Finite Group Databasebeta

← #365482

Subgroups — #365482

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2C2×#1996C211yesmaximal, characteristic
3#14159C211yesmaximal, characteristic
4#14149C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic
6#11344C214yesmaximal
7#10918C214yesmaximal
8#10854C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11#2368—22no—
12#2360—22no—
13#2345—22no—
14#2345—21no—
15Q8×C22⋊C4—21no—
16#2329—22no—
17#2315C2212yes—
18C4×C22⋊Q8—22no—
19C2×#385C2214yes—
20C2×#385C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#2273C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C22×#95C2211yescharacteristic
28#1996C418yescomplemented
29C2×#342C2212yes—
30C2×#342C2214yes—
31C2×#342C2214yes—
32C2×#342C2212yes—
33#1446—21no—
34#1445—21no—
35#1444—22no—
36#1441—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95—22no—
40C22⋊#95C2212yes—
41C22⋊#95C2214yes—
42C23×Q8D411yescharacteristic
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C414yes—
52C2×C22⋊Q8Q814yes—
53C2×C4×Q8—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2312yes—
64C22×C22⋊C4C2312yes—
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2312yes—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D412yes—
70#391—24no—
71#391—22no—
72#389—24no—
73#389—22no—
74#387—24no—
75#387—24no—
76#387—24no—
77#385—24no—
78#385—24no—
79#385—28no—
80#385—44no—
81#385—44no—
82#380—22no—
83#380—22no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380—22no—
89#380C2×C414yes—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—24no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4C2×C414yes—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—21no—
109C2×#95—24no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—24no—
119C2×#95D412yes—
120C2×#95D412yes—
121#342—24no—
122#342—44no—
123#342C2×C4116yes—
124#342Q818yescomplemented
125C25C2411yescharacteristic
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—82no—
130C22×Q8C4⋊C414yescomplemented
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—44no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22×C412yes—
143C23×C4C2×D414yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×Q812yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C4×Q8—84no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C4⋊C414yescomplemented
186C2×C4⋊C4C4⋊C418yescomplemented
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4C22×C414yes—
214C2×C22⋊C4C22×C414yes—
215C2×C22⋊C4C22×C414yes—
216C2×C22⋊C4C2×Q814yes—
217C2×C22⋊C4C2×Q814yes—
218C2×C22⋊C4C2×Q814yes—
219C2×C22⋊C4C2×Q814yes—
220C2×C22⋊C4#4114yes—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—42no—
227C2×C42—82no—
228C2×C42—82no—
229C2×C42—82no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—48no—
241#95—48no—
242#95—44no—
243#95—44no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—42no—
249#95—82no—
250#95—84no—
251C24—42no—
252C24—42no—
253C24C23×C411yescharacteristic
254C24C22×D411yescharacteristic
255C24C22×D411yescharacteristic
256C24C22×D411yescharacteristic
257C24C22×Q811yescharacteristic
258C24C22×Q812yes—
259C24C2×#4111yescharacteristic
260C24C2×#4111yescharacteristic
261C24C2×#4112yes—
262C24H(5,𝔽2)12yes—
263C242−1+412yes—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—44no—
267C2×Q8—82no—
268C2×Q8—84no—
269C2×Q8—84no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
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—44no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—88no—
333C22×C4—88no—
334C22×C4C2×C4⋊C414yes—
335C22×C4C2×C4⋊C414yes—
336C22×C4C2×C4⋊C412yes—
337C22×C4C2×C4⋊C412yes—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4C22⋊Q812yes—
350C22×C4#12312yes—
351C22×C4#12312yes—
352C22×C4C4⋊Q814yes—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—44no—
358C4⋊C4—82no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—816no—
363C4⋊C4—88no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—28no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—44no—
383C22⋊C4—42no—
384C22⋊C4C4×Q818yescomplemented
385C42—44no—
386C42—82no—
387C42—84no—
388C42—84no—
389C42—84no—
390C42—84no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—44no—
402C23—44no—
403C23—41no—
404C23—41no—
405C23—44no—
406C23—41no—
407C23—41no—
408C23—44no—
409C23C22×C4⋊C411yescharacteristic
410C23C2×C4×D411yescharacteristic
411C23C2×C4×D411yescharacteristic
412C23C2×C4×Q812yes—
413C23#52012yes—
414C23C2×#12011yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23C2×C22⋊Q811yescharacteristic
417C23C2×#12311yescharacteristic
418C23C2×C4⋊Q811yescharacteristic
419C23#53211yescharacteristic
420C23#53312yes—
421C23#53511yescharacteristic
422C23#53611yescharacteristic
423C23#54212yes—
424C23#54612yes—
425C23D4×Q812yes—
426C23D4×Q812yes—
427C23#55012yes—
428C23#55212yes—
429C23#55412yes—
430C23#55712yes—
431C23Q8212yes—
432Q8—48no—
433Q8—84no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—48no—
456C2×C4—44no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—84no—
494C2×C4—82no—
495C2×C4—84no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4#38512yes—
502C2×C4#38512yes—
503C2×C4#38512yes—
504C2×C4#38512yes—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—22no—
518C22—24no—
519C22—22no—
520C22—24no—
521C22—22no—
522C22—24no—
523C22—24no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22C2×#38511yescharacteristic
534C22C4×C4⋊Q811yescharacteristic
535C22#232611yescharacteristic
536C22Q8×C4⋊C412yes—
537C22#236012yes—
538C22#242312yes—
539C22#243111yescharacteristic
540C22#243411yescharacteristic
541C22#245112yes—
542C22#256511yescharacteristic
543C22#256711yescharacteristic
544C22#302512yes—
545C22#302612yes—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—48no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—84no—
555C4—84no—
556C4—84no—
557C4—82no—
558C4—84no—
559C2—22no—
560C2—21no—
561C2—22no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—22no—
566C2—22no—
567C2#1397611yescharacteristic
568C2#2088811yescharacteristic
569C2#2088911yescharacteristic
570C2#2558712yes—
571C2#2585412yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1996C4semidirect
124. #342Q8semidirect
130. C22×Q8C4⋊C4semidirect
185. C2×C4⋊C4C4⋊C4semidirect
186. C2×C4⋊C4C4⋊C4semidirect
384. C22⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342122
C2×#19962
#141592
#141492
#139652
#113442
#109182
#108542
C4×#3422