Finite Group Databasebeta

← #411504

Subgroups — #411504

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1972C211yesmaximal, characteristic
2C2×#1746C211yesmaximal, characteristic, complemented
3#14260C212yesmaximal, complemented
4#11327C211yesmaximal, characteristic, complemented
5#11257C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11022C211yesmaximal, characteristic
9#11022C211yesmaximal, characteristic
10#10925C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#376C2211yescharacteristic
24#1972—24no—
25#1746—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1440—22no—
36#1440—22no—
37#1439—22no—
38#1439—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×D4C2312yes—
42C22×D8—24no—
43C22×D8—22no—
44C22×D8—22no—
45C22×D8—22no—
46C22×D8—42no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#106—22no—
65C2×#106—42no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#97—22no—
76C2×#97—22no—
77C2×#97D414yes—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81#390—24no—
82#390—24no—
83#390—24no—
84#390—24no—
85#390—42no—
86#390—42no—
87#389—42no—
88#389—42no—
89#376—22no—
90#376—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D412yes—
97C2×#95D412yes—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—42no—
102#340—42no—
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107C25—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4#4114yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C2×D8—28no—
131C2×D8—48no—
132C2×D8—48no—
133C2×D8—44no—
134C2×D8—44no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8#4112yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154C22×C8#4112yes—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
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—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4#4114yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208#106—44no—
209#102—48no—
210#102—48no—
211#102—48no—
212#97—28no—
213C4×C8—28no—
214#95—24no—
215#95—24no—
216#95—44no—
217#95—82no—
218#95—82no—
219#95—82no—
220#95—82no—
221C24—24no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24C22×D411yescharacteristic
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4H(5,𝔽2)12yes—
319C22×C4H(5,𝔽2)12yes—
320D8—416no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8#12314yes—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—88no—
359C22⋊C4—88no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—41no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—42no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23C2×#12111yescharacteristic
406C23#52512yes—
407C23#53211yescharacteristic
408C23#53211yescharacteristic
409C23#53512yes—
410C23#57612yes—
411C23#57612yes—
412C23#57712yes—
413C23#57712yes—
414D4—44no—
415D4—44no—
416D4—44no—
417D4—44no—
418D4—82no—
419D4—82no—
420D4—82no—
421D4—82no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4C2×#12311yescharacteristic
468C2×C4#52511yescharacteristic
469C2×C4#52511yescharacteristic
470C2×C4#52911yescharacteristic
471C2×C4#52912yes—
472C2×C4#52911yescharacteristic
473C2×C4#53211yescharacteristic
474C2×C4#53511yescharacteristic
475C2×C4#53511yescharacteristic
476C2×C4#54012yes—
477C2×C4#55112yes—
478C2×C4#55112yes—
479C2×C4#55112yes—
480C2×C4#55112yes—
481C2×C4#55312yes—
482C2×C4#55312yes—
483C2×C4#55512yes—
484C2×C4#55512yes—
485C8—24no—
486C8—42no—
487C8—42no—
488C8—42no—
489C22—84no—
490C22—84no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—162no—
497C22—162no—
498C22—162no—
499C22—162no—
500C22—162no—
501C22—162no—
502C22—162no—
503C22—162no—
504C22#239211yescharacteristic
505C22#260312yes—
506C22#261012yes—
507C22#261911yescharacteristic
508C22#285411yescharacteristic
509C22#306212yes—
510C22#306212yes—
511C22#311812yes—
512C22#311912yes—
513C22#312212yes—
514C22#312512yes—
515C22#317311yescharacteristic
516C22#317311yescharacteristic
517C22#317611yescharacteristic
518C22#317611yescharacteristic
519C22#317912yes—
520C22#317912yes—
521C22#322012yes—
522C22#322112yes—
523C22#322412yes—
524C22#322412yes—
525C4—42no—
526C4—42no—
527C4—84no—
528C4—82no—
529C4—82no—
530C4—162no—
531C4—162no—
532C4#248211yescharacteristic
533C4#250011yescharacteristic
534C4#252612yes—
535C4#253112yes—
536C4#259811yescharacteristic
537C4#261011yescharacteristic
538C2—164no—
539C2#1512011yescharacteristic
540C2#2371212yes—
541C2#2950512yes—
542C2#2951012yes—
543C2#2956412yes—
544C2#2957012yes—
545C2#2981612yes—
546C2#3314011yescharacteristic
547C2#3314011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×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×#1746C2semidirect
3. #14260C2semidirect
4. #11327C2semidirect
5. #11257C2semidirect
6. #11189C2semidirect
7. #11066C2semidirect
10. #10925C2semidirect
11. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19722
C2×#17462
#142602
#113272
#112572
#111892
#110662
#110222
#109252
#108212