Finite Group Databasebeta

← #411225

Subgroups — #411225

Order 512.

1102 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
1#14700C211yesmaximal, characteristic
2C4×#382C211yesmaximal, characteristic, complemented
3#11316C212yesmaximal, complemented
4#11268C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal
7#11051C211yesmaximal, characteristic
8#11051C211yesmaximal, characteristic
9#10931C212yesmaximal
10#10884C212yesmaximal, complemented
11#10795C211yesmaximal, characteristic, complemented
12C22×#102C2212yes—
13C22×#102C2212yes—
14C2×#390C2211yescharacteristic
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#388—22no—
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C43C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1437—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37#1434—22no—
38C4×#95—22no—
39C23×D4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C42—22no—
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#102—24no—
59C2×#102—24no—
60C2×#102—24no—
61C2×#102—42no—
62C2×#102—42no—
63C2×#102—42no—
64#390—22no—
65#390—22no—
66#390—22no—
67#390—22no—
68#390—24no—
69#389—24no—
70#388—48no—
71#382—24no—
72#382—24no—
73C4×C4⋊C4—42no—
74C4×C4⋊C4—42no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C43—22no—
84C43—22no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—42no—
89#340—42no—
90#340—42no—
91#336—22no—
92#336—22no—
93#336—24no—
94#336—42no—
95C25—21no—
96C22×D4—21no—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4#4112yes—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C2×D412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
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—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
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—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4#4112yes—
154C2×C4⋊C4#4114yes—
155C2×C4⋊C4#4114yes—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—44no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42#4114yes—
189#102—48no—
190#102—48no—
191#95—28no—
192#95—28no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—42no—
198#95—44no—
199C24—22no—
200C24—41no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24—41no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24C22×D411yescharacteristic
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—42no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4#12314yes—
276C22×C4#12412yes—
277C22×C4#12412yes—
278C22×C4#13514yes—
279C22×C4AGL(1,ℤ/8)14yes—
280C22×C4C2×#4112yes—
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+411yescharacteristic
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
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—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—42no—
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—84no—
319C22⋊C4—416no—
320C22⋊C4—88no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—48no—
328C42—48no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—88no—
334C23—44no—
335C23—42no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—81no—
343C23—82no—
344C23—81no—
345C23—82no—
346C23—82no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—82no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23C2×#12311yescharacteristic
358C23#52511yescharacteristic
359C23#52511yescharacteristic
360C23#52511yescharacteristic
361C23C2×#12411yescharacteristic
362C23#53511yescharacteristic
363C23#53711yescharacteristic
364C23C2×#13512yes—
365C23C2×AGL(1,ℤ/8)12yes—
366C23#57512yes—
367C23#57512yes—
368D4—44no—
369D4—44no—
370D4—82no—
371D4—82no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4#49014yes—
425C2×C4#52912yes—
426C2×C4#52912yes—
427C2×C4#52912yes—
428C2×C4#52912yes—
429C2×C4#53812yes—
430C2×C4#53912yes—
431C2×C4#55112yes—
432C2×C4#55211yescharacteristic
433C2×C4#55312yes—
434C2×C4#55312yes—
435C2×C4#55312yes—
436C2×C4#55311yescharacteristic
437C2×C4#55511yescharacteristic
438C2×C4#55612yes—
439C2×C4#55611yescharacteristic
440C2×C4#55912yes—
441C8—82no—
442C8—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—84no—
447C22—84no—
448C22—161no—
449C22—162no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—162no—
454C22#239811yescharacteristic
455C22#248511yescharacteristic
456C22#250411yescharacteristic
457C22#259211yescharacteristic
458C22#259811yescharacteristic
459C22#263611yescharacteristic
460C22#285311yescharacteristic
461C22#309112yes—
462C22#309412yes—
463C22#310212yes—
464C22#310512yes—
465C22#310612yes—
466C22#311112yes—
467C22#311112yes—
468C22#311512yes—
469C22#313111yescharacteristic
470C22C2×#49011yescharacteristic
471C22#314011yescharacteristic
472C22#314211yescharacteristic
473C22#321212yes—
474C22#321312yes—
475C22#325012yes—
476C22#325112yes—
477C4—24no—
478C4—42no—
479C4—44no—
480C4—44no—
481C4—82no—
482C4—84no—
483C4—84no—
484C4—161no—
485C4—161no—
486C4#257312yes—
487C4#259312yes—
488C4#259412yes—
489C4#259512yes—
490C2—162no—
491C2#1511011yescharacteristic
492C2#2381911yescharacteristic
493C2#2382611yescharacteristic
494C2#2763212yes—
495C2#2797611yescharacteristic
496C2#2798211yescharacteristic
497C2#2935111yescharacteristic
498C2#2935611yescharacteristic
499C2#2937212yes—
500C2#3013412yes—
501C2#3309712yes—
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. C4×#382C2semidirect
3. #11316C2semidirect
4. #11268C2semidirect
5. #11268C2semidirect
10. #10884C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
#147002
C4×#3822
#113162
#112682
#111892
#110512
#109312
#108842
#107952