Finite Group Databasebeta

← #413399

Subgroups — #413399

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2099C212yesmaximal
2#13910C211yesmaximal, characteristic
3#13910C211yesmaximal, characteristic
4#11292C212yesmaximal
5#11292C212yesmaximal
6#11207C212yesmaximal
7#10849C211yesmaximal, characteristic
8#10849C211yesmaximal, characteristic
9#10841C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389—21no—
15C2×#389—21no—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#384—21no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2099—28no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444—22no—
39#1444—22no—
40C4⋊#95—21no—
41C4⋊#95C2212yes—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71#389—24no—
72#389—24no—
73#389—42no—
74#389—42no—
75#389—44no—
76#389—44no—
77#384—24no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384—22no—
82#384—42no—
83#384—44no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4Q814yes—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—42no—
104#340—42no—
105#340D414yes—
106#340Q814yes—
107#336—24no—
108#336—42no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
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—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×D412yes—
140C22×C8C2×Q812yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
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×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4D814yes—
198C2×C4⋊C4SD1614yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×Q812yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C4⋊C4C2×Q812yes—
205C2×C4⋊C4#4112yes—
206C2×C4⋊C4#4112yes—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C4⋊C8—44no—
226C4×C8—44no—
227#95—24no—
228#95—24no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—44no—
233#95—44no—
234C24C22×D411yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
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—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
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—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C22⋊Q812yes—
294C22×C4C22⋊Q812yes—
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4#12412yes—
298C22×C4#12412yes—
299C22×C4C2×D814yes—
300C22×C4C2×D812yes—
301C22×C4C2×SD1614yes—
302C22×C4C2×SD1612yes—
303C22×C4#13712yes—
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316C22×C4H(5,𝔽2)11yescharacteristic
317C22×C42−1+411yescharacteristic
318C22×C42−1+411yescharacteristic
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—28no—
347C4⋊C4—28no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—88no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C4⋊C4—84no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4#12514yes—
373C4⋊C4C4⋊Q814yes—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—48no—
379C42—84no—
380C42#12514yes—
381C42#12514yes—
382C42#12614yes—
383C42#12614yes—
384C23C2×C22⋊Q811yescharacteristic
385C23C2×#12311yescharacteristic
386C23C2×#12411yescharacteristic
387C23D4211yescharacteristic
388C23#54611yescharacteristic
389C23#54811yescharacteristic
390C23D4×Q811yescharacteristic
391C23C22×D811yescharacteristic
392C23C22×SD1611yescharacteristic
393C23C2×#13711yescharacteristic
394C23C2×#13711yescharacteristic
395C23#57611yescharacteristic
396C23#57711yescharacteristic
397C23#57711yescharacteristic
398C23#57811yescharacteristic
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#40014yes—
462C2×C4#40014yes—
463C2×C4#47414yes—
464C2×C4#47514yes—
465C2×C4#48014yes—
466C2×C4#48314yes—
467C2×C4#48614yes—
468C2×C4#49114yes—
469C2×C4C2×#12511yescharacteristic
470C2×C4C2×#12511yescharacteristic
471C2×C4C2×#12511yescharacteristic
472C2×C4C2×#12611yescharacteristic
473C2×C4C2×#12611yescharacteristic
474C2×C4C2×C4⋊Q811yescharacteristic
475C2×C4#53211yescharacteristic
476C2×C4#53211yescharacteristic
477C2×C4#53311yescharacteristic
478C2×C4#53311yescharacteristic
479C2×C4#54011yescharacteristic
480C2×C4#54111yescharacteristic
481C2×C4#54211yescharacteristic
482C2×C4#54211yescharacteristic
483C2×C4#54411yescharacteristic
484C2×C4#54411yescharacteristic
485C2×C4#54711yescharacteristic
486C2×C4#54711yescharacteristic
487C2×C4#55011yescharacteristic
488C2×C4#55211yescharacteristic
489C2×C4#55311yescharacteristic
490C2×C4#55411yescharacteristic
491C2×C4#55411yescharacteristic
492C2×C4#55511yescharacteristic
493C2×C4#55611yescharacteristic
494C2×C4#55611yescharacteristic
495C2×C4#55711yescharacteristic
496C2×C4#55911yescharacteristic
497C8—42no—
498C8—42no—
499C8—82no—
500C22C2×#40011yescharacteristic
501C22#244011yescharacteristic
502C22#244211yescharacteristic
503C22#244511yescharacteristic
504C22#244611yescharacteristic
505C22#249111yescharacteristic
506C22#249211yescharacteristic
507C22C2×#47411yescharacteristic
508C22C2×#47511yescharacteristic
509C22#308311yescharacteristic
510C22#308411yescharacteristic
511C22C2×#48011yescharacteristic
512C22C2×#48311yescharacteristic
513C22#309611yescharacteristic
514C22#309711yescharacteristic
515C22C2×#48611yescharacteristic
516C22C2×#49111yescharacteristic
517C22#313611yescharacteristic
518C22#313711yescharacteristic
519C22#327611yescharacteristic
520C22#327811yescharacteristic
521C22D4×D811yescharacteristic
522C22D4×SD1611yescharacteristic
523C22#329411yescharacteristic
524C22#329811yescharacteristic
525C22#333111yescharacteristic
526C22#333311yescharacteristic
527C22#335811yescharacteristic
528C22#336211yescharacteristic
529C22#337211yescharacteristic
530C22#337411yescharacteristic
531C22Q8×D811yescharacteristic
532C22Q8×SD1611yescharacteristic
533C22#338511yescharacteristic
534C22#338711yescharacteristic
535C4—24no—
536C4—44no—
537C4—44no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—162no—
543C4—162no—
544C4#209918yescomplemented
545C4#249611yescharacteristic
546C4#249711yescharacteristic
547C4#251111yescharacteristic
548C4#251211yescharacteristic
549C4#255211yescharacteristic
550C4#257111yescharacteristic
551C4#264711yescharacteristic
552C4#285911yescharacteristic
553C2#1514611yescharacteristic
554C2C2×#209911yescharacteristic
555C2#2374811yescharacteristic
556C2#2604011yescharacteristic
557C2#2604211yescharacteristic
558C2#2610611yescharacteristic
559C2#2610811yescharacteristic
560C2#2623011yescharacteristic
561C2#2623411yescharacteristic
562C2#2627611yescharacteristic
563C2#2628011yescharacteristic
564C2#2776711yescharacteristic
565C2#2777711yescharacteristic
566C2#2781311yescharacteristic
567C2#2782411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
544. C4#2099semidirect

Maximal subgroups

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

subgroupindex
C2×#20992
#139102
#112922
#112072
#108492
#108412
#108342
C4×#3402