Finite Group Databasebeta

← #455191

Subgroups — #455191

Order 512.

1444 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3C2×#2056C212yesmaximal
4C2×#1865C211yesmaximal, characteristic
5#15188C211yesmaximal, characteristic
6#13935C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#11067C212yesmaximal
10#11024C212yesmaximal
11#11010C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#462—28no—
16C2×#462—24no—
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30#2066—28no—
31#2056—28no—
32#1865—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1442—22no—
39#1441—21no—
40#1441—21no—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44#1434—21no—
45C4×#95C2211yescharacteristic
46C23×Q8C2312yes—
47C23×Q8C2311yescharacteristic
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—42no—
54C22×Q16—42no—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×C4×Q8D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#462—232no—
79C2×#106—24no—
80C2×#106—22no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—42no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103D414yes—
99C2×#103D418yes—
100#391—24no—
101#391—24no—
102#391—24no—
103#391—42no—
104#391—42no—
105#391—42no—
106#389—24no—
107#389—22no—
108#389—22no—
109#389—42no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122C2×#95D412yes—
123#340—24no—
124#340—24no—
125#340—22no—
126#336—22no—
127#336—42no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C2×D414yes—
149C22×Q8C2×D412yes—
150C22×Q8C2×D412yes—
151C22×Q8#4112yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C2×Q16—216no—
164C2×Q16—48no—
165C2×Q16—48no—
166C2×Q16—48no—
167C2×Q16—48no—
168C2×Q16—48no—
169C2×Q16—48no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×D414yes—
186C22×C8C2×D412yes—
187C4⋊Q8—28no—
188C4⋊Q8—48no—
189C4⋊Q8—44no—
190C4×Q8—28no—
191C4×Q8—44no—
192C4×Q8—48no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—44no—
235C2×C42C2×D412yes—
236C2×C42#4112yes—
237#106—44no—
238C4⋊C8—28no—
239C4⋊C8—48no—
240#103—28no—
241#103—216no—
242#103—48no—
243#103—44no—
244#103—48no—
245#95—22no—
246#95—24no—
247#95—22no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—81no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8#12114yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#13512yes—
363C22×C4#13714yes—
364C22×C4#13712yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+412yes—
374C22×C42−1+411yescharacteristic
375Q16—432no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—88no—
410C4⋊C4—84no—
411C4⋊C4#12014yes—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—48no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C23C2×#12111yescharacteristic
421C23#53712yes—
422C23D4212yes—
423C23D4211yescharacteristic
424C23#54811yescharacteristic
425C23C2×#13511yescharacteristic
426C23C2×#13711yescharacteristic
427C23C2×#13712yes—
428C23#57711yescharacteristic
429C23#57811yescharacteristic
430C23#57812yes—
431Q8—24no—
432Q8—24no—
433Q8—42no—
434Q8—44no—
435Q8—44no—
436Q8—42no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—82no—
442Q8—82no—
443Q8—82no—
444Q8—82no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
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—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4#46414yes—
505C2×C4C2×#12011yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4#53411yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#53612yes—
511C2×C4#53712yes—
512C2×C4#53711yescharacteristic
513C2×C4#53712yes—
514C2×C4#53912yes—
515C2×C4#54212yes—
516C2×C4D4212yes—
517C2×C4#54612yes—
518C2×C4#54612yes—
519C2×C4#54711yescharacteristic
520C2×C4#55311yescharacteristic
521C2×C4#56012yes—
522C2×C4#56212yes—
523C8—42no—
524C8—44no—
525C8—81no—
526C22#241011yescharacteristic
527C22#241611yescharacteristic
528C22#262011yescharacteristic
529C22#267512yes—
530C22#268112yes—
531C22C2×#46411yescharacteristic
532C22#303911yescharacteristic
533C22#304411yescharacteristic
534C22#304611yescharacteristic
535C22#324812yes—
536C22#325512yes—
537C22#325812yes—
538C22#326112yes—
539C22#326612yes—
540C22#327012yes—
541C22#327012yes—
542C22#327612yes—
543C22#327611yescharacteristic
544C22#328411yescharacteristic
545C22#328512yes—
546C22#336411yescharacteristic
547C22#336911yescharacteristic
548C22#337412yes—
549C4—24no—
550C4—42no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—84no—
555C4—82no—
556C4—82no—
557C4—161no—
558C4—162no—
559C4#240811yescharacteristic
560C4#261611yescharacteristic
561C4#261811yescharacteristic
562C4#267212yes—
563C4#269112yes—
564C4#281311yescharacteristic
565C2#1626111yescharacteristic
566C2#2436111yescharacteristic
567C2#2437611yescharacteristic
568C2#2472911yescharacteristic
569C2#2474611yescharacteristic
570C2#2983111yescharacteristic
571C2#2983711yescharacteristic
572C2#3085412yes—
573C2#3086412yes—
574C2#3114912yes—
575C2#3117312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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
C22×#4622
C2×#20662
C2×#20562
C2×#18652
#151882
#139352
#113222
#112272
#110672
#110242
#110102