Finite Group Databasebeta

← #189658

Subgroups — #189658

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2335C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3#21930C211yesmaximal, characteristic, complemented
4C2×#1468C212yesmaximal
5#8562C212yesmaximal, complemented
6#8561C212yesmaximal, complemented
7#8560C212yesmaximal, complemented
8#8557C212yesmaximal, complemented
9#7915C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C2×#407C2212yes—
12C22×C22⋊C8C2212yes—
13#2335—24no—
14#2335—24no—
15C2×#390C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C411yescharacteristic
19C2×#388C411yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#386C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1851—24no—
30#1850—24no—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1468—28no—
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C23×D4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C23×C8—22no—
45C2×#123—22no—
46C2×#123—22no—
47C2×#123—22no—
48C2×#123—22no—
49C2×#123D414yes—
50C2×#123Q814yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2311yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77#407—24no—
78#407—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2314yes—
84C2×C22⋊C8C2314yes—
85#390—24no—
86#389—24no—
87#388—24no—
88#388—24no—
89#388—22no—
90#388—22no—
91#388—24no—
92#387—22no—
93#387—22no—
94#387—24no—
95#386—24no—
96#385—24no—
97#380—22no—
98#380—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4D414yes—
104C4×C22⋊C4D414yes—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2311yescharacteristic
114C2×#95C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#324—28no—
120#323—28no—
121C25C22×C411yescharacteristic
122C25C22×C411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—81no—
126C22×D4—81no—
127C22×D4C4⋊C414yes—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—44no—
154C2×M16—82no—
155C2×M16—82no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8—44no—
159C22×C8—82no—
160C22×C8—82no—
161#123—28no—
162#123—28no—
163#123—28no—
164#123—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4—81no—
180C2×C4⋊C4—81no—
181C2×C4⋊C4C4⋊C414yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×Q812yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—81no—
201C2×C22⋊C4—81no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4—81no—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4C22⋊C414yes—
206C2×C22⋊C4C4⋊C414yes—
207C2×C22⋊C4C4⋊C414yes—
208C2×C22⋊C4C2×D412yes—
209C2×C22⋊C4C2×Q812yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42—81no—
217C2×C42—81no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C4⋊C8—82no—
223C4⋊C8—82no—
224C4⋊C8—82no—
225C4⋊C8—82no—
226C22⋊C8—24no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95C22⋊C414yes—
243#95C22⋊C414yes—
244C24—22no—
245C24—22no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24C2×C22⋊C412yes—
259C24C2×C4⋊C412yes—
260C24C2×C4⋊C412yes—
261C24#11712yes—
262C24C23×C411yescharacteristic
263C2×D4—28no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—84no—
267C2×D4—84no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—44no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—88no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C4⋊C412yes—
329C22×C4C2×C4⋊C412yes—
330C22×C4C2×C4⋊C412yes—
331C22×C4C2×C4⋊C412yes—
332C22×C4#11712yes—
333C22×C4#12114yes—
334C22×C4#12114yes—
335C22×C4C22⋊Q814yes—
336C22×C4C22⋊Q814yes—
337C22×C4#13114yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q811yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344M16—88no—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—88no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—28no—
357C4⋊C4—44no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C22⋊C4—28no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—28no—
368C22⋊C4—28no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—44no—
380C42—82no—
381C42—82no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23#38014yes—
420C23C22×C22⋊C411yescharacteristic
421C23C22×C4⋊C411yescharacteristic
422C23C2×#11711yescharacteristic
423C23C2×#13112yes—
424C23#56812yes—
425D4—88no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—88no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#38014yes—
490C2×C4#40814yes—
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×C22⋊Q811yescharacteristic
495C8—84no—
496C8—84no—
497C22—22no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22C2×#38011yescharacteristic
524C22C2×#40811yescharacteristic
525C22#287911yescharacteristic
526C22#288311yescharacteristic
527C22#288311yescharacteristic
528C22#288811yescharacteristic
529C22#288911yescharacteristic
530C22#290612yes—
531C22#292512yes—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—81no—
545C4—81no—
546C4—81no—
547C4—81no—
548C2—22no—
549C2—22no—
550C2—44no—
551C2—82no—
552C2#1738611yescharacteristic
553C2#1738811yescharacteristic
554C2#1740611yescharacteristic
555C2#1740811yescharacteristic
556C2#2062911yescharacteristic
557C2#2063311yescharacteristic
558C2#2064011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21934C2semidirect
3. #21930C2semidirect
5. #8562C2semidirect
6. #8561C2semidirect
7. #8560C2semidirect
8. #8557C2semidirect
9. #7915C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23352
#219342
#219302
C2×#14682
#85622
#85612
#85602
#85572
#79152