Finite Group Databasebeta

← #411485

Subgroups — #411485

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1927C212yesmaximal
2#14714C211yesmaximal, characteristic
3#13755C211yesmaximal, characteristic
4#11323C211yesmaximal, characteristic
5#11267C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11122C212yesmaximal
8#11010C211yesmaximal, characteristic
9#11010C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376—21no—
24C2×C43C2211yescharacteristic
25#1927—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42#1434—21no—
43C4×#95—21no—
44C23×Q8C2312yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42—21no—
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#106—24no—
70C2×#106—42no—
71C2×#106—42no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84#391—24no—
85#391—24no—
86#391—42no—
87#391—42no—
88#384—42no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4D414yes—
93#376—44no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C43—22no—
109C43—22no—
110#340—24no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—24no—
115#336—22no—
116#336—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8#4114yes—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8#4114yes—
158C4⋊Q8—48no—
159C4⋊Q8—48no—
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×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
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—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4#4112yes—
193C2×C4⋊C4#4112yes—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42SD1614yes—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226#106—48no—
227#103—216no—
228#103—48no—
229#103—48no—
230#103—48no—
231#95—24no—
232#95—24no—
233#95—42no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—42no—
245C24C22×D411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4#12312yes—
350C22×C4C2×SD1612yes—
351C22×C4C2×SD1612yes—
352C22×C4C2×SD1612yes—
353C22×C4#13512yes—
354C22×C4#13712yes—
355C22×C4#13712yes—
356C22×C4#13712yes—
357C22×C4#13712yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4H(5,𝔽2)12yes—
367C22×C4H(5,𝔽2)12yes—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—24no—
403C42—24no—
404C42—28no—
405C42—44no—
406C42—44no—
407C42—48no—
408C42—42no—
409C42—42no—
410C42—48no—
411C42—44no—
412C23C2×#12111yescharacteristic
413C23C2×#12311yescharacteristic
414C23#52511yescharacteristic
415C23#52511yescharacteristic
416C23#53211yescharacteristic
417C23#53511yescharacteristic
418C23#53511yescharacteristic
419C23C22×SD1611yescharacteristic
420C23C2×#13511yescharacteristic
421C23C2×#13711yescharacteristic
422C23C2×#13711yescharacteristic
423C23C2×#13711yescharacteristic
424C23C2×#13711yescharacteristic
425C23#57511yescharacteristic
426C23#57511yescharacteristic
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—82no—
432Q8—82no—
433Q8—82no—
434Q8—82no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4#46114yes—
517C2×C4#48314yes—
518C2×C4#48514yes—
519C2×C4#52512yes—
520C2×C4#52911yescharacteristic
521C2×C4#52912yes—
522C2×C4#52911yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53211yescharacteristic
525C2×C4#53512yes—
526C2×C4#54012yes—
527C2×C4#55112yes—
528C2×C4#55112yes—
529C2×C4#55112yes—
530C2×C4#55112yes—
531C2×C4#55312yes—
532C2×C4#55312yes—
533C2×C4#55512yes—
534C2×C4#55512yes—
535C8—44no—
536C8—82no—
537C22#239211yescharacteristic
538C22#248211yescharacteristic
539C22#250011yescharacteristic
540C22#259811yescharacteristic
541C22#261011yescharacteristic
542C22#261911yescharacteristic
543C22#285411yescharacteristic
544C22C2×#46111yescharacteristic
545C22#303511yescharacteristic
546C22#303711yescharacteristic
547C22#303911yescharacteristic
548C22C2×#48311yescharacteristic
549C22C2×#48511yescharacteristic
550C22#309511yescharacteristic
551C22#309511yescharacteristic
552C22#310111yescharacteristic
553C22#310311yescharacteristic
554C22#310711yescharacteristic
555C22#310811yescharacteristic
556C22#311311yescharacteristic
557C22#311311yescharacteristic
558C22#311611yescharacteristic
559C22#311711yescharacteristic
560C22#316711yescharacteristic
561C22#317211yescharacteristic
562C22#318111yescharacteristic
563C22#318511yescharacteristic
564C22#320611yescharacteristic
565C22#320611yescharacteristic
566C22#321011yescharacteristic
567C22#321111yescharacteristic
568C22#321511yescharacteristic
569C22#321811yescharacteristic
570C22#321811yescharacteristic
571C22#321911yescharacteristic
572C4—24no—
573C4—24no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—84no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—162no—
584C4#252612yes—
585C4#253112yes—
586C4#260312yes—
587C4#261012yes—
588C2#1512011yescharacteristic
589C2#2367711yescharacteristic
590C2#2368511yescharacteristic
591C2#2758311yescharacteristic
592C2#2758911yescharacteristic
593C2#2788911yescharacteristic
594C2#2789611yescharacteristic
595C2#2936811yescharacteristic
596C2#2937111yescharacteristic
597C2#2955411yescharacteristic
598C2#2955911yescharacteristic
599C2#2979811yescharacteristic
600C2#2980111yescharacteristic
601C2#3310811yescharacteristic
602C2#3311411yescharacteristic
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×#19272
#147142
#137552
#113232
#112672
#111902
#111222
#110102
#109322
#107952