Finite Group Databasebeta

← #410431

Subgroups — #410431

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1925C212yesmaximal
2#14260C211yesmaximal, characteristic
3#13779C211yesmaximal, characteristic, complemented
4#11324C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11219C212yesmaximal, complemented
7#11189C212yesmaximal
8#11051C211yesmaximal, characteristic
9#11049C211yesmaximal, characteristic
10#10807C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C22×#107C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#1925—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1440—22no—
34#1439—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×D4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C42—22no—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—42no—
61C2×#102—24no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—24no—
65C2×#102—24no—
66C2×#102—42no—
67#390—22no—
68#390—22no—
69#390—22no—
70#390—42no—
71#390D414yes—
72#389—42no—
73#384—24no—
74#382—24no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—24no—
77#376—22no—
78#376—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95D412yes—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—42no—
92#340—42no—
93#336—24no—
94#336—24no—
95#336—22no—
96#336—22no—
97C25—21no—
98C22×D4—21no—
99C22×D4—21no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—21no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4C2×D412yes—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8#4114yes—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4#4114yes—
160C2×C4⋊C4#4114yes—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4—82no—
169C2×C22⋊C4—82no—
170C2×C22⋊C4—82no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—44no—
182C2×C42—82no—
183C2×C42—82no—
184C2×C42C2×D412yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188#107—48no—
189#102—216no—
190#102—48no—
191#95—28no—
192#95—24no—
193#95—24no—
194#95—42no—
195#95—44no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200C24—22no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24—41no—
206C24—41no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24C22×D411yescharacteristic
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
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—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4#12114yes—
280C22×C4#12314yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4C2×#4112yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+411yescharacteristic
290C22×C42−1+411yescharacteristic
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—88no—
320C4⋊C4—84no—
321C22⋊C4—88no—
322C22⋊C4—88no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C42—24no—
326C42—24no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—42no—
332C42—42no—
333C42—88no—
334C23—42no—
335C23—42no—
336C23—44no—
337C23—44no—
338C23—42no—
339C23—82no—
340C23—82no—
341C23—81no—
342C23—81no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—81no—
348C23—82no—
349C23—82no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23C2×#12111yescharacteristic
358C23C2×#12311yescharacteristic
359C23#52511yescharacteristic
360C23#52511yescharacteristic
361C23#53211yescharacteristic
362C23#53511yescharacteristic
363C23#53611yescharacteristic
364C23#57612yes—
365C23#57612yes—
366C23#57712yes—
367C23#57812yes—
368D4—44no—
369D4—44no—
370D4—82no—
371D4—82no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4#52512yes—
426C2×C4#52912yes—
427C2×C4#52912yes—
428C2×C4#52912yes—
429C2×C4#54212yes—
430C2×C4#54212yes—
431C2×C4#54612yes—
432C2×C4#54711yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#55212yes—
435C2×C4#55211yescharacteristic
436C2×C4#55211yescharacteristic
437C2×C4#55312yes—
438C2×C4#55312yes—
439C2×C4#55512yes—
440C2×C4#55612yes—
441C8—44no—
442C8—82no—
443C22—82no—
444C22—84no—
445C22—82no—
446C22—84no—
447C22—82no—
448C22—162no—
449C22—162no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22#239211yescharacteristic
455C22#248611yescharacteristic
456C22#250011yescharacteristic
457C22#253511yescharacteristic
458C22#260411yescharacteristic
459C22#263211yescharacteristic
460C22#285311yescharacteristic
461C22#304112yes—
462C22#304612yes—
463C22#309612yes—
464C22#309812yes—
465C22#311912yes—
466C22#312112yes—
467C22#312212yes—
468C22#312412yes—
469C22#317512yes—
470C22#317912yes—
471C22#322011yescharacteristic
472C22#322011yescharacteristic
473C22#322211yescharacteristic
474C22#322311yescharacteristic
475C22#324112yes—
476C22#324312yes—
477C4—44no—
478C4—44no—
479C4—84no—
480C4—82no—
481C4—84no—
482C4—81no—
483C4—81no—
484C4—161no—
485C4—161no—
486C4#246812yes—
487C4#247412yes—
488C4#252512yes—
489C4#252912yes—
490C2—162no—
491C2#1507811yescharacteristic
492C2#2369212yes—
493C2#2765512yes—
494C2#2789911yescharacteristic
495C2#2790311yescharacteristic
496C2#2816511yescharacteristic
497C2#2817311yescharacteristic
498C2#2952712yes—
499C2#3007111yescharacteristic
500C2#3007211yescharacteristic
501C2#3310312yes—
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
3. #13779C2semidirect
4. #11324C2semidirect
5. #11268C2semidirect
6. #11219C2semidirect
10. #10807C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19252
#142602
#137792
#113242
#112682
#112192
#111892
#110512
#110492
#108072
#107962