Finite Group Databasebeta

← #403168

Subgroups — #403168

Order 512.

1076 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C211yesmaximal, characteristic
2C2×C4×C4⋊C8C211yesmaximal, characteristic
3#14712C211yesmaximal, characteristic
4#14699C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic
6#11267C211yesmaximal, characteristic
7#11227C212yesmaximal
8#11227C212yesmaximal
9#11207C211yesmaximal, characteristic
10#10880C212yesmaximal
11#10820C212yesmaximal
12C22×C4⋊C8C2212yes—
13C22×C4×C8C2212yes—
14C2×#389—21no—
15C2×#389—21no—
16C2×#389C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2211yescharacteristic
19C2×#387—21no—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#2099—24no—
24C4×C4⋊C8—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1436—21no—
36#1436—21no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×C4×C8—24no—
71C2×C4×C8—24no—
72C2×C4×C8—42no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—24no—
77#389—24no—
78#389—44no—
79#389—44no—
80#389Q814yes—
81#387—44no—
82#384—24no—
83#384—22no—
84#384—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C43—22no—
99C43—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—42no—
107#340—42no—
108#336—22no—
109#336D414yes—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
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—
137C2×C4⋊C4—21no—
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—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
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—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C2×Q812yes—
186C2×C4⋊C4C2×Q812yes—
187C2×C4⋊C4#4114yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42D814yes—
208C2×C42SD1614yes—
209C2×C42C2×D412yes—
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4114yes—
214C4⋊C8—216no—
215C4×C8—48no—
216#95—24no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229C24C22×D411yescharacteristic
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4#12412yes—
303C22×C4#12412yes—
304C22×C4C2×D812yes—
305C22×C4C2×D812yes—
306C22×C4C2×D812yes—
307C22×C4C2×SD1612yes—
308C22×C4C2×SD1612yes—
309C22×C4C2×SD1612yes—
310C22×C4#13512yes—
311C22×C4#13512yes—
312C22×C4#13712yes—
313C22×C4#13712yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×Q811yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C42−1+412yes—
323C22×C42−1+411yescharacteristic
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—88no—
363C42—24no—
364C42—24no—
365C42—22no—
366C42—24no—
367C42—22no—
368C42—24no—
369C42—22no—
370C42—22no—
371C42—28no—
372C42—28no—
373C42—44no—
374C42—44no—
375C23C2×#12111yescharacteristic
376C23C2×C22⋊Q811yescharacteristic
377C23#52511yescharacteristic
378C23C2×#12411yescharacteristic
379C23#53211yescharacteristic
380C23#53611yescharacteristic
381C23#53711yescharacteristic
382C23C22×D811yescharacteristic
383C23C22×SD1611yescharacteristic
384C23C2×#13511yescharacteristic
385C23C2×#13511yescharacteristic
386C23C2×#13711yescharacteristic
387C23C2×#13711yescharacteristic
388C23#57511yescharacteristic
389C23#57511yescharacteristic
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#46514yes—
470C2×C4#46614yes—
471C2×C4#47414yes—
472C2×C4#47514yes—
473C2×C4#48614yes—
474C2×C4#52512yes—
475C2×C4#52912yes—
476C2×C4#52912yes—
477C2×C4#53312yes—
478C2×C4#53912yes—
479C2×C4#54112yes—
480C2×C4#54611yescharacteristic
481C2×C4#54712yes—
482C2×C4D4×Q811yescharacteristic
483C2×C4#55212yes—
484C2×C4#55212yes—
485C2×C4#55211yescharacteristic
486C2×C4#55312yes—
487C2×C4#55411yescharacteristic
488C2×C4#55612yes—
489C2×C4#55712yes—
490C8—44no—
491C8—44no—
492C22#239611yescharacteristic
493C22#241311yescharacteristic
494C22#241711yescharacteristic
495C22#256211yescharacteristic
496C22#256811yescharacteristic
497C22#266411yescharacteristic
498C22#285011yescharacteristic
499C22C2×#46511yescharacteristic
500C22C2×#46611yescharacteristic
501C22#305611yescharacteristic
502C22#305611yescharacteristic
503C22C2×#47411yescharacteristic
504C22C2×#47511yescharacteristic
505C22#307611yescharacteristic
506C22#307611yescharacteristic
507C22#310011yescharacteristic
508C22#310111yescharacteristic
509C22#310811yescharacteristic
510C22#310811yescharacteristic
511C22C2×#48611yescharacteristic
512C22#313111yescharacteristic
513C22#313911yescharacteristic
514C22#314211yescharacteristic
515C22#316711yescharacteristic
516C22#316911yescharacteristic
517C22#317211yescharacteristic
518C22#317211yescharacteristic
519C22#322611yescharacteristic
520C22#322911yescharacteristic
521C22#323211yescharacteristic
522C22#323211yescharacteristic
523C22#324511yescharacteristic
524C22#324611yescharacteristic
525C22#324811yescharacteristic
526C22#324811yescharacteristic
527C4—24no—
528C4—24no—
529C4—24no—
530C4—24no—
531C4—24no—
532C4—42no—
533C4—82no—
534C4—84no—
535C4—82no—
536C4—162no—
537C4—161no—
538C4—161no—
539C4#246212yes—
540C4#246612yes—
541C4#253312yes—
542C4#259012yes—
543C2#1486911yescharacteristic
544C2#2377411yescharacteristic
545C2#2377611yescharacteristic
546C2#2460011yescharacteristic
547C2#2460111yescharacteristic
548C2#2483511yescharacteristic
549C2#2483811yescharacteristic
550C2#2912111yescharacteristic
551C2#2912611yescharacteristic
552C2#2921911yescharacteristic
553C2#2922111yescharacteristic
554C2#3037611yescharacteristic
555C2#3037711yescharacteristic
556C2#3304211yescharacteristic
557C2#3305911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20992
C2×C4×C4⋊C82
#147122
#146992
#113732
#112672
#112272
#112072
#108802
#108202