Finite Group Databasebeta

← #405631

Subgroups — #405631

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×#1864C212yesmaximal
3#13785C211yesmaximal, characteristic
4#11323C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11066C212yesmaximal
8#11051C211yesmaximal, characteristic
9#11049C211yesmaximal, characteristic
10#10884C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2647—28no—
17C2×#400—22no—
18C2×#398—22no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1864—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1437—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22⋊#95—22no—
41C23×D4C2311yescharacteristic
42C24×C4—21no—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—42no—
73#400—216no—
74#398—216no—
75#390—22no—
76#390—22no—
77#390—22no—
78#390Q814yes—
79#389—24no—
80#389—24no—
81#389—24no—
82#389D418yes—
83#384—24no—
84#382—24no—
85C4×C4⋊C4—42no—
86#376—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#336—24no—
101#336—24no—
102#336—22no—
103#336—22no—
104C25—21no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C2×Q812yes—
113C22×D4#4112yes—
114C22×D4#4112yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8#4114yes—
144C4×D4—48no—
145C4×D4—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
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—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×Q812yes—
177C2×C4⋊C4#4112yes—
178C2×C4⋊C4#4112yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—82no—
199C2×C42C2×D414yes—
200C2×C42C2×Q812yes—
201C2×C42#4112yes—
202#106—48no—
203#102—216no—
204#102—48no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—24no—
209#95—24no—
210#95—42no—
211#95—82no—
212#95—82no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24C22×D411yescharacteristic
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4#12514yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—48no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—88no—
293C22×C4—88no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4#12012yes—
297C22×C4#12012yes—
298C22×C4C22⋊Q814yes—
299C22×C4C22×D412yes—
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)12yes—
308C22×C42−1+412yes—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—88no—
338C4⋊C4—88no—
339C4⋊C4—88no—
340C22⋊C4—48no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—84no—
350C23—24no—
351C23—42no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—42no—
357C23—44no—
358C23—44no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23C2×#12011yescharacteristic
374C23C2×C22⋊Q811yescharacteristic
375C23#52511yescharacteristic
376C23#52511yescharacteristic
377C23#53211yescharacteristic
378C23#53511yescharacteristic
379C23#53611yescharacteristic
380C23#57612yes—
381C23#57712yes—
382C23#57712yes—
383C23#57712yes—
384D4—28no—
385D4—44no—
386D4—44no—
387D4—44no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—816no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#52512yes—
443C2×C4C2×#12511yescharacteristic
444C2×C4#52911yescharacteristic
445C2×C4#52911yescharacteristic
446C2×C4#53212yes—
447C2×C4#53311yescharacteristic
448C2×C4#53912yes—
449C2×C4#53912yes—
450C2×C4#54612yes—
451C2×C4#54612yes—
452C2×C4#54812yes—
453C2×C4D4×Q812yes—
454C2×C4#55212yes—
455C2×C4#55212yes—
456C2×C4#55312yes—
457C2×C4#55412yes—
458C8—44no—
459C8—82no—
460C22—48no—
461C22—44no—
462C22—82no—
463C22—82no—
464C22—84no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—84no—
470C22—84no—
471C22#238811yescharacteristic
472C22#240411yescharacteristic
473C22#240911yescharacteristic
474C22#256611yescharacteristic
475C22#261111yescharacteristic
476C22#263311yescharacteristic
477C22#285111yescharacteristic
478C22#301012yes—
479C22#301212yes—
480C22#308112yes—
481C22#308412yes—
482C22#311812yes—
483C22#311912yes—
484C22#312312yes—
485C22#312512yes—
486C22#317312yes—
487C22#317912yes—
488C22#322011yescharacteristic
489C22#322111yescharacteristic
490C22#322211yescharacteristic
491C22#322211yescharacteristic
492C22#324112yes—
493C22#324412yes—
494C4—42no—
495C4—42no—
496C4—44no—
497C4—82no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4—84no—
502C4—82no—
503C4#246012yes—
504C4#247312yes—
505C4#255612yes—
506C4#255812yes—
507C2—84no—
508C2#1491611yescharacteristic
509C2#2358312yes—
510C2#2394111yescharacteristic
511C2#2394211yescharacteristic
512C2#2430912yes—
513C2#2919911yescharacteristic
514C2#2920411yescharacteristic
515C2#2959012yes—
516C2#3008411yescharacteristic
517C2#3009011yescharacteristic
518C2#3307612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
1. C2×#2647C2semidirect
4. #11323C2semidirect
5. #11268C2semidirect
6. #11220C2semidirect
10. #10884C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#18642
#137852
#113232
#112682
#112202
#110662
#110512
#110492
#108842
#107962