Finite Group Databasebeta

← #410557

Subgroups — #410557

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#1925C212yesmaximal
4#14260C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic, complemented
6#11232C212yesmaximal, complemented
7#11208C211yesmaximal, characteristic, complemented
8#11076C212yesmaximal
9#11051C211yesmaximal, characteristic
10#10833C211yesmaximal, characteristic, complemented
11#10807C212yesmaximal, complemented
12C22×#107C2212yes—
13C22×C4⋊C8C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×#376C2211yescharacteristic
24#2083—24no—
25#2071—24no—
26#1925—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1440—22no—
36#1439—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C23×D4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C42—22no—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—24no—
62C2×#107—24no—
63C2×#107—42no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—42no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—42no—
73#390—22no—
74#390—22no—
75#390—22no—
76#390—24no—
77#390D414yes—
78#389—22no—
79#389—22no—
80#384—22no—
81#384D414yes—
82#382—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85#376—24no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D412yes—
95C2×#95D412yes—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#336—22no—
102#336—22no—
103#336—24no—
104C25—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4C2×D412yes—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
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#4114yes—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4#4114yes—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—44no—
189C2×C42—82no—
190C2×C42—82no—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195C2×C42#4114yes—
196#107—48no—
197C4⋊C8—48no—
198#102—216no—
199#102—48no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—44no—
205#95—42no—
206#95—82no—
207#95—82no—
208#95—82no—
209#95—82no—
210C24—22no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24C22×D411yescharacteristic
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—42no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
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—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4#12114yes—
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12314yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C42−1+411yescharacteristic
300C22×C42−1+412yes—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—88no—
345C23—42no—
346C23—42no—
347C23—44no—
348C23—44no—
349C23—42no—
350C23—82no—
351C23—82no—
352C23—81no—
353C23—82no—
354C23—82no—
355C23—81no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C2×#12111yescharacteristic
369C23C2×#12111yescharacteristic
370C23C2×#12311yescharacteristic
371C23#52511yescharacteristic
372C23#53211yescharacteristic
373C23#53611yescharacteristic
374C23#53711yescharacteristic
375C23#57612yes—
376C23#57612yes—
377C23#57712yes—
378C23#57812yes—
379D4—44no—
380D4—44no—
381D4—82no—
382D4—82no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#52512yes—
437C2×C4#52912yes—
438C2×C4#52912yes—
439C2×C4#53212yes—
440C2×C4#54112yes—
441C2×C4#54212yes—
442C2×C4D4211yescharacteristic
443C2×C4#54712yes—
444C2×C4#54711yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4#54812yes—
447C2×C4#55112yes—
448C2×C4#55212yes—
449C2×C4#55212yes—
450C2×C4#55511yescharacteristic
451C2×C4#55612yes—
452C8—44no—
453C8—82no—
454C22—84no—
455C22—82no—
456C22—82no—
457C22—84no—
458C22—82no—
459C22—161no—
460C22—161no—
461C22—162no—
462C22—162no—
463C22—161no—
464C22—161no—
465C22#239211yescharacteristic
466C22#248611yescharacteristic
467C22#249811yescharacteristic
468C22#253711yescharacteristic
469C22#254511yescharacteristic
470C22#263711yescharacteristic
471C22#285411yescharacteristic
472C22#304112yes—
473C22#304111yescharacteristic
474C22#304111yescharacteristic
475C22#304311yescharacteristic
476C22#304411yescharacteristic
477C22#304612yes—
478C22#309612yes—
479C22#309812yes—
480C22#311912yes—
481C22#312412yes—
482C22#317512yes—
483C22#317912yes—
484C22#324112yes—
485C22#324312yes—
486C22#325812yes—
487C22#325912yes—
488C4—44no—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—82no—
493C4—84no—
494C4—84no—
495C4—161no—
496C4—161no—
497C4#246112yes—
498C4#247812yes—
499C4#251912yes—
500C4#253212yes—
501C2—162no—
502C2#1508111yescharacteristic
503C2#2369212yes—
504C2#2765512yes—
505C2#2787412yes—
506C2#2825811yescharacteristic
507C2#2825911yescharacteristic
508C2#2866311yescharacteristic
509C2#2866711yescharacteristic
510C2#3016112yes—
511C2#3311811yescharacteristic
512C2#3312011yescharacteristic
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×#2083C2semidirect
5. #13779C2semidirect
6. #11232C2semidirect
7. #11208C2semidirect
10. #10833C2semidirect
11. #10807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
C2×#20712
C2×#19252
#142602
#137792
#112322
#112082
#110762
#110512
#108332
#108072