Finite Group Databasebeta

← #198237

Subgroups — #198237

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic
3#17667C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5#17394C211yesmaximal, characteristic
6#9246C218yesmaximal, complemented
7#7988C211yesmaximal, characteristic
8#7935C211yesmaximal, characteristic
9C22×#125C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#479—22no—
12C2×#479—21no—
13C2×#479—21no—
14C2×#479—21no—
15C2×#479—21no—
16C2×#479—22no—
17C2×#479C414yes—
18C2×#479C2214yes—
19#2984—22no—
20#2984C2214yes—
21#2983—22no—
22#2983—21no—
23#2982—21no—
24Q8×M16—22no—
25Q8×M16—22no—
26#2920—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C22×#107C411yescharacteristic
32C22×#106C411yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C22×#103C411yescharacteristic
35C22×#103C411yescharacteristic
36#1534—22no—
37#1528—22no—
38#1528C2214yes—
39#1491C2214yes—
40#1487—22no—
41#1479—21no—
42#1479—21no—
43#1470—22no—
44#1470C2214yes—
45#1463—22no—
46#1290C2214yes—
47#1283C2214yes—
48C23×Q8C2×C411yescharacteristic
49C23×C8C2×C411yescharacteristic
50C23×C8C2×C411yescharacteristic
51C2×#125—21no—
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125C2×C412yes—
62C2×#125C2312yes—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D412yes—
76C2×C4×Q8C2312yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#479—24no—
86#479—24no—
87#479—24no—
88#479—24no—
89#479—24no—
90#479—24no—
91#479—24no—
92#479—24no—
93#446—28no—
94#446—24no—
95#446—24no—
96#446—28no—
97C8×Q8—24no—
98C8×Q8—24no—
99#432—24no—
100#432—21no—
101#432—24no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432D412yes—
106#432D412yes—
107#432C2312yes—
108#432C2312yes—
109#431—24no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—24no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431C2312yes—
119C2×#107—22no—
120C2×#107—21no—
121C2×#107—21no—
122C2×#107C2×C414yes—
123C2×#106—22no—
124C2×#106—21no—
125C2×#106—21no—
126C2×#106C2×C414yes—
127#423—24no—
128#423—21no—
129#423—21no—
130#423—21no—
131#423—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8C2×C412yes—
137C2×C4⋊C8C2312yes—
138C2×#103—21no—
139C2×#103—22no—
140C2×#103—21no—
141C2×#103—21no—
142C2×#103—22no—
143C2×#103—21no—
144C2×#103C2×C414yes—
145C2×#103C2×C414yes—
146#407—21no—
147#407—21no—
148#407—21no—
149#407D412yes—
150C4×M16—24no—
151C4×M16—21no—
152C4×M16—21no—
153C4×M16D412yes—
154C4×M16C2312yes—
155#330—28no—
156Q8⋊C8—24no—
157Q8⋊C8—24no—
158#324—28no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—24no—
163C22×Q8—42no—
164C22×Q8C22⋊C411yescharacteristic
165C22×Q8C22⋊C412yes—
166C22×Q8C22⋊C411yescharacteristic
167C22×Q8C22×C412yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—41no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8C22×C412yes—
186C22×C8C22×C412yes—
187#125—22no—
188#125—22no—
189#125—22no—
190#125—22no—
191#125—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—41no—
196#125—41no—
197#125—41no—
198#125—41no—
199#125—41no—
200#125—41no—
201#125—41no—
202#125—41no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—21no—
206C4×Q8—22no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—22no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—22no—
214C4×Q8—21no—
215C4×Q8—22no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8C22⋊C412yes—
220C4×Q8C22⋊C412yes—
221C4×Q8C22⋊C412yes—
222C4×Q8C22⋊C412yes—
223C4×Q8C2×D412yes—
224C4×Q8C2×D412yes—
225C4×Q8C2×D412yes—
226C4×Q8C2×D412yes—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42C2411yescharacteristic
286#107—24no—
287#107—24no—
288#106—24no—
289#106—24no—
290C4⋊C8—28no—
291C4⋊C8—28no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303C4⋊C8—41no—
304#103—24no—
305#103—24no—
306#103—24no—
307#103—24no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8C2×D414yes—
316#97—28no—
317#97—22no—
318#97—22no—
319#97—22no—
320#97C2×D414yes—
321C4×C8—28no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—41no—
326C24C4×D411yescharacteristic
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—44no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8#10114yes—
342C2×Q8C2×C22⋊C412yes—
343C2×Q8C2×C22⋊C412yes—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4#12011yescharacteristic
374C22×C4#12011yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12311yescharacteristic
380C22×C4#12311yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384M16—44no—
385M16—44no—
386M16—44no—
387M16—82no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4C2×C22⋊C412yes—
429C4⋊C4C2×C22⋊C412yes—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42C2×C22⋊C411yescharacteristic
439C42C2×C22⋊C411yescharacteristic
440C42C4×D411yescharacteristic
441C42C4×D411yescharacteristic
442C42#12011yescharacteristic
443C42#12011yescharacteristic
444C42#12111yescharacteristic
445C42#12111yescharacteristic
446C42#12111yescharacteristic
447C42#12111yescharacteristic
448C42#12311yescharacteristic
449C42#12311yescharacteristic
450C42C22×D411yescharacteristic
451C42C22×D411yescharacteristic
452C42C22×D411yescharacteristic
453C42C2×#4111yescharacteristic
454C23—22no—
455C23—22no—
456C23—22no—
457C23#38611yescharacteristic
458C23#38611yescharacteristic
459C23C2×C4×D411yescharacteristic
460Q8—22no—
461Q8—24no—
462Q8—22no—
463Q8—22no—
464Q8—44no—
465Q8—44no—
466Q8—42no—
467Q8C2×#10114yes—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#38611yescharacteristic
508C2×C4#38611yescharacteristic
509C2×C4#38611yescharacteristic
510C2×C4#38611yescharacteristic
511C2×C4#38611yescharacteristic
512C2×C4#38611yescharacteristic
513C2×C4C2×#10112yes—
514C2×C4#41312yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×#12011yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C8—44no—
522C8—44no—
523C8—42no—
524C8—42no—
525C8—82no—
526C8—82no—
527C22—28no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22#190112yes—
535C22#190512yes—
536C22C2×#38611yescharacteristic
537C22#295712yes—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—24no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4#190112yes—
555C4C22×#10111yescharacteristic
556C4C2×#41311yescharacteristic
557C2—22no—
558C2—22no—
559C2—22no—
560C2#2091911yescharacteristic
561C2C2×#190111yescharacteristic
562C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9246C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
#602482
#176672
#174492
#173942
#92462
#79882
#79352