Finite Group Databasebeta

← #474169

Subgroups — #474169

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1984C211yesmaximal, characteristic, complemented
2C2×#1934C211yesmaximal, characteristic
3#15202C211yesmaximal, characteristic, complemented
4#15157C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11228C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11123C212yesmaximal
9#11076C211yesmaximal, characteristic
10#11066C212yesmaximal
11#10862C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#393—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24#1984—24no—
25#1934—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1446—21no—
35#1444—21no—
36#1444—22no—
37#1439—22no—
38#1439—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8D411yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—42no—
51C2×C4×Q8—42no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—42no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102Q814yescomplemented
80C2×C4×C8—42no—
81C2×C4×C8D414yes—
82#393—44no—
83#390—22no—
84#390—22no—
85#390—42no—
86#390—42no—
87#389—22no—
88#389—22no—
89#389—42no—
90#389—42no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—24no—
109C25—21no—
110C22×Q8—22no—
111C22×Q8—24no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—42no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4C2×Q812yes—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—48no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4C2×Q812yes—
190C2×C4⋊C4#4114yes—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#107—48no—
213C4⋊C8—44no—
214#102—28no—
215#102—48no—
216#102—44no—
217C4×C8—44no—
218#95—44no—
219#95—44no—
220#95—42no—
221#95—44no—
222#95—42no—
223#95—42no—
224#95—81no—
225#95—81no—
226#95—82no—
227#95—82no—
228C24—22no—
229C24—41no—
230C24—42no—
231C24—42no—
232C24—41no—
233C24—42no—
234C24—41no—
235C24—41no—
236C24—42no—
237C24—41no—
238C24—41no—
239C24C22×D411yescharacteristic
240C2×Q8—28no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#13514yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×Q811yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4112yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)12yes—
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C42−1+412yes—
325C22×C42−1+411yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8C22⋊Q814yes—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—84no—
364C4⋊C4—88no—
365C22⋊C4—88no—
366C22⋊C4—84no—
367C22⋊C4—88no—
368C22⋊C4—84no—
369C42—22no—
370C42—22no—
371C42—24no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—42no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—81no—
386C23—82no—
387C23—81no—
388C23—81no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23C2×#12111yescharacteristic
400C23#53412yes—
401C23#54812yes—
402C23D4×Q811yescharacteristic
403C23#55011yescharacteristic
404C23C2×#13512yes—
405C23C2×AGL(1,ℤ/8)11yescharacteristic
406C23C2×#13711yescharacteristic
407C23#57611yescharacteristic
408C23#57612yes—
409C23#57811yescharacteristic
410Q8—416no—
411D4—42no—
412D4—42no—
413D4—44no—
414D4—81no—
415D4—82no—
416D4—81no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#46814yes—
470C2×C4C2×C22⋊Q811yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4#53411yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53711yescharacteristic
476C2×C4#53812yes—
477C2×C4#53812yes—
478C2×C4#54212yes—
479C2×C4#54212yes—
480C2×C4#54412yes—
481C2×C4#55112yes—
482C2×C4#55412yes—
483C2×C4#55411yescharacteristic
484C2×C4#55611yescharacteristic
485C2×C4#55612yes—
486C2×C4#56212yes—
487C2×C4#56312yes—
488C8—24no—
489C8—42no—
490C8—82no—
491C22—82no—
492C22—84no—
493C22—84no—
494C22—82no—
495C22—82no—
496C22—161no—
497C22—161no—
498C22—162no—
499C22—162no—
500C22—161no—
501C22—161no—
502C22#253811yescharacteristic
503C22#254511yescharacteristic
504C22#261711yescharacteristic
505C22#272912yes—
506C22#273012yes—
507C22#305011yescharacteristic
508C22C2×#46811yescharacteristic
509C22#305811yescharacteristic
510C22#306111yescharacteristic
511C22#318812yes—
512C22#319012yes—
513C22#319912yes—
514C22#320312yes—
515C22#336112yes—
516C22#336412yes—
517C22#336612yes—
518C22#337512yes—
519C22#338012yes—
520C22#338511yescharacteristic
521C22#338611yescharacteristic
522C22#339612yes—
523C22#339811yescharacteristic
524C22#339911yescharacteristic
525C4—24no—
526C4—44no—
527C4—42no—
528C4—82no—
529C4—84no—
530C4—82no—
531C4—161no—
532C4—162no—
533C4—161no—
534C4#256411yescharacteristic
535C4#262811yescharacteristic
536C4#266211yescharacteristic
537C4#275912yes—
538C4#278812yes—
539C4#281411yescharacteristic
540C2—162no—
541C2#1690111yescharacteristic
542C2#2826711yescharacteristic
543C2#2827211yescharacteristic
544C2#2864111yescharacteristic
545C2#2864511yescharacteristic
546C2#2973511yescharacteristic
547C2#2973811yescharacteristic
548C2#3257312yes—
549C2#3257612yes—
550C2#3260612yes—
551C2#3261112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1984C2semidirect
3. #15202C2semidirect
5. #11316C2semidirect
6. #11228C2semidirect
7. #11219C2semidirect
11. #10862C2semidirect
79. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
C2×#19342
#152022
#151572
#113162
#112282
#112192
#111232
#110762
#110662
#108622