Finite Group Databasebeta

← #475546

Subgroups — #475546

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×#1846C211yesmaximal, characteristic, complemented
3#15233C212yesmaximal
4#11372C212yesmaximal, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11256C211yesmaximal, characteristic, complemented
7#11231C212yesmaximal, complemented
8#11096C211yesmaximal, characteristic
9#11096C211yesmaximal, characteristic
10#11095C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#382C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1923—24no—
24#1846—24no—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1443—22no—
35#1443—22no—
36#1440—22no—
37#1440—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—42no—
59C2×#107—42no—
60C2×#106—24no—
61C2×#106—42no—
62C2×#106—42no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—24no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8D414yes—
73#382—24no—
74#382—24no—
75#380—42no—
76#380—42no—
77#380—42no—
78#380—42no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93#340—22no—
94#340—22no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—22no—
101#336—22no—
102C25C2×D411yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8#4112yes—
141C22×C8#4112yes—
142C22×C8#4114yes—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
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—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C2×D412yes—
187C2×C42#4112yes—
188#107—44no—
189#107—44no—
190#106—48no—
191C4⋊C8—44no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194C4×C8—28no—
195#95—22no—
196#95—22no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207C24—21no—
208C24—22no—
209C24—21no—
210C24—22no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—21no—
215C24—22no—
216C24—21no—
217C24#13512yes—
218C24C22×D411yescharacteristic
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
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—42no—
266C22×C4—42no—
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—42no—
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#12112yes—
290C22×C4#12112yes—
291C22×C4#13512yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4111yescharacteristic
297C22×C4H(5,𝔽2)12yes—
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+412yes—
300C22×C42−1+411yescharacteristic
301C22×C42−1+411yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8#12314yes—
321C2×C8#12414yes—
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—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C22⋊C4—28no—
339C22⋊C4—44no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—41no—
347C42—41no—
348C42—42no—
349C23—22no—
350C23—24no—
351C23—22no—
352C23—24no—
353C23—22no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23C2×#12111yescharacteristic
373C23#53512yes—
374C23#54711yescharacteristic
375C23#54712yes—
376C23#54711yescharacteristic
377C23C2×#13511yescharacteristic
378C23C2×#13511yescharacteristic
379C23#57512yes—
380C23#57611yescharacteristic
381C23#57712yes—
382C23#57811yescharacteristic
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
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—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4C2×#12311yescharacteristic
445C2×C4C2×#12411yescharacteristic
446C2×C4#53511yescharacteristic
447C2×C4#53511yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#53611yescharacteristic
450C2×C4#53912yes—
451C2×C4#53912yes—
452C2×C4#54012yes—
453C2×C4#54212yes—
454C2×C4#54212yes—
455C2×C4#55211yescharacteristic
456C2×C4#55211yescharacteristic
457C2×C4#55312yes—
458C2×C4#55312yes—
459C2×C4#55612yes—
460C2×C4#56312yes—
461C2×C4#56312yes—
462C8—24no—
463C8—24no—
464C8—81no—
465C8—81no—
466C22—42no—
467C22—44no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22—81no—
472C22—82no—
473C22—82no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22#248611yescharacteristic
478C22#254711yescharacteristic
479C22#261911yescharacteristic
480C22#271812yes—
481C22#271812yes—
482C22#305011yescharacteristic
483C22#305411yescharacteristic
484C22#306011yescharacteristic
485C22#306111yescharacteristic
486C22#321112yes—
487C22#321212yes—
488C22#322412yes—
489C22#322512yes—
490C22#333611yescharacteristic
491C22#333712yes—
492C22#333712yes—
493C22#333911yescharacteristic
494C22#334111yescharacteristic
495C22#334211yescharacteristic
496C22#334812yes—
497C22#335012yes—
498C22#335212yes—
499C22#335512yes—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—81no—
504C4—81no—
505C4—162no—
506C4—162no—
507C4—162no—
508C4—162no—
509C4#250411yescharacteristic
510C4#262311yescharacteristic
511C4#264011yescharacteristic
512C4#279312yes—
513C4#279312yes—
514C4#282111yescharacteristic
515C2—82no—
516C2#1706011yescharacteristic
517C2#2766011yescharacteristic
518C2#2766511yescharacteristic
519C2#2870711yescharacteristic
520C2#2870911yescharacteristic
521C2#2980911yescharacteristic
522C2#2981311yescharacteristic
523C2#3228612yes—
524C2#3228912yes—
525C2#3229112yes—
526C2#3229212yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1846C2semidirect
4. #11372C2semidirect
5. #11256C2semidirect
6. #11256C2semidirect
7. #11231C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#18462
#152332
#113722
#112562
#112312
#110962
#110952
#108202