Finite Group Databasebeta

← #403057

Subgroups — #403057

Order 512.

1214 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×#2025C211yesmaximal, characteristic
2C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
3#14709C211yesmaximal, characteristic, complemented
4#14700C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic, complemented
6#11268C211yesmaximal, characteristic, complemented
7#11228C212yesmaximal, complemented
8#11076C212yesmaximal
9#11051C211yesmaximal, characteristic
10#10925C212yesmaximal
11#10820C212yesmaximal, complemented
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C4×C8C2212yes—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#388—22no—
22C2×#382C2211yescharacteristic
23C2×C43C2211yescharacteristic
24#2025—24no—
25C4×C4⋊C8—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1437—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×D4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—42no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#102—42no—
73C2×C4×C8—24no—
74C2×C4×C8—24no—
75C2×C4×C8—42no—
76#390—22no—
77#390—22no—
78#390—22no—
79#390—24no—
80#390Q814yes—
81#389—24no—
82#388—48no—
83#382—42no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C43—22no—
93C43—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336D414yes—
102C25—21no—
103C22×D4—21no—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—22no—
111C22×D4—21no—
112C22×D4C2×Q812yes—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
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—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C2×D414yes—
136C22×C8#4114yes—
137#125—48no—
138#125—48no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C2×Q812yes—
167C2×C4⋊C4#4114yes—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—44no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—42no—
193C2×C42C2×D412yes—
194C2×C42C2×Q812yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42#4114yes—
198C4⋊C8—216no—
199#102—48no—
200#102—48no—
201C4×C8—48no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—42no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
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—22no—
225C2×D4—24no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
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—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#12114yes—
285C22×C4C22⋊Q814yes—
286C22×C4#12412yes—
287C22×C4#12412yes—
288C22×C4#13514yes—
289C22×C4#13514yes—
290C22×C4AGL(1,ℤ/8)14yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×Q811yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)12yes—
300C22×C42−1+411yescharacteristic
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
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—416no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—84no—
331C22⋊C4—416no—
332C22⋊C4—84no—
333C22⋊C4—88no—
334C22⋊C4—84no—
335C42—216no—
336C42—24no—
337C42—24no—
338C42—22no—
339C42—24no—
340C42—22no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—48no—
345C23—44no—
346C23—42no—
347C23—44no—
348C23—42no—
349C23—42no—
350C23—82no—
351C23—81no—
352C23—81no—
353C23—82no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—82no—
359C23—82no—
360C23—81no—
361C23—81no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23C2×#12111yescharacteristic
369C23C2×C22⋊Q811yescharacteristic
370C23#52511yescharacteristic
371C23C2×#12411yescharacteristic
372C23#53211yescharacteristic
373C23#53511yescharacteristic
374C23#53711yescharacteristic
375C23C2×#13512yes—
376C23C2×#13512yes—
377C23C2×AGL(1,ℤ/8)12yes—
378C23#57512yes—
379D4—44no—
380D4—44no—
381D4—82no—
382D4—82no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#47818yes—
435C2×C4#48814yes—
436C2×C4#52512yes—
437C2×C4#52912yes—
438C2×C4#52912yes—
439C2×C4#53312yes—
440C2×C4#53812yes—
441C2×C4#54212yes—
442C2×C4#54611yescharacteristic
443C2×C4#54612yes—
444C2×C4D4×Q811yescharacteristic
445C2×C4#55112yes—
446C2×C4#55211yescharacteristic
447C2×C4#55411yescharacteristic
448C2×C4#55412yes—
449C2×C4#55512yes—
450C2×C4#55612yes—
451C2×C4#55912yes—
452C8—44no—
453C8—44no—
454C22—84no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—84no—
459C22—161no—
460C22—161no—
461C22—162no—
462C22—161no—
463C22—161no—
464C22—162no—
465C22#239611yescharacteristic
466C22#241111yescharacteristic
467C22#241711yescharacteristic
468C22#256311yescharacteristic
469C22#259211yescharacteristic
470C22#266211yescharacteristic
471C22#285011yescharacteristic
472C22#305012yes—
473C22#305512yes—
474C22C2×#47812yes—
475C22#307612yes—
476C22#310212yes—
477C22#310612yes—
478C22C2×#48811yescharacteristic
479C22#313111yescharacteristic
480C22#313811yescharacteristic
481C22#314211yescharacteristic
482C22#316812yes—
483C22#317112yes—
484C22#320712yes—
485C22#321212yes—
486C22#324712yes—
487C22#325012yes—
488C4—28no—
489C4—24no—
490C4—42no—
491C4—44no—
492C4—84no—
493C4—82no—
494C4—161no—
495C4—162no—
496C4—161no—
497C4#245912yes—
498C4#246512yes—
499C4#253212yes—
500C4#258912yes—
501C2—162no—
502C2#1486611yescharacteristic
503C2#2377812yes—
504C2#2450412yes—
505C2#2483912yes—
506C2#2914011yescharacteristic
507C2#2914311yescharacteristic
508C2#2934811yescharacteristic
509C2#2935111yescharacteristic
510C2#3034012yes—
511C2#3304411yescharacteristic
512C2#3305811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊C8C2semidirect
3. #14709C2semidirect
5. #11385C2semidirect
6. #11268C2semidirect
7. #11228C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20252
C2×C4×C4⋊C82
#147092
#147002
#113852
#112682
#112282
#110762
#110512
#109252
#108202