Finite Group Databasebeta

← #121028

Subgroups — #121028

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13908C211yesmaximal, characteristic
2#10854C211yesmaximal, characteristic
3#8023C211yesmaximal, characteristic
4C2×#1294C211yesmaximal, characteristic
5C2×#1294C211yesmaximal, characteristic
6C2×#1289C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C22×C22⋊C8C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×#380—21no—
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15C22×#95C2211yescharacteristic
16C2×#342—21no—
17C2×#342—21no—
18#1444—21no—
19#1442—21no—
20#1436—22no—
21C22⋊#95—21no—
22#1294—21no—
23#1294—21no—
24#1294—22no—
25#1294C414yescomplemented
26#1289—22no—
27#1282—22no—
28C24×C4—21no—
29C24×C4C2311yescharacteristic
30C23×C8—21no—
31C23×C8—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4Q812yes—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4D411yescharacteristic
40C22×C22⋊C4D411yescharacteristic
41C22×C42—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8C2×C412yes—
53C2×C22⋊C8D412yes—
54#380—21no—
55#380—21no—
56#380—21no—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65#380—42no—
66#380—42no—
67#380C2×C412yes—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95C2×C412yes—
82C2×#95D412yes—
83C2×#95D411yescharacteristic
84C2×#95D411yescharacteristic
85C2×#95Q812yes—
86#342—42no—
87#342—42no—
88C25C2×D411yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—44no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×Q811yescharacteristic
110C23×C4C2×Q811yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C23×C4#4111yescharacteristic
114C23×C4#4111yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—82no—
126C22×C8—82no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4—82no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4D812yes—
168C2×C22⋊C4SD1612yes—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—82no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180#95—22no—
181#95—22no—
182#95—21no—
183#95—21no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—44no—
188#95—44no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194#95—82no—
195#95C4⋊C412yes—
196#95C4⋊C412yes—
197C24—42no—
198C24—42no—
199C24C4×D411yescharacteristic
200C24#12011yescharacteristic
201C24#12111yescharacteristic
202C24C22⋊Q811yescharacteristic
203C24C22⋊Q811yescharacteristic
204C24#12311yescharacteristic
205C24#12411yescharacteristic
206C24C2×D811yescharacteristic
207C24C2×SD1611yescharacteristic
208C24#13511yescharacteristic
209C24#13511yescharacteristic
210C24AGL(1,ℤ/8)11yescharacteristic
211C24#13711yescharacteristic
212C24#13711yescharacteristic
213C24#13711yescharacteristic
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4#11711yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×Q811yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4C22⋊Q811yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4#12311yescharacteristic
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12411yescharacteristic
295C22×C4#12511yescharacteristic
296C22×C4#12511yescharacteristic
297C22×C4#12511yescharacteristic
298C22×C4#12511yescharacteristic
299C22×C4#12511yescharacteristic
300C22×C4#12611yescharacteristic
301C22×C4#12611yescharacteristic
302C22×C4#12611yescharacteristic
303C22×C4#12611yescharacteristic
304C22×C4C4⋊Q811yescharacteristic
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—82no—
308C2×C8—82no—
309C2×C8—82no—
310C2×C8—82no—
311C2×C8—82no—
312C2×C8—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—88no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—81no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C42—82no—
341C42—82no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23#38511yescharacteristic
365C23#38711yescharacteristic
366C23#38811yescharacteristic
367C23#39311yescharacteristic
368C23#39411yescharacteristic
369C23#39811yescharacteristic
370C23#40011yescharacteristic
371C23C4×D811yescharacteristic
372C23C4×SD1611yescharacteristic
373C23#44011yescharacteristic
374C23#44111yescharacteristic
375C23#44711yescharacteristic
376C23#44811yescharacteristic
377C23#45211yescharacteristic
378C23#45211yescharacteristic
379C23#46511yescharacteristic
380C23#46611yescharacteristic
381C23#46811yescharacteristic
382C23#47011yescharacteristic
383C23#47411yescharacteristic
384C23#47411yescharacteristic
385C23#47511yescharacteristic
386C23#47611yescharacteristic
387C23#47811yescharacteristic
388C23#47911yescharacteristic
389C23#47911yescharacteristic
390C23#47911yescharacteristic
391C23#48011yescharacteristic
392C23#48211yescharacteristic
393C23#48311yescharacteristic
394C23#48511yescharacteristic
395C23#48611yescharacteristic
396C23#48811yescharacteristic
397C23#48911yescharacteristic
398C23#49111yescharacteristic
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—162no—
465C2×C4—162no—
466C2×C4—162no—
467C2×C4—162no—
468C2×C4#38211yescharacteristic
469C2×C4#38711yescharacteristic
470C2×C4#38811yescharacteristic
471C2×C4#38911yescharacteristic
472C2×C4#39711yescharacteristic
473C2×C4#39811yescharacteristic
474C2×C4#40011yescharacteristic
475C2×C4#40011yescharacteristic
476C8—82no—
477C8—82no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22#144211yescharacteristic
519C22#186211yescharacteristic
520C22#186511yescharacteristic
521C22#191811yescharacteristic
522C22#192011yescharacteristic
523C22#193411yescharacteristic
524C22#193511yescharacteristic
525C22#202311yescharacteristic
526C22#202611yescharacteristic
527C22#203112yes—
528C22#203212yes—
529C22#203311yescharacteristic
530C22#203511yescharacteristic
531C22#208611yescharacteristic
532C22#208811yescharacteristic
533C22#209911yescharacteristic
534C22#210111yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—162no—
548C4—162no—
549C4—162no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2#1037412yes—
563C2#1037712yes—
564C2#1126211yescharacteristic
565C2#1127911yescharacteristic
566C2#1129211yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ 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
25. #1294C4semidirect

Maximal subgroups

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

subgroupindex
#139082
#108542
#80232
C2×#12942
C2×#12892
C2×#12822