Finite Group Databasebeta

← #364097

Subgroups — #364097

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1840C211yesmaximal, characteristic, complemented
3#13797C211yesmaximal, characteristic, complemented
4#11373C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11009C211yesmaximal, characteristic
7#11009C211yesmaximal, characteristic
8#10932C212yesmaximal, complemented
9#10931C212yesmaximal
10#10789C212yesmaximal, complemented
11#10788C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2326—24no—
20#2326C418yescomplemented
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#388—22no—
24C2×#385—22no—
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2211yescharacteristic
28#1840—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—22no—
41#1434—22no—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#124—24no—
48C2×#124—24no—
49C2×#124—22no—
50C2×#124C2×C414yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—42no—
53C2×C4×Q8—42no—
54C2×C4×Q8C2×C414yes—
55C2×C4×D4—22no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4C2×C414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85#391—22no—
86#391C2×C414yes—
87#390—22no—
88#390C2×C414yes—
89#388—216no—
90#385—216no—
91#384—22no—
92#384C2×C414yes—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95#376C2×C414yes—
96#376D414yes—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104C2×#95D412yes—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—22no—
111#336—22no—
112#336—24no—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8C22×C412yes—
121C22×Q8#4114yes—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4C22×C412yes—
129C22×D4#4114yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159#124—216no—
160#124—28no—
161#124—28no—
162C4×Q8—48no—
163C4×Q8—48no—
164C4×D4—48no—
165C4×D4—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4#4114yes—
191C2×C4⋊C4#4114yes—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—28no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—82no—
212C2×C42—82no—
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219#103—48no—
220#103—48no—
221#102—48no—
222#102—48no—
223#95—28no—
224#95—28no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229#95C22⋊C418yes—
230#95C22⋊C418yes—
231C24—22no—
232C24—24no—
233C24—42no—
234C24—41no—
235C24—41no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8#11714yes—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4#11714yes—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
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—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—48no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—88no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—88no—
327C22×C4C2×C22⋊C414yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C414yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C4H(5,𝔽2)12yes—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C22⋊C4—232no—
368C22⋊C4—48no—
369C22⋊C4—88no—
370C42—22no—
371C42—22no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—22no—
376C42—22no—
377C42—24no—
378C42—48no—
379C42—88no—
380C23—24no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—42no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23—82no—
401C23C22×C22⋊C411yescharacteristic
402C23#52512yes—
403C23#52512yes—
404C23#53511yescharacteristic
405C23#53511yescharacteristic
406C23#57612yes—
407C23#57712yes—
408C23#57712yes—
409C23#57812yes—
410Q8—28no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414D4—28no—
415D4—44no—
416D4—44no—
417D4—44no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—816no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4C2×#11711yescharacteristic
473C2×C4C2×#11711yescharacteristic
474C2×C4C4×#4112yes—
475C2×C4#51812yes—
476C2×C4#52512yes—
477C2×C4#53211yescharacteristic
478C2×C4#53211yescharacteristic
479C2×C4#53512yes—
480C2×C4#55112yes—
481C2×C4#55112yes—
482C2×C4#55112yes—
483C2×C4#55112yes—
484C2×C4#55312yes—
485C2×C4#55312yes—
486C2×C4#55312yes—
487C2×C4#55312yes—
488C8—82no—
489C8—82no—
490C22—48no—
491C22—44no—
492C22—82no—
493C22—82no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—84no—
498C22—84no—
499C22#228011yescharacteristic
500C22#229711yescharacteristic
501C22#230911yescharacteristic
502C22#259812yes—
503C22#259812yes—
504C22#289712yes—
505C22#289812yes—
506C22#311812yes—
507C22#311912yes—
508C22#312012yes—
509C22#312112yes—
510C22#312412yes—
511C22#312412yes—
512C22#312512yes—
513C22#312512yes—
514C22#322011yescharacteristic
515C22#322111yescharacteristic
516C22#322211yescharacteristic
517C22#322311yescharacteristic
518C22#322412yes—
519C22#322512yes—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—44no—
525C4—82no—
526C4—82no—
527C4—84no—
528C4—84no—
529C4#235312yes—
530C4#235312yes—
531C4#260312yes—
532C4#260312yes—
533C2—84no—
534C2#1395911yescharacteristic
535C2#2074612yes—
536C2#2139712yes—
537C2#2181311yescharacteristic
538C2#2181411yescharacteristic
539C2#2938012yes—
540C2#2938112yes—
541C2#2938212yes—
542C2#2938312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1840C2semidirect
3. #13797C2semidirect
4. #11373C2semidirect
5. #11371C2semidirect
8. #10932C2semidirect
10. #10789C2semidirect
20. #2326C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#18402
#137972
#113732
#113712
#110092
#109322
#109312
#107892
#107882