Finite Group Databasebeta

← #99739

Subgroups — #99739

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2#14702C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1272C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C4⋊C8—21no—
9C22×C22⋊C8—21no—
10C22×C22⋊C8—21no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#380C411yescharacteristic
14#1797—24no—
15C2×#336—21no—
16C2×#336C2212yes—
17C2×#336C2212yes—
18#1436—21no—
19#1436—21no—
20C4⋊#95C411yescharacteristic
21C4⋊#95C2211yescharacteristic
22C22⋊#95—21no—
23C22⋊#95—21no—
24#1272—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C24×C4D411yescharacteristic
29C23×C8—22no—
30C23×C8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—42no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8C814yescomplemented
35C22×C4⋊C4—21no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C22⋊C4—21no—
40C22×C22⋊C4—21no—
41C22×C42—22no—
42C22×C42—22no—
43C22×C42—42no—
44C22×C42—42no—
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—24no—
50C2×C4⋊C8—42no—
51C2×C4⋊C8—42no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—24no—
54C2×C22⋊C8—42no—
55C2×C22⋊C8—42no—
56C2×C22⋊C8—42no—
57C2×C22⋊C8—42no—
58#391—22no—
59#391—22no—
60#391—42no—
61#380—42no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71#336—22no—
72#336—22no—
73#336—22no—
74#336—42no—
75#336D414yes—
76C25C22⋊C411yescharacteristic
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8C2×C812yes—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C2×C812yes—
115C23×C4M1612yes—
116C23×C4SD1612yes—
117C23×C4Q1612yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C22×C8—24no—
122C22×C8—24no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22⋊Q8—48no—
136C22⋊Q8—48no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4C2×C812yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—48no—
165C2×C42—48no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C2×D412yes—
171C2×C42#4112yes—
172C4⋊C8—48no—
173C22⋊C8—48no—
174C22⋊C8—48no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179#95—42no—
180#95—41no—
181#95—42no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24C22⋊C812yes—
200C24#10312yes—
201C24#10412yes—
202C24C2×C22⋊C411yescharacteristic
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—82no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4C22⋊C812yes—
306C22×C4#9912yes—
307C22×C4#9912yes—
308C22×C4#10012yes—
309C22×C4#10312yes—
310C22×C4C4×D412yes—
311C22×C4#12011yescharacteristic
312C22×C4#12112yes—
313C22×C4#12311yescharacteristic
314C22×C4C22×C811yescharacteristic
315C22×C4C2×M1611yescharacteristic
316C22×C4#13112yes—
317C22×C4C2×SD1611yescharacteristic
318C22×C4C2×Q1611yescharacteristic
319C22×C4AGL(1,ℤ/8)11yescharacteristic
320C22×C4AGL(1,ℤ/8)11yescharacteristic
321C22×C4#13711yescharacteristic
322C22×C4#13712yes—
323C22×C4#13711yescharacteristic
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8—82no—
333C2×C8—82no—
334C2×C8—82no—
335C2×C8—82no—
336C2×C8—82no—
337C2×C8—82no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C22⋊C4—48no—
344C22⋊C4—84no—
345C42—24no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—44no—
350C42—44no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23Q8⋊C814yes—
386C23#38611yescharacteristic
387C23C2×C22⋊C811yescharacteristic
388C23#40811yescharacteristic
389C23C2×#9911yescharacteristic
390C23C2×#9911yescharacteristic
391C23C2×#10011yescharacteristic
392C23#41311yescharacteristic
393C23C2×#10311yescharacteristic
394C23#41711yescharacteristic
395C23#41711yescharacteristic
396C23#41911yescharacteristic
397C23C2×#10411yescharacteristic
398C23#42111yescharacteristic
399C23#42111yescharacteristic
400C23#42111yescharacteristic
401Q8—84no—
402Q8—84no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4#32314yes—
488C2×C4C8×D412yes—
489C2×C4#43512yes—
490C2×C4#45011yescharacteristic
491C2×C4#45111yescharacteristic
492C2×C4#46812yes—
493C2×C4#47012yes—
494C2×C4#48311yescharacteristic
495C2×C4#48411yescharacteristic
496C8—84no—
497C8—84no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22C2×#32311yescharacteristic
514C22#145911yescharacteristic
515C22#146811yescharacteristic
516C22#147111yescharacteristic
517C22C2×Q8⋊C811yescharacteristic
518C22#147711yescharacteristic
519C22#147711yescharacteristic
520C22#147911yescharacteristic
521C22#148111yescharacteristic
522C22#148311yescharacteristic
523C22#149511yescharacteristic
524C22#149711yescharacteristic
525C22#150411yescharacteristic
526C22#150811yescharacteristic
527C22#151111yescharacteristic
528C22#151311yescharacteristic
529C22#151311yescharacteristic
530C22#151311yescharacteristic
531C22#152011yescharacteristic
532C22#152011yescharacteristic
533C22#152711yescharacteristic
534C22#153211yescharacteristic
535C22#153511yescharacteristic
536C22#153611yescharacteristic
537C22#154811yescharacteristic
538C22#155011yescharacteristic
539C22#155911yescharacteristic
540C22#156011yescharacteristic
541C22#186911yescharacteristic
542C22#187111yescharacteristic
543C22#187211yescharacteristic
544C22#189411yescharacteristic
545C22#189711yescharacteristic
546C22#189811yescharacteristic
547C22#190011yescharacteristic
548C4—24no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—161no—
562C4—161no—
563C2—44no—
564C2#757511yescharacteristic
565C2#758911yescharacteristic
566C2#759711yescharacteristic
567C2#760011yescharacteristic
568C2#770111yescharacteristic
569C2#771011yescharacteristic
570C2#771511yescharacteristic
571C2#772711yescharacteristic
572C2#893711yescharacteristic
573C2#907311yescharacteristic
574C2#907911yescharacteristic
575C2#911511yescharacteristic
576C2#912311yescharacteristic
577C2#913511yescharacteristic
578C2#915111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1272C2semidirect
5. #7025C2semidirect
34. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
#147022
#107122
C2×#12722
#70252