Finite Group Databasebeta

← #119997

Subgroups — #119997

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21930C211yesmaximal, characteristic
2#13782C211yesmaximal, characteristic
3C2×#1282C212yesmaximal
4#7907C211yesmaximal, characteristic
5C2×#1280C212yesmaximal
6C22×C22⋊C8C2212yes—
7C22×C22⋊C8C2212yes—
8C2×#385C411yescharacteristic
9C2×#385C411yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12C22×#95C2211yescharacteristic
13#1850—24no—
14#1797—21no—
15#1797—21no—
16#1436—21no—
17#1436—21no—
18C4⋊#95—21no—
19#1434—21no—
20C22⋊#95C411yescharacteristic
21C22⋊#95C411yescharacteristic
22#1282—24no—
23#1280—24no—
24#1280—21no—
25#1280—21no—
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4C2311yescharacteristic
29C23×C8—22no—
30C23×C8—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C42—21no—
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8D414yes—
53C2×C22⋊C8Q814yes—
54#385—22no—
55#385—22no—
56#385—22no—
57#385C814yescomplemented
58C4×C22⋊C4—22no—
59C4×C22⋊C4—21no—
60C4×C22⋊C4—22no—
61C4×C22⋊C4—21no—
62C4×C22⋊C4D412yes—
63C4×C22⋊C4Q812yes—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95D411yescharacteristic
76C2×#95D411yescharacteristic
77#336—22no—
78#336—22no—
79C25C22×C411yescharacteristic
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—81no—
91C23×C4—81no—
92C23×C4—81no—
93C23×C4—81no—
94C23×C4—81no—
95C23×C4—81no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C2×C812yes—
101C23×C4M1612yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q812yes—
107C23×C4C2×Q811yescharacteristic
108C23×C4#4112yes—
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4#4112yes—
112C22×C8—24no—
113C22×C8—24no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—82no—
119C22×C8—82no—
120C22×C8—82no—
121C22×C8—82no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—81no—
125C2×C4⋊C4—81no—
126C2×C4⋊C4—81no—
127C2×C4⋊C4—81no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4C4⋊C412yes—
141C2×C22⋊C4C4⋊C412yes—
142C2×C22⋊C4C2×C812yes—
143C2×C22⋊C4C2×C812yes—
144C2×C22⋊C4M1612yes—
145C2×C22⋊C4M1612yes—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—81no—
153C2×C42—81no—
154C2×C42—81no—
155C2×C42—81no—
156C4⋊C8—82no—
157C4⋊C8—82no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—24no—
162C22⋊C8—42no—
163C22⋊C8—42no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166#95—24no—
167#95—24no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—41no—
173#95—42no—
174#95—41no—
175#95—81no—
176#95—81no—
177#95—81no—
178#95—81no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24C2×C22⋊C411yescharacteristic
184C24C2×C4⋊C411yescharacteristic
185C24#11711yescharacteristic
186C24C4×D412yes—
187C24C4×Q812yes—
188C24C22×C811yescharacteristic
189C24C2×M1611yescharacteristic
190C24C2×M1611yescharacteristic
191C24C2×M1611yescharacteristic
192C24#13112yes—
193C24#13112yes—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—82no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C22⋊C812yes—
269C22×C4C22⋊C812yes—
270C22×C4#12011yescharacteristic
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4C22⋊Q812yes—
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q812yes—
278C22×C4C22⋊Q812yes—
279C22×C4C22⋊Q811yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12312yes—
282C22×C4#12411yescharacteristic
283C22×C4#12512yes—
284C22×C4#12512yes—
285C22×C4#12612yes—
286C22×C4C4⋊Q812yes—
287C22×C4C4⋊Q811yescharacteristic
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—84no—
295C2×C8—84no—
296C2×C8—84no—
297C2×C8—84no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4C4⋊C814yescomplemented
310C42—42no—
311C42—42no—
312C42—81no—
313C42—81no—
314C42—82no—
315C42—82no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—21no—
321C23—22no—
322C23—21no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23#38011yescharacteristic
343C23#38611yescharacteristic
344C23#38712yes—
345C23#38912yes—
346C23#39111yescharacteristic
347C23C2×C22⋊C811yescharacteristic
348C23#40711yescharacteristic
349C23#40811yescharacteristic
350C23#40811yescharacteristic
351C23C2×C4⋊C811yescharacteristic
352C23#42311yescharacteristic
353C23#42412yes—
354C23#43111yescharacteristic
355C23#43211yescharacteristic
356C23#43312yes—
357C23C8×D412yes—
358C23#43512yes—
359C23#43512yes—
360C23#43612yes—
361C23C8×Q812yes—
362C23#44612yes—
363C23#44612yes—
364C23#44612yes—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4#39411yescharacteristic
439C2×C4#39511yescharacteristic
440C2×C4#39612yes—
441C2×C4#39812yes—
442C2×C4#39811yescharacteristic
443C2×C4#40011yescharacteristic
444C8—84no—
445C8—84no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—21no—
463C22—22no—
464C22—22no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—21no—
470C22—41no—
471C22—41no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22#144111yescharacteristic
481C22#179711yescharacteristic
482C22#179811yescharacteristic
483C22#186911yescharacteristic
484C22#187011yescharacteristic
485C22#191311yescharacteristic
486C22#191311yescharacteristic
487C22#191411yescharacteristic
488C22#191511yescharacteristic
489C22#191612yes—
490C22#191712yes—
491C22#193912yes—
492C22#194012yes—
493C22#198411yescharacteristic
494C22#198511yescharacteristic
495C4—21no—
496C4—22no—
497C4—21no—
498C4—21no—
499C4—22no—
500C4—21no—
501C4—44no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—81no—
507C4—82no—
508C4—81no—
509C2—21no—
510C2—21no—
511C2—22no—
512C2—22no—
513C2—22no—
514C2—22no—
515C2—21no—
516C2—21no—
517C2#910711yescharacteristic
518C2#910811yescharacteristic
519C2#910911yescharacteristic
520C2#911011yescharacteristic
521C2#1120211yescharacteristic
522C2#1120511yescharacteristic
523C2#1120611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C22×C8
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
57. #385C8semidirect
309. C22⋊C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
#219302
#137822
C2×#12822
#79072
C2×#12802