Finite Group Databasebeta

← #99778

Subgroups — #99778

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1728C211yesmaximal, characteristic
2#14697C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C23×M16—21no—
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—21no—
9C2×#389C411yescharacteristic
10C2×#389C2211yescharacteristic
11C2×#385—21no—
12C2×#380C411yescharacteristic
13C2×#380C411yescharacteristic
14#1728—24no—
15C2×#336—21no—
16C2×#336—21no—
17C2×#336C2212yes—
18C2×#336C2212yes—
19#1436—21no—
20#1436—21no—
21C4⋊#95C411yescharacteristic
22C4⋊#95C2211yescharacteristic
23C22⋊#95—21no—
24#1271—24no—
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C22×M16—24no—
29C22×M16—24no—
30C22×M16—24no—
31C22×M16—42no—
32C22×M16—42no—
33C22×M16—42no—
34C22×M16—42no—
35C23×C8—22no—
36C23×C8—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—42no—
48C22×C42—42no—
49C22×C42C2×C411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42C2311yescharacteristic
53C2×C22⋊C8—24no—
54C2×C22⋊C8—42no—
55C2×C22⋊C8—42no—
56#389—22no—
57#389—22no—
58#389—42no—
59#385—44no—
60#380—22no—
61#380—22no—
62#380—42no—
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—24no—
72#336—22no—
73#336—22no—
74#336—22no—
75#336—42no—
76#336D414yes—
77C25C22⋊C411yescharacteristic
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—41no—
103C23×C4—41no—
104C23×C4—41no—
105C23×C4—41no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4SD1612yes—
110C23×C4Q1612yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C2×M16—216no—
115C2×M16—48no—
116C2×M16—48no—
117C2×M16—48no—
118C2×M16—48no—
119C2×M16—48no—
120C2×M16—48no—
121C22×C8—24no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8—24no—
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—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4M1612yescomplemented
155C2×C4⋊C4M1612yescomplemented
156C2×C22⋊C4—24no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42—48no—
175C2×C42—48no—
176C2×C42C2×D412yes—
177C2×C42#4112yes—
178C22⋊C8—48no—
179#95—24no—
180#95—42no—
181#95—42no—
182#95—42no—
183#95—42no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—42no—
188#95—41no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24#9912yes—
202C24#10312yes—
203C24#10412yes—
204C24C2×C22⋊C411yescharacteristic
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
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—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
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—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
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—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4#9912yes—
298C22×C4#10112yes—
299C22×C4#10312yes—
300C22×C4#10412yes—
301C22×C4#10412yes—
302C22×C4C4×D412yes—
303C22×C4#12011yescharacteristic
304C22×C4#12112yes—
305C22×C4#12311yescharacteristic
306C22×C4C2×M1611yescharacteristic
307C22×C4C2×M1611yescharacteristic
308C22×C4#13112yes—
309C22×C4C2×SD1611yescharacteristic
310C22×C4C2×Q1611yescharacteristic
311C22×C4AGL(1,ℤ/8)11yescharacteristic
312C22×C4AGL(1,ℤ/8)12yes—
313C22×C4AGL(1,ℤ/8)11yescharacteristic
314C22×C4#13711yescharacteristic
315C22×C4#13711yescharacteristic
316M16—432no—
317C2×C8—28no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—82no—
328C2×C8—82no—
329C2×C8—82no—
330C2×C8—82no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—84no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C22⋊C4—48no—
341C22⋊C4—84no—
342C42—42no—
343C42—42no—
344C42—44no—
345C42—44no—
346C42—42no—
347C42—42no—
348C23—22no—
349C23—22no—
350C23—22no—
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—41no—
367C23—41no—
368C23—41no—
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#32814yes—
383C23#38611yescharacteristic
384C23#40711yescharacteristic
385C23#40811yescharacteristic
386C23C2×#9911yescharacteristic
387C23C2×#9911yescharacteristic
388C23C2×#10111yescharacteristic
389C23#41311yescharacteristic
390C23C2×#10311yescharacteristic
391C23#41711yescharacteristic
392C23#41711yescharacteristic
393C23#41811yescharacteristic
394C23C2×#10411yescharacteristic
395C23C2×#10411yescharacteristic
396C23C2×#10411yescharacteristic
397C23#42111yescharacteristic
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—48no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
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—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#32814yes—
483C2×C4#43512yes—
484C2×C4#43612yes—
485C2×C4#45011yescharacteristic
486C2×C4#45111yescharacteristic
487C2×C4#46812yes—
488C2×C4#46912yes—
489C2×C4#48311yescharacteristic
490C2×C4#48411yescharacteristic
491C8—48no—
492C8—84no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22#146311yescharacteristic
509C22#146511yescharacteristic
510C22#146811yescharacteristic
511C22#147211yescharacteristic
512C22#148311yescharacteristic
513C22#148411yescharacteristic
514C22#148711yescharacteristic
515C22#148911yescharacteristic
516C22#148911yescharacteristic
517C22#149011yescharacteristic
518C22#149311yescharacteristic
519C22#149511yescharacteristic
520C22C2×#32811yescharacteristic
521C22C2×#32811yescharacteristic
522C22#150411yescharacteristic
523C22#150411yescharacteristic
524C22#150411yescharacteristic
525C22#150711yescharacteristic
526C22#151011yescharacteristic
527C22#151311yescharacteristic
528C22#152411yescharacteristic
529C22#152611yescharacteristic
530C22#152811yescharacteristic
531C22#152911yescharacteristic
532C22#154911yescharacteristic
533C22#155011yescharacteristic
534C22#155711yescharacteristic
535C22#156111yescharacteristic
536C22#187011yescharacteristic
537C22#187111yescharacteristic
538C22#187211yescharacteristic
539C22#189511yescharacteristic
540C22#189611yescharacteristic
541C22#189811yescharacteristic
542C22#190111yescharacteristic
543C4—22no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—21no—
548C4—24no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—161no—
557C4—161no—
558C2—44no—
559C2#760711yescharacteristic
560C2#761711yescharacteristic
561C2#763211yescharacteristic
562C2#763311yescharacteristic
563C2#768111yescharacteristic
564C2#768811yescharacteristic
565C2#769411yescharacteristic
566C2#771111yescharacteristic
567C2#894211yescharacteristic
568C2#907811yescharacteristic
569C2#908511yescharacteristic
570C2#911711yescharacteristic
571C2#912211yescharacteristic
572C2#913711yescharacteristic
573C2#914311yescharacteristic
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×#1271C2semidirect
5. #7025C2semidirect
154. C2×C4⋊C4M16semidirect
155. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#17282
#146972
#107122
C2×#12712
#70252