Finite Group Databasebeta

← #414880

Subgroups — #414880

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic
7#11257C212yesmaximal
8#11231C212yesmaximal
9#11190C212yesmaximal
10#11077C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#475—24no—
15C2×#475—24no—
16C2×#475—24no—
17C22×#106C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C2×#391C2212yes—
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1993—24no—
29#1940—24no—
30#1763—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1446—21no—
39#1446—21no—
40#1441—22no—
41C4⋊#95—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—21no—
46#1434—21no—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#475—232no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—42no—
85C2×C4⋊C8D414yes—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103Q814yescomplemented
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98#391—24no—
99#391—42no—
100#382—22no—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—42no—
125#336—24no—
126C22×Q8—24no—
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×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C22×Q8—82no—
142C22×Q8C2×D412yes—
143C22×Q8C2×Q812yes—
144C22×Q8#4112yes—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D414yes—
175C22×C8C2×Q812yes—
176C22×C8#4112yes—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179C4⋊Q8—44no—
180C4⋊Q8—44no—
181C4×Q8—28no—
182C4×Q8—28no—
183C4×Q8—44no—
184C4×Q8—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×Q812yes—
223C2×C4⋊C4#4112yes—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42—42no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#106—216no—
242C4⋊C8—28no—
243C4⋊C8—48no—
244#103—28no—
245#103—216no—
246#103—28no—
247#97—44no—
248#95—22no—
249#95—22no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—82no—
258#95—82no—
259#95—82no—
260#95—82no—
261C24C22×D411yescharacteristic
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—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—48no—
290C2×Q8—88no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—88no—
294C2×Q8C22⋊Q814yes—
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—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#13714yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×Q811yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+411yescharacteristic
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—48no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—48no—
418C42—84no—
419C42#12014yes—
420C42#12314yes—
421C42#12314yes—
422C42#12414yes—
423C23C2×#12111yescharacteristic
424C23C2×#12111yescharacteristic
425C23C2×#12111yescharacteristic
426C23#54611yescharacteristic
427C23#54711yescharacteristic
428C23#54811yescharacteristic
429C23D4×Q811yescharacteristic
430C23C2×#13712yes—
431C23#57512yes—
432C23#57712yes—
433C23#57712yes—
434Q8—24no—
435Q8—24no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—816no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
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#39414yes—
502C2×C4#39414yes—
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C2×C4C2×#12311yescharacteristic
507C2×C4#52511yescharacteristic
508C2×C4C2×#12411yescharacteristic
509C2×C4#52911yescharacteristic
510C2×C4#52911yescharacteristic
511C2×C4#53311yescharacteristic
512C2×C4#53311yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53711yescharacteristic
515C2×C4#54111yescharacteristic
516C2×C4#54111yescharacteristic
517C2×C4#54211yescharacteristic
518C2×C4#54211yescharacteristic
519C2×C4#54611yescharacteristic
520C2×C4#54711yescharacteristic
521C2×C4#54811yescharacteristic
522C2×C4#54811yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4D4×Q811yescharacteristic
525C2×C4D4×Q811yescharacteristic
526C2×C4#55011yescharacteristic
527C2×C4#55211yescharacteristic
528C2×C4#55211yescharacteristic
529C2×C4#55411yescharacteristic
530C2×C4#55611yescharacteristic
531C8—44no—
532C8—42no—
533C8—42no—
534C22C2×#39411yescharacteristic
535C22#241511yescharacteristic
536C22#241711yescharacteristic
537C22#242611yescharacteristic
538C22#242711yescharacteristic
539C22#242911yescharacteristic
540C22#242911yescharacteristic
541C22#303712yes—
542C22#303911yescharacteristic
543C22#304011yescharacteristic
544C22#304212yes—
545C22#304311yescharacteristic
546C22#304311yescharacteristic
547C22#305612yes—
548C22#305712yes—
549C22#331612yes—
550C22#332112yes—
551C22#334412yes—
552C22#334812yes—
553C22#336912yes—
554C22#337612yes—
555C22#338712yes—
556C22#338812yes—
557C4—24no—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—82no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4—82no—
566C4—162no—
567C4#240711yescharacteristic
568C4#240911yescharacteristic
569C4#242111yescharacteristic
570C4#242311yescharacteristic
571C4#248911yescharacteristic
572C4#249211yescharacteristic
573C4#262511yescharacteristic
574C4#285511yescharacteristic
575C2#1517911yescharacteristic
576C2#2354111yescharacteristic
577C2#2354711yescharacteristic
578C2#2469511yescharacteristic
579C2#2469811yescharacteristic
580C2#2482811yescharacteristic
581C2#2483111yescharacteristic
582C2#2528712yes—
583C2#2535612yes—
584C2#2546412yes—
585C2#2548712yes—
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
94. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#19932
C2×#19402
C2×#17632
#141612
#139422
#112572
#112312
#111902
#110772
#108502