Finite Group Databasebeta

← #381013

Subgroups — #381013

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#14682C211yesmaximal, characteristic
3#13753C211yesmaximal, characteristic
4#11381C214yesmaximal
5#11353C214yesmaximal
6#10854C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C22⋊Q8C2212yes—
9C23×C42C2211yescharacteristic
10#2857—21no—
11#2857—21no—
12#2857—21no—
13#2857—21no—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18C2×#391C412yescomplemented
19C2×#389C412yescomplemented
20C2×#389C412yescomplemented
21C2×#384C411yescharacteristic, complemented
22C2×#382—21no—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2212yes—
27C2×#376—21no—
28C2×#342C22116yes—
29C4⋊#95—21no—
30#1434—21no—
31#1434—21no—
32#1434—21no—
33#1434—21no—
34C22⋊#95C2212yes—
35C22⋊#95C2212yes—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C4×#95—21no—
39C23×Q8C2×C412yes—
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—44no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—48no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C414yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C22⋊C4C2314yes—
61C22×C22⋊C4C2314yes—
62C22×C22⋊C4C2314yes—
63C22×C42—22no—
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77#391—22no—
78#391—22no—
79#391—22no—
80#391—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#382—44no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—28no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4Q818yescomplemented
105#376—44no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95C2×C414yes—
117C2×#95C2×C414yes—
118#342—416no—
119C25C2411yescharacteristic
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—44no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C414yes—
139C23×C4C22×C414yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×Q814yes—
156C23×C4#4114yes—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—44no—
160C4⋊Q8—44no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164C22⋊Q8—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—216no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4C2×Q818yes—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—28no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—28no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241C24—41no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24C23×C411yescharacteristic
246C24C22×D411yescharacteristic
247C24C22×D411yescharacteristic
248C24C22×Q812yes—
249C24C2×#4112yes—
250C24H(5,𝔽2)14yes—
251C242−1+414yes—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—816no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
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—28no—
292C22×C4—28no—
293C22×C4—28no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—416no—
315C22×C4—416no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C4×Q814yes—
346C22×C4C4×Q814yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22⋊Q814yes—
349C22×C4#12412yes—
350C22×C4#12412yes—
351C22×C4C4⋊Q812yes—
352C22×C4C4⋊Q812yes—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—88no—
366C22⋊C4—216no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—416no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—416no—
387C42#9914yes—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23C22×C22⋊C411yescharacteristic
409C23C2×C4×Q812yes—
410C23#51812yes—
411C23#51912yes—
412C23C2×C22⋊Q812yes—
413C23C2×#12411yescharacteristic
414C23C2×C4⋊Q811yescharacteristic
415C23#53512yes—
416C23#53512yes—
417C23#53612yes—
418C23#53612yes—
419C23#54218yes—
420C23#54418yes—
421Q8—88no—
422Q8—88no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—88no—
483C2×C4—88no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4#39112yes—
492C2×C4#39112yes—
493C2×C4#39112yes—
494C2×C4#39112yes—
495C2×C4C2×#9912yes—
496C2×C4C2×#9912yes—
497C2×C4C2×#9912yes—
498C2×C4#41012yes—
499C2×C4#41012yes—
500C2×C4#41012yes—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—22no—
516C22—21no—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—21no—
521C22—48no—
522C22—48no—
523C22C2×#39111yescharacteristic
524C22#230911yescharacteristic
525C22#231011yescharacteristic
526C22#233114yes—
527C22#262814yes—
528C22#264012yes—
529C22#266012yes—
530C22C22×#9911yescharacteristic
531C22C2×#41011yescharacteristic
532C22C2×#41011yescharacteristic
533C22C2×#41011yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—88no—
546C4—84no—
547C4—84no—
548C4—164no—
549C4#199614yes—
550C2—24no—
551C2—22no—
552C2—24no—
553C2—22no—
554C2#1428211yescharacteristic
555C2C2×#199611yescharacteristic
556C2#2133211yescharacteristic
557C2#2179311yescharacteristic
558C2#2179411yescharacteristic
559C2#2182511yescharacteristic
560C2#2182611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C22×C4⋊Q8C4semidirect
18. C2×#391C4semidirect
19. C2×#389C4semidirect
20. C2×#389C4semidirect
21. C2×#384C4semidirect
104. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#146822
#137532
#113812
#113532
#108542