Finite Group Databasebeta

← #381871

Subgroups — #381871

Order 512.

1354 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×#2001C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic, complemented
3C2×#1750C211yesmaximal, characteristic
4#14258C212yesmaximal
5#11257C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11024C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10926C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#380C2212yes—
26#2001C418yescomplemented
27#1971—24no—
28#1750—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1441—22no—
37#1441—22no—
38#1440—22no—
39#1440—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8C2312yes—
43C24×C4C2311yescharacteristic
44C22×Q16—22no—
45C22×Q16—22no—
46C22×Q16—42no—
47C22×Q16—42no—
48C22×Q16C2×C414yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—42no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8C2×C418yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—42no—
62C2×#117—42no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#106—22no—
74C2×#106—42no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103C2×C418yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2×C414yes—
90C2×#97—42no—
91C2×#97D414yes—
92C2×C4×C8—22no—
93C2×C4×C8—42no—
94#391—24no—
95#391—24no—
96#391—42no—
97#391—42no—
98#380—42no—
99#380—42no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109#340—22no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340C2×C414yes—
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C414yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×Q16—28no—
150C2×Q16—48no—
151C2×Q16—44no—
152C2×Q16—44no—
153C2×Q16—48no—
154C22×C8—22no—
155C22×C8—22no—
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—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8#4112yes—
173C22×C8#4112yes—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C22⋊Q8—48no—
178#117—44no—
179#117—44no—
180#117—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C22×C414yes—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D412yes—
221C2×C42#4112yes—
222#106—44no—
223#103—48no—
224#103—48no—
225#103—48no—
226C22⋊C8—28no—
227C22⋊C8—28no—
228#97—44no—
229C4×C8—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—82no—
236#95—82no—
237#95—82no—
238#95—82no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24C4×D412yes—
248C24C22×D411yescharacteristic
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
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×C4C4×D412yes—
341C22×C4#12114yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4H(5,𝔽2)12yes—
349C22×C4H(5,𝔽2)12yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352Q16—416no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8C4×D414yes—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—416no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C22⋊C4—48no—
388C22⋊C4—416no—
389C22⋊C4—88no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—41no—
394C42—41no—
395C42—42no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23C2×C4×D411yescharacteristic
418C23C2×#12111yescharacteristic
419C23#53211yescharacteristic
420C23#53412yes—
421C23#53512yes—
422C23#57712yes—
423C23#57712yes—
424C23#57812yes—
425C23#57812yes—
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—82no—
431Q8—82no—
432Q8—82no—
433Q8—82no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C4×#4111yescharacteristic
489C2×C4#51812yes—
490C2×C4#51812yes—
491C2×C4#52511yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53411yescharacteristic
494C2×C4#53511yescharacteristic
495C2×C4#53511yescharacteristic
496C2×C4#53511yescharacteristic
497C2×C4#53812yes—
498C2×C4#53812yes—
499C2×C4#53812yes—
500C2×C4#53812yes—
501C2×C4#53812yes—
502C2×C4#54012yes—
503C2×C4#54012yes—
504C2×C4#54012yes—
505C8—24no—
506C8—42no—
507C8—42no—
508C8—42no—
509C22—42no—
510C22—42no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C4×#12111yescharacteristic
519C22#232112yes—
520C22#261711yescharacteristic
521C22#261911yescharacteristic
522C22#266112yes—
523C22#294712yes—
524C22#295011yescharacteristic
525C22#295211yescharacteristic
526C22#306312yes—
527C22#306312yes—
528C22#317311yescharacteristic
529C22#317711yescharacteristic
530C22#317812yes—
531C22#320112yes—
532C22#320112yes—
533C22#320212yes—
534C22#320412yes—
535C22#322212yes—
536C22#322312yes—
537C22#322512yes—
538C22#322512yes—
539C4—44no—
540C4—42no—
541C4—84no—
542C4—82no—
543C4—84no—
544C4—162no—
545C4—162no—
546C4#232111yescharacteristic
547C4#232811yescharacteristic
548C4#233312yes—
549C4#263111yescharacteristic
550C4#263511yescharacteristic
551C4#264812yes—
552C2—82no—
553C2#1431111yescharacteristic
554C2#2153011yescharacteristic
555C2#2154211yescharacteristic
556C2#2205412yes—
557C2#2205912yes—
558C2#2974612yes—
559C2#2981712yes—
560C2#3033112yes—
561C2#3033412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1971C2semidirect
5. #11257C2semidirect
6. #11190C2semidirect
8. #11024C2semidirect
10. #10926C2semidirect
11. #10821C2semidirect
26. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#19712
C2×#17502
#142582
#112572
#111902
#110952
#110242
#109702
#109262
#108212