Finite Group Databasebeta

← #198205

Subgroups — #198205

Order 512.

1049 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#17672C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6#9246C218yesmaximal, complemented
7#7990C211yesmaximal, characteristic
8#7934C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3078—21no—
12#3078—21no—
13#3078—22no—
14#3078—21no—
15#3078—21no—
16#3078—22no—
17C2×#479C2214yes—
18C2×#477C412yes—
19C2×#475C412yes—
20#2984—22no—
21#2984—22no—
22#2984—22no—
23#2984C2214yes—
24C2×#446—22no—
25C2×#446—21no—
26C2×C8×Q8—21no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432—21no—
30C2×#432C2211yescharacteristic
31C2×#431—21no—
32C2×#431C2211yescharacteristic
33C2×#428C412yes—
34C2×#423C2211yescharacteristic
35C2×#419C412yes—
36#1537—22no—
37#1531—22no—
38#1528C2214yes—
39#1491C2214yes—
40#1487—22no—
41#1479—21no—
42#1479—21no—
43#1472—21no—
44#1472—21no—
45#1470C2214yes—
46#1461—22no—
47#1290C2214yes—
48#1283C2214yes—
49C23×Q8C2×C411yescharacteristic
50C22×M16C2×C412yes—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#125C2312yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D412yes—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×#117C2×C412yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#479—24no—
86#479—24no—
87#479—24no—
88#479—24no—
89#477—22no—
90#477—22no—
91#477—22no—
92#477—22no—
93#475—22no—
94#475—22no—
95#475—22no—
96#475—22no—
97#446—28no—
98#446—24no—
99#446—24no—
100#446—28no—
101C8×Q8—24no—
102C8×Q8—24no—
103#432—24no—
104#432—21no—
105#432—21no—
106#432—24no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432D412yes—
111#432C2312yes—
112#432C2312yes—
113#431—24no—
114#431—21no—
115#431—21no—
116#431—24no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431D412yes—
122#431C2312yes—
123#428—24no—
124#428—22no—
125#428—22no—
126C2×#107C2×C414yes—
127C2×#106C2×C414yes—
128#423—21no—
129#423—21no—
130#423—21no—
131#423—21no—
132#423—21no—
133#423—21no—
134C2×C4⋊C8—22no—
135C2×C4⋊C8—22no—
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×C412yes—
139C2×C4⋊C8C2312yes—
140#419—22no—
141#419—24no—
142#419—22no—
143C2×#103C2×C414yes—
144C2×#103C2×C414yes—
145#407—21no—
146#407—21no—
147C2×C22⋊C8—21no—
148C2×C22⋊C8D411yescharacteristic
149C2×C22⋊C8D411yescharacteristic
150C4×M16—21no—
151C4×M16C2312yes—
152C2×#97—22no—
153C2×#97D412yes—
154C2×C4×C8—22no—
155C2×C4×C8—21no—
156#330—28no—
157Q8⋊C8—24no—
158Q8⋊C8—24no—
159#324—24no—
160#324—24no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—24no—
167C22×Q8—42no—
168C22×Q8C22⋊C411yescharacteristic
169C22×Q8C22⋊C412yes—
170C22×Q8C22⋊C411yescharacteristic
171C22×Q8C22×C412yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—44no—
181C2×M16—41no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8C22×C414yes—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194#125—22no—
195#125—22no—
196#125—22no—
197#125—22no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—21no—
209C4×Q8—22no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—22no—
214C4×Q8—21no—
215C4×Q8—22no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8C22⋊C412yes—
231C4×Q8C22⋊C412yes—
232C4×Q8C22⋊C412yes—
233C4×Q8C22⋊C412yes—
234C4×Q8C2×D412yes—
235C4×Q8C2×D412yes—
236C4×Q8C2×D412yes—
237C4×Q8C2×D412yes—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—22no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42#4111yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42C2411yescharacteristic
285#107—24no—
286#107—24no—
287#106—24no—
288#106—24no—
289C4⋊C8—28no—
290C4⋊C8—28no—
291C4⋊C8—22no—
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—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303#103—24no—
304#103—24no—
305#103—24no—
306#103—24no—
307C22⋊C8—22no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8C2×D412yes—
316C22⋊C8C2×D412yes—
317#97—28no—
318#97—22no—
319#97—22no—
320#97—22no—
321#97C2×D414yes—
322C4×C8—28no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8—41no—
327C24C4×D411yescharacteristic
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—44no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—82no—
348C2×Q8#10114yes—
349C2×Q8C2×C22⋊C412yes—
350C2×Q8C2×C22⋊C412yes—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4#12011yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12311yescharacteristic
381C22×C4#12311yescharacteristic
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385M16—44no—
386M16—44no—
387M16—44no—
388M16—82no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—22no—
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—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4C2×C22⋊C412yes—
424C4⋊C4C2×C22⋊C412yes—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—42no—
436C42—42no—
437C42C2×C22⋊C411yescharacteristic
438C42C2×C22⋊C411yescharacteristic
439C42C4×D411yescharacteristic
440C42C4×D411yescharacteristic
441C42#12011yescharacteristic
442C42#12011yescharacteristic
443C42#12111yescharacteristic
444C42#12111yescharacteristic
445C42#12111yescharacteristic
446C42#12111yescharacteristic
447C42#12311yescharacteristic
448C42#12311yescharacteristic
449C42C22×D411yescharacteristic
450C42C22×D411yescharacteristic
451C42C22×D411yescharacteristic
452C42C2×#4111yescharacteristic
453C23—22no—
454C23—22no—
455C23—41no—
456C23#38611yescharacteristic
457C23#38611yescharacteristic
458C23C2×C4×D411yescharacteristic
459Q8—22no—
460Q8—24no—
461Q8—22no—
462Q8—22no—
463Q8—44no—
464Q8—44no—
465Q8—42no—
466Q8—82no—
467Q8—82no—
468Q8—82no—
469Q8—82no—
470Q8C2×#10114yes—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—82no—
506C2×C4#38611yescharacteristic
507C2×C4#38611yescharacteristic
508C2×C4#38611yescharacteristic
509C2×C4#38611yescharacteristic
510C2×C4#38611yescharacteristic
511C2×C4#38611yescharacteristic
512C2×C4C2×#10112yes—
513C2×C4#41312yes—
514C2×C4C22×C22⋊C411yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×#12011yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12111yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C8—42no—
521C8—42no—
522C8—44no—
523C8—44no—
524C8—82no—
525C8—82no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22#190112yes—
535C22C2×#38611yescharacteristic
536C22#295512yes—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—24no—
547C4—42no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4#190112yes—
553C4#190512yes—
554C4C22×#10111yescharacteristic
555C4C2×#41311yescharacteristic
556C2—22no—
557C2—22no—
558C2—41no—
559C2#2091711yescharacteristic
560C2C2×#190111yescharacteristic
561C2C2×#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
C2×#30782
C2×#29842
#176722
#174492
#173852
#92462
#79902
#79342