Finite Group Databasebeta

← #99726

Subgroups — #99726

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊C8C211yesmaximal, characteristic
2#14702C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1272C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—21no—
9C22×C22⋊C8—21no—
10C22×C4×C8—21no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#380C411yescharacteristic
14C4×C22⋊C8—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×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—24no—
52C2×C22⋊C8—24no—
53C2×C22⋊C8—42no—
54C2×C22⋊C8—42no—
55C2×C4×C8—24no—
56C2×C4×C8—42no—
57C2×C4×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—22no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—22no—
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×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C2×C812yes—
113C23×C4M1612yes—
114C23×C4SD1612yes—
115C23×C4Q1612yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
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⋊Q8—48no—
134C22⋊Q8—48no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4C2×C812yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—48no—
163C2×C42—48no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C2×D412yes—
169C2×C42#4112yes—
170C22⋊C8—216no—
171C22⋊C8—48no—
172C4×C8—48no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—41no—
179#95—42no—
180#95—41no—
181#95—42no—
182#95—42no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24C22⋊C812yes—
196C24#10312yes—
197C24#10412yes—
198C24C2×C22⋊C411yescharacteristic
199C2×Q8—24no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
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—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
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×C4C22⋊C812yes—
294C22×C4#9912yes—
295C22×C4#9912yes—
296C22×C4#10012yes—
297C22×C4#10312yes—
298C22×C4#10412yes—
299C22×C4#10412yes—
300C22×C4C4×D412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12112yes—
303C22×C4#12311yescharacteristic
304C22×C4C22×C811yescharacteristic
305C22×C4C2×M1611yescharacteristic
306C22×C4#13112yes—
307C22×C4C2×SD1611yescharacteristic
308C22×C4C2×Q1611yescharacteristic
309C22×C4#13512yes—
310C22×C4#13512yes—
311C22×C4#13712yes—
312C2×C8—28no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8—82no—
325C2×C8—82no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—48no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C22⋊C4—48no—
332C22⋊C4—84no—
333C42—24no—
334C42—24no—
335C42—42no—
336C42—42no—
337C42—44no—
338C42—44no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23Q8⋊C814yes—
374C23#38611yescharacteristic
375C23C2×C22⋊C811yescharacteristic
376C23#40811yescharacteristic
377C23C2×#9911yescharacteristic
378C23C2×#9911yescharacteristic
379C23C2×#10011yescharacteristic
380C23#41311yescharacteristic
381C23C2×#10311yescharacteristic
382C23#41611yescharacteristic
383C23#41611yescharacteristic
384C23#41911yescharacteristic
385C23C2×#10411yescharacteristic
386C23C2×#10411yescharacteristic
387C23C2×#10411yescharacteristic
388C23#42111yescharacteristic
389Q8—84no—
390Q8—84no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
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—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#32314yes—
469C2×C4#32814yes—
470C2×C4C8×D412yes—
471C2×C4#43512yes—
472C2×C4#45212yes—
473C2×C4#46512yes—
474C2×C4#46712yes—
475C2×C4#48512yes—
476C8—48no—
477C8—84no—
478C22—24no—
479C22—24no—
480C22—24no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22C2×#32311yescharacteristic
494C22#145911yescharacteristic
495C22#146811yescharacteristic
496C22#147111yescharacteristic
497C22C2×Q8⋊C811yescharacteristic
498C22#147611yescharacteristic
499C22#147611yescharacteristic
500C22#147911yescharacteristic
501C22#148211yescharacteristic
502C22#148511yescharacteristic
503C22#149311yescharacteristic
504C22#149411yescharacteristic
505C22C2×#32811yescharacteristic
506C22#150111yescharacteristic
507C22#150111yescharacteristic
508C22#150111yescharacteristic
509C22#150811yescharacteristic
510C22#150811yescharacteristic
511C22#151611yescharacteristic
512C22#152011yescharacteristic
513C22#152411yescharacteristic
514C22#152511yescharacteristic
515C22#152711yescharacteristic
516C22#152911yescharacteristic
517C22#154211yescharacteristic
518C22#154311yescharacteristic
519C22#154811yescharacteristic
520C22#155411yescharacteristic
521C22#186911yescharacteristic
522C22#187111yescharacteristic
523C22#187311yescharacteristic
524C22#189211yescharacteristic
525C22#189611yescharacteristic
526C22#189811yescharacteristic
527C22#190011yescharacteristic
528C4—24no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—24no—
533C4—22no—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—82no—
538C4—82no—
539C4—161no—
540C4—161no—
541C2—44no—
542C2#756311yescharacteristic
543C2#756611yescharacteristic
544C2#757511yescharacteristic
545C2#758311yescharacteristic
546C2#766111yescharacteristic
547C2#767011yescharacteristic
548C2#767511yescharacteristic
549C2#768711yescharacteristic
550C2#893811yescharacteristic
551C2#907111yescharacteristic
552C2#907911yescharacteristic
553C2#911111yescharacteristic
554C2#912011yescharacteristic
555C2#912811yescharacteristic
556C2#914611yescharacteristic
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×C4×C22⋊C82
#147022
#107122
C2×#12722
#70252