2×C4#46811yescharacteristic
453C2×C4#47011yescharacteristic
454C2×C4#47511yescharacteristic
455C2×C4#47711yescharacteristic
456C2×C4#47911yescharacteristic
457C2×C4#47911yescharacteristic
458C2×C4#48711yescharacteristic
459C2×C4#48811yescharacteristic
460C2×C4#48911yescharacteristic
461C2×C4#49111yescharacteristic
462C8—82no—
463C8—82no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22#143911yescharacteristic
478C22#187711yescharacteristic
479C22#187811yescharacteristic
480C22#200011yescharacteristic
481C22#200111yescharacteristic
482C22#200311yescharacteristic
483C22#200411yescharacteristic
484C22#200811yescharacteristic
485C22#200911yescharacteristic
486C22#202411yescharacteristic
487C22#202611yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—84no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—162no—
500C4#186012yes—
501C4#186311yescharacteristic
502C4#186511yescharacteristic
503C4#193311yescharacteristic
504C4#193511yescharacteristic
505C4#204012yes—
506C4#204211yescharacteristic
507C4#204311yescharacteristic
508C4#206611yescharacteristic
509C4#206811yescharacteristic
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2#1024711yescharacteristic
515C2#1024911yescharacteristic
516C2#1025011yescharacteristic
517C2#1025211yescharacteristic
518C2#1106711yescharacteristic
519C2#1107911yescharacteristic
520C2#1109311yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
26. #1503C4semidirect

Maximal subgroups

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

subgroupindex
#218602
#213122
C2×#15032
#139112
C2×#13032
#80432
#78922