Finite Group Databasebeta

← #99755

Subgroups — #99755

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1817C211yesmaximal, 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—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#380C411yescharacteristic
13#1817—24no—
14C2×#336—21no—
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—22no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—42no—
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—42no—
52C2×C22⋊C8—42no—
53C2×C22⋊C8—42no—
54C2×C22⋊C8—42no—
55#391—22no—
56#391—22no—
57#391—42no—
58#380—42no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68#336—24no—
69#336—22no—
70#336—22no—
71#336—22no—
72#336—42no—
73#336D414yes—
74C25C22⋊C411yescharacteristic
75C22×Q8—21no—
76C22×Q8—22no—
77C22×Q8—22no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8M1612yes—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
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×C4SD1612yes—
113C23×C4Q1612yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C22×C8—24no—
118C22×C8—24no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8—42no—
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⋊Q8—48no—
132C22⋊Q8—48no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4M1612yes—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—48no—
165C2×C42—48no—
166C2×C42C2×D412yes—
167C2×C42#4112yes—
168C22⋊C8—48no—
169C22⋊C8—48no—
170#95—42no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95—41no—
176#95—42no—
177#95—41no—
178#95—42no—
179#95—42no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24#9912yes—
193C24#10312yes—
194C24#10412yes—
195C24C2×C22⋊C411yescharacteristic
196C2×Q8—24no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8—82no—
205C2×Q8—82no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
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—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
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—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
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—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
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×C4—41no—
294C22×C4#9912yes—
295C22×C4#10012yes—
296C22×C4#10312yes—
297C22×C4C4×D412yes—
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12311yescharacteristic
301C22×C4C2×M1611yescharacteristic
302C22×C4C2×M1611yescharacteristic
303C22×C4#13112yes—
304C22×C4C2×SD1611yescharacteristic
305C22×C4C2×Q1611yescharacteristic
306C22×C4#13512yes—
307C22×C4#13512yes—
308C22×C4#13712yes—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—82no—
316C2×C8—82no—
317C2×C8—82no—
318C2×C8—82no—
319C2×C8—82no—
320C2×C8—82no—
321C2×C8—82no—
322C2×C8—82no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C22⋊C4—48no—
329C22⋊C4—84no—
330C42—42no—
331C42—42no—
332C42—44no—
333C42—44no—
334C42—42no—
335C42—42no—
336C23—22no—
337C23—22no—
338C23—22no—
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—41no—
355C23—41no—
356C23—41no—
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#32814yes—
371C23#38611yescharacteristic
372C23#40711yescharacteristic
373C23#40811yescharacteristic
374C23C2×#9911yescharacteristic
375C23C2×#9911yescharacteristic
376C23C2×#10011yescharacteristic
377C23#41311yescharacteristic
378C23C2×#10311yescharacteristic
379C23#41611yescharacteristic
380C23#41611yescharacteristic
381C23#41911yescharacteristic
382C23C2×#10411yescharacteristic
383C23#42111yescharacteristic
384C23#42111yescharacteristic
385C23#42111yescharacteristic
386Q8—84no—
387Q8—84no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—48no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
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#43512yes—
468C2×C4#43612yes—
469C2×C4#45212yes—
470C2×C4#46512yes—
471C2×C4#46712yes—
472C2×C4#48512yes—
473C8—84no—
474C8—84no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22#146211yescharacteristic
491C22#146511yescharacteristic
492C22#146811yescharacteristic
493C22#147111yescharacteristic
494C22#148111yescharacteristic
495C22#148211yescharacteristic
496C22#148711yescharacteristic
497C22#148811yescharacteristic
498C22#148811yescharacteristic
499C22#149111yescharacteristic
500C22#149411yescharacteristic
501C22#149711yescharacteristic
502C22C2×#32811yescharacteristic
503C22#150111yescharacteristic
504C22#151111yescharacteristic
505C22#151111yescharacteristic
506C22#151611yescharacteristic
507C22#151611yescharacteristic
508C22#151611yescharacteristic
509C22#152011yescharacteristic
510C22#152711yescharacteristic
511C22#153211yescharacteristic
512C22#153511yescharacteristic
513C22#153611yescharacteristic
514C22#154211yescharacteristic
515C22#154311yescharacteristic
516C22#154811yescharacteristic
517C22#155411yescharacteristic
518C22#187011yescharacteristic
519C22#187111yescharacteristic
520C22#187311yescharacteristic
521C22#189211yescharacteristic
522C22#189711yescharacteristic
523C22#189811yescharacteristic
524C22#190011yescharacteristic
525C4—22no—
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—22no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—82no—
535C4—82no—
536C4—161no—
537C4—161no—
538C2—44no—
539C2#760911yescharacteristic
540C2#762311yescharacteristic
541C2#764511yescharacteristic
542C2#764811yescharacteristic
543C2#765511yescharacteristic
544C2#767011yescharacteristic
545C2#772111yescharacteristic
546C2#772711yescharacteristic
547C2#894111yescharacteristic
548C2#907511yescharacteristic
549C2#908311yescharacteristic
550C2#911111yescharacteristic
551C2#912311yescharacteristic
552C2#912811yescharacteristic
553C2#915411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×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

Maximal subgroups

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

subgroupindex
C2×#18172
#147022
#107122
C2×#12722
#70252