Finite Group Databasebeta

← #196068

Subgroups — #196068

Order 512.

1378 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic, complemented
4C2×#1469C211yesmaximal, characteristic
5C2×#1469C211yesmaximal, characteristic
6#9084C214yesmaximal, complemented
7#9080C214yesmaximal, complemented
8#7912C211yesmaximal, characteristic, complemented
9#7908C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C411yescharacteristic
11C22×#124C411yescharacteristic
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#2984—24no—
15#2984—24no—
16C2×#446—22no—
17C2×#446—21no—
18C2×C8×Q8—21no—
19C2×#432C2211yescharacteristic
20C2×#432C2212yes—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#431C2212yes—
24C2×#431C2211yescharacteristic
25#2545—21no—
26#2545—22no—
27#2545—21no—
28#2545—21no—
29#2545—21no—
30#2545—22no—
31C2×#396C414yes—
32C2×#391C411yescharacteristic
33C2×#390C411yescharacteristic
34C2×#388C2214yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1985—22no—
37#1984—21no—
38#1984—21no—
39#1817C2214yes—
40#1469—24no—
41#1469—24no—
42C2×#323C2212yes—
43C2×#323C2214yes—
44C2×#323C2212yes—
45#1280C2212yes—
46#1280C2214yes—
47#1280C2212yes—
48C23×Q8C2×C411yescharacteristic
49C23×Q8C2×C411yescharacteristic
50C23×D4C2×C411yescharacteristic
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—44no—
55C2×#124—21no—
56C2×#124—22no—
57C2×#124—21no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—21no—
61C2×#124—21no—
62C2×#124—21no—
63C2×#124—21no—
64C2×#124—22no—
65C2×#124—22no—
66C2×#124—44no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D412yes—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D412yes—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C412yes—
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C414yes—
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C414yes—
83C22×C22⋊C4C2×C414yes—
84C22×C22⋊C4C2312yes—
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42C2312yes—
88C22×C42C2311yescharacteristic
89#446—216no—
90#446—28no—
91C8×Q8—28no—
92#432—22no—
93#432—24no—
94#432—24no—
95#432—22no—
96#432—22no—
97#432D414yes—
98#431—22no—
99#431—24no—
100#431—24no—
101#431—22no—
102#431—22no—
103#431D414yes—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C22⋊C8C2312yes—
111C2×C22⋊C8C2312yes—
112C2×C22⋊C8C2311yescharacteristic
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2311yescharacteristic
115C2×#97—22no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—22no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121#396—24no—
122#396—24no—
123#396—24no—
124#396—24no—
125#391—22no—
126#391—21no—
127#391—21no—
128#390—22no—
129#390—21no—
130#390—21no—
131#388—24no—
132#388—24no—
133#388—24no—
134#388—24no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C2×#95C2×C414yes—
140C2×#95C2312yes—
141#336—24no—
142#336—24no—
143#323—28no—
144#323—24no—
145#323—24no—
146C25C22×C412yes—
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8C22⋊C411yescharacteristic
158C22×Q8C22⋊C412yes—
159C22×Q8C22⋊C411yescharacteristic
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C4⋊Q8—48no—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182#124—48no—
183#124—44no—
184#124—44no—
185#124—42no—
186#124—42no—
187#124—42no—
188#124—42no—
189#124—42no—
190#124—42no—
191#124—42no—
192#124—42no—
193C4×Q8—22no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—24no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—24no—
202C4×Q8—22no—
203C4×Q8—24no—
204C4×Q8—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—48no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4112yes—
261C2×C42#4112yes—
262C4⋊C8—28no—
263C4⋊C8—28no—
264C4⋊C8—24no—
265C4⋊C8—24no—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8C2×D412yes—
276C22⋊C8C2×D414yes—
277C22⋊C8C2×D412yes—
278#97—28no—
279#97—24no—
280#97—42no—
281C4×C8—28no—
282C4×C8—24no—
283C4×C8—42no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—24no—
292C24—24no—
293C24—42no—
294C24—42no—
295C24—44no—
296C24C23×C411yescharacteristic
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—42no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8#10014yes—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—84no—
323C2×D4—84no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#13114yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—42no—
374C2×C8—44no—
375C2×C8—42no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42#12012yes—
414C42#12012yes—
415C42#12112yes—
416C42#12112yes—
417C42#12112yes—
418C42#12112yes—
419C42#12312yes—
420C42#12312yes—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—82no—
433C23—82no—
434C23—84no—
435C23—84no—
436C23—84no—
437C23—84no—
438C23—82no—
439C23—82no—
440C23C22×C22⋊C411yescharacteristic
441C23C2×C4×D411yescharacteristic
442C23C2×C4×D411yescharacteristic
443C23C2×#13112yes—
444C23#56812yes—
445Q8—28no—
446Q8—48no—
447Q8—44no—
448Q8—84no—
449Q8—84no—
450D4—84no—
451D4—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#38612yes—
497C2×C4#38612yes—
498C2×C4#38612yes—
499C2×C4#38612yes—
500C2×C4C2×#10014yes—
501C2×C4C2×#10012yes—
502C2×C4#41312yes—
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×#12311yescharacteristic
507C8—42no—
508C8—44no—
509C8—42no—
510C8—82no—
511C8—82no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—162no—
526C22—162no—
527C22—164no—
528C22C2×#38611yescharacteristic
529C22#288012yes—
530C22#288412yes—
531C22C22×#10011yescharacteristic
532C22C2×#41311yescharacteristic
533C22#293111yescharacteristic
534C22#293211yescharacteristic
535C22#293512yes—
536C4—22no—
537C4—22no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—22no—
543C4—24no—
544C4—44no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—82no—
549C4—82no—
550C4#190014yes—
551C2—21no—
552C2—21no—
553C2—22no—
554C2—162no—
555C2#1742512yes—
556C2#1742812yes—
557C2#2090011yescharacteristic
558C2#2098611yescharacteristic
559C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2984C2semidirect
3. #21820C2semidirect
6. #9084C2semidirect
7. #9080C2semidirect
8. #7912C2semidirect
9. #7908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
C2×#25452
#218202
C2×#14692
#90842
#90802
#79122
#79082