Finite Group Databasebeta

← #364843

Subgroups — #364843

Order 512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C4×#419C211yesmaximal, characteristic
6#11210C212yesmaximal
7#11055C212yesmaximal
8#10957C212yesmaximal
9#10891C212yesmaximal
10#10852C211yesmaximal, characteristic
11#10851C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#419C2211yescharacteristic
16C2×#419C2212yes—
17C2×#419C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2567—24no—
20#2567C418yescomplemented
21C2×#395—22no—
22#2369—22no—
23#2365—22no—
24#2345—22no—
25Q8×C22⋊C4—22no—
26#2331—22no—
27#2316C2212yes—
28C4×C4⋊Q8—22no—
29C2×#391C2212yes—
30C2×#389—22no—
31#2285C2211yescharacteristic
32C2×#384C2212yes—
33#2274C2212yes—
34C4×#117C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1997—24no—
37#1805—22no—
38#1803—22no—
39#1763—22no—
40C4×#103—22no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2211yescharacteristic
44#1736C2212yes—
45#1736C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C23×Q8D412yes—
51C23×Q8C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8D414yes—
63C2×C4×Q8—24no—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8C2×C414yes—
72C2×#117C2311yescharacteristic
73C2×#117C2312yes—
74C2×#117C2311yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#423—22no—
85#423—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#419—22no—
89#419—22no—
90#419—22no—
91#419—24no—
92#419—42no—
93#419Q814yescomplemented
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97—21no—
100C2×C4×C8—21no—
101#395—216no—
102#391—24no—
103#391—28no—
104#391—24no—
105#391C2×C418yes—
106#389—24no—
107#389—216no—
108#389—24no—
109#389—24no—
110#387—28no—
111#384—24no—
112#384C2×C418yes—
113#384D412yes—
114#382D412yes—
115#380—24no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C414yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—24no—
129C4×C22⋊C4—21no—
130#376—21no—
131C2×#95—21no—
132C2×#95—24no—
133C43—21no—
134#343—24no—
135#343—24no—
136#340—24no—
137#340—22no—
138#340—22no—
139#339—24no—
140#339—24no—
141#339—42no—
142#339—42no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—24no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—84no—
158C22×Q8C22⋊C418yes—
159C22×Q8C22×C412yes—
160C22×Q8C2×Q812yes—
161C22×Q8#4112yes—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D414yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16C2×Q812yes—
186C2×M16#4112yes—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—24no—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—88no—
207#117—21no—
208#117—21no—
209#117—22no—
210#117—22no—
211#117—21no—
212#117—21no—
213#117—21no—
214#117—22no—
215#117—22no—
216#117—21no—
217#117C2×Q812yes—
218#117#4112yes—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—88no—
242C2×C4⋊C4C22⋊C418yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D412yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—42no—
267C2×C42C22×C414yes—
268C2×C42C22×C412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C4⋊C8—42no—
277C4⋊C8—42no—
278#103—28no—
279#103—28no—
280#103—48no—
281#97—42no—
282C4×C8—42no—
283#95—28no—
284#95—28no—
285#95—24no—
286#95—216no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291C24C22×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—28no—
296C2×Q8—28no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—48no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—84no—
323C2×Q8—88no—
324C2×Q8—88no—
325C2×Q8C4×Q814yescomplemented
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—48no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C418yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12011yescharacteristic
357C22×C4#12312yes—
358C22×C4#12312yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
372C22×C42−1+411yescharacteristic
373M16—22no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—42no—
382M16—42no—
383M16—42no—
384M16—42no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—216no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—28no—
401C4⋊C4—28no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—44no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—816no—
415C4⋊C4—88no—
416C4⋊C4C22⋊Q814yes—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—42no—
420C22⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C42—22no—
424C42—22no—
425C42—24no—
426C42—24no—
427C42—44no—
428C42—42no—
429C42—42no—
430C42—41no—
431C42—41no—
432C42—42no—
433C42—42no—
434C42—44no—
435C42—44no—
436C42—44no—
437C42—88no—
438C42C4×D412yes—
439C42C4×D412yes—
440C42C4×D412yes—
441C42C4×D412yes—
442C42#12112yes—
443C42#12112yes—
444C42#12112yes—
445C42#12112yes—
446C23—22no—
447C23—22no—
448C23C22×C22⋊C411yescharacteristic
449C23C2×#12011yescharacteristic
450C23C2×#12311yescharacteristic
451C23#54811yescharacteristic
452C23#54811yescharacteristic
453C23D4×Q811yescharacteristic
454C23#55011yescharacteristic
455Q8—24no—
456Q8—24no—
457Q8—416no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—88no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4#38612yes—
502C2×C4#38612yes—
503C2×C4#38612yes—
504C2×C4#38612yes—
505C2×C4#45614yes—
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×Q811yescharacteristic
509C2×C4C4×#4111yescharacteristic
510C2×C4#51911yescharacteristic
511C2×C4#52011yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×C22⋊Q811yescharacteristic
515C2×C4#52511yescharacteristic
516C2×C4#52911yescharacteristic
517C2×C4#53311yescharacteristic
518C2×C4#53611yescharacteristic
519C2×C4#53711yescharacteristic
520C2×C4#54111yescharacteristic
521C2×C4#54211yescharacteristic
522C2×C4#54611yescharacteristic
523C2×C4#54711yescharacteristic
524C2×C4#54811yescharacteristic
525C2×C4#54811yescharacteristic
526C2×C4#54811yescharacteristic
527C2×C4D4×Q811yescharacteristic
528C2×C4D4×Q811yescharacteristic
529C2×C4#55011yescharacteristic
530C2×C4#55211yescharacteristic
531C2×C4#55211yescharacteristic
532C2×C4#55411yescharacteristic
533C2×C4#55611yescharacteristic
534C8—42no—
535C8—42no—
536C8—82no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22C2×#38611yescharacteristic
546C22Q8×C22⋊C411yescharacteristic
547C22#234111yescharacteristic
548C22#240711yescharacteristic
549C22#240911yescharacteristic
550C22#242111yescharacteristic
551C22#242311yescharacteristic
552C22C2×#45612yes—
553C22#301912yes—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—48no—
558C4—44no—
559C4—44no—
560C4—48no—
561C4—82no—
562C4—84no—
563C4—82no—
564C4#188614yes—
565C4C4×C22⋊Q811yescharacteristic
566C4#232311yescharacteristic
567C4#236011yescharacteristic
568C4#236211yescharacteristic
569C4#242611yescharacteristic
570C4#242911yescharacteristic
571C4#253611yescharacteristic
572C4#253811yescharacteristic
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2#1396911yescharacteristic
578C2C2×#188611yescharacteristic
579C2#2095311yescharacteristic
580C2#2417312yes—
581C2#2432312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #2567C4semidirect
93. #419Q8semidirect
325. C2×Q8C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#342162
C2×#19972
#206472
C4×#4192
#112102
#110552
#109572
#108912
#108522
#108512