Finite Group Databasebeta

← #412507

Subgroups — #412507

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#13954C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic
8#11228C212yesmaximal
9#11220C212yesmaximal
10#11067C212yesmaximal
11#10843C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2066—28no—
28#1948—24no—
29#1940—24no—
30#1805—24no—
31#1763—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1444—22no—
40#1441—21no—
41#1441—21no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44C4⋊#95—22no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#106—24no—
72C2×#106—24no—
73C2×#106—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8D414yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103D414yes—
88C2×#103Q814yes—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—22no—
92#391—24no—
93#391—42no—
94#389—24no—
95#389—24no—
96#389—42no—
97#389—42no—
98#384—22no—
99#384—22no—
100#382—22no—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—42no—
123#336—24no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C2×D412yes—
136C22×Q8C2×Q812yes—
137C22×Q8C2×Q812yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8C2×D414yes—
168C22×C8C2×D412yes—
169C22×C8C2×Q812yes—
170C4×Q8—28no—
171C4×Q8—28no—
172C4×Q8—44no—
173C4×Q8—44no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4112yes—
219C2×C4⋊C4#4112yes—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—82no—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237#106—48no—
238C4⋊C8—28no—
239C4⋊C8—48no—
240#103—28no—
241#103—28no—
242#103—48no—
243#97—44no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—82no—
254#95—82no—
255C24C22×D411yescharacteristic
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C4⋊Q814yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4C22⋊Q812yes—
342C22×C4C22⋊Q812yes—
343C22×C4#13714yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q811yescharacteristic
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—41no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—48no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—88no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—88no—
403C4⋊C4#12114yes—
404C4⋊C4C22⋊Q814yes—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—48no—
410C42—84no—
411C42#12114yes—
412C42C22⋊Q814yes—
413C42C22⋊Q814yes—
414C42#12614yes—
415C23C2×#12111yescharacteristic
416C23C2×#12111yescharacteristic
417C23C2×C22⋊Q811yescharacteristic
418C23D4211yescharacteristic
419C23#54611yescharacteristic
420C23#54811yescharacteristic
421C23D4×Q811yescharacteristic
422C23C2×#13712yes—
423C23#57512yes—
424C23#57712yes—
425C23#57712yes—
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433Q8—42no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#39614yes—
493C2×C4#39614yes—
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4C2×C22⋊Q811yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C2×C4C2×#12611yescharacteristic
500C2×C4C2×C4⋊Q811yescharacteristic
501C2×C4#53211yescharacteristic
502C2×C4#53311yescharacteristic
503C2×C4#53311yescharacteristic
504C2×C4#53711yescharacteristic
505C2×C4#53711yescharacteristic
506C2×C4#54011yescharacteristic
507C2×C4#54111yescharacteristic
508C2×C4#54411yescharacteristic
509C2×C4#54411yescharacteristic
510C2×C4D4211yescharacteristic
511C2×C4#54711yescharacteristic
512C2×C4#54711yescharacteristic
513C2×C4#54711yescharacteristic
514C2×C4D4×Q811yescharacteristic
515C2×C4#55011yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55311yescharacteristic
518C2×C4#55411yescharacteristic
519C2×C4#55411yescharacteristic
520C2×C4#55511yescharacteristic
521C2×C4#55611yescharacteristic
522C8—42no—
523C8—44no—
524C8—42no—
525C22C2×#39611yescharacteristic
526C22#241611yescharacteristic
527C22#241711yescharacteristic
528C22#242811yescharacteristic
529C22#242911yescharacteristic
530C22#243911yescharacteristic
531C22#244011yescharacteristic
532C22#303911yescharacteristic
533C22#304011yescharacteristic
534C22#304211yescharacteristic
535C22#304311yescharacteristic
536C22#305612yes—
537C22#306012yes—
538C22#307812yes—
539C22#308412yes—
540C22#326612yes—
541C22#327112yes—
542C22#331112yes—
543C22#331612yes—
544C22#337112yes—
545C22#337212yes—
546C22#338712yes—
547C22#338812yes—
548C4—24no—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—82no—
556C4—162no—
557C4—162no—
558C4#244011yescharacteristic
559C4#244111yescharacteristic
560C4#249611yescharacteristic
561C4#249711yescharacteristic
562C4#253711yescharacteristic
563C4#253811yescharacteristic
564C4#262911yescharacteristic
565C4#285711yescharacteristic
566C2#1513111yescharacteristic
567C2#2361011yescharacteristic
568C2#2361911yescharacteristic
569C2#2474711yescharacteristic
570C2#2474811yescharacteristic
571C2#2483111yescharacteristic
572C2#2483411yescharacteristic
573C2#2540312yes—
574C2#2548812yes—
575C2#2603512yes—
576C2#2605412yes—
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

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#19482
C2×#19402
C2×#18052
C2×#17632
#139542
#139262
#112282
#112202
#110672
#108432