Finite Group Databasebeta

← #375483

Subgroups — #375483

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11358C212yesmaximal, complemented
6#10881C212yesmaximal, complemented
7#10880C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2628—24no—
18#2628C418yescomplemented
19C2×#400—21no—
20C2×#399—21no—
21C2×#395—21no—
22C2×#393—21no—
23C2×#389C2212yes—
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26#1750—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—21no—
41C22⋊#95—21no—
42C23×Q8—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—41no—
52C2×C4⋊Q8C2×C414yescomplemented
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—41no—
56C2×#117—41no—
57C2×#117—41no—
58C2×#117—41no—
59C2×#117—41no—
60C2×#117—41no—
61C2×#117C2×C414yes—
62C2×#117D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—41no—
69C22×C4⋊C4—41no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8Q818yescomplemented
86C2×#97—22no—
87C2×#97—41no—
88C2×#97—41no—
89C2×C4×C8—22no—
90C2×C4×C8—41no—
91C2×C4×C8—41no—
92#400—28no—
93#399—28no—
94#395—28no—
95#393—28no—
96#389—24no—
97#389C2×C418yescomplemented
98#382—22no—
99#382C2×C414yescomplemented
100#380—24no—
101#380C2×C418yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336—24no—
119C25C2×D411yescharacteristic
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—82no—
124C22×Q8—82no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×Q814yes—
148C23×C4#4114yes—
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C2×Q814yes—
163C22×C8C2×Q814yes—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168#117—44no—
169#117—44no—
170#117—44no—
171#117—44no—
172#117—48no—
173#117—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4C22⋊C414yes—
211C2×C4⋊C4C22×C414yes—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4#4114yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4C22⋊C414yes—
226C2×C22⋊C4C22⋊C414yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C412yes—
239C2×C42C22×C412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C22⋊C8—216no—
243#97—44no—
244C4×C8—44no—
245#95—28no—
246#95—24no—
247#95—24no—
248#95—28no—
249#95—82no—
250#95—82no—
251#95—82no—
252#95—82no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24C2×C22⋊C412yes—
262C24#12412yes—
263C24C4⋊Q812yes—
264C24C22×D411yescharacteristic
265C2×Q8—48no—
266C2×Q8—88no—
267C2×Q8—88no—
268C2×Q8—84no—
269C2×Q8—84no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—48no—
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—88no—
334C22×C4—88no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C4×Q818yes—
343C22×C4C22⋊Q818yes—
344C22×C4#12412yes—
345C22×C4C4⋊Q812yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×Q812yes—
349C22×C4C2×#4112yes—
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—88no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—48no—
394C22⋊C4—88no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—48no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23#39114yes—
427C23C22×C22⋊C411yescharacteristic
428C23#52511yescharacteristic
429C23#52511yescharacteristic
430C23C2×#12411yescharacteristic
431C23C2×C4⋊Q811yescharacteristic
432C23#53411yescharacteristic
433C23#53411yescharacteristic
434C23#57612yes—
435C23#57712yes—
436C23#57712yes—
437C23#57812yes—
438Q8—816no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—816no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#39114yes—
499C2×C4C2×C4×Q812yes—
500C2×C4C4×#4112yes—
501C2×C4#51811yescharacteristic
502C2×C4#51811yescharacteristic
503C2×C4C2×C22⋊Q812yes—
504C2×C4#53212yes—
505C2×C4#53411yescharacteristic
506C2×C4#53411yescharacteristic
507C2×C4#55112yes—
508C2×C4#55112yes—
509C2×C4#55112yes—
510C2×C4#55112yes—
511C2×C4#55412yes—
512C2×C4#55412yes—
513C2×C4#55412yes—
514C2×C4#55412yes—
515C8—44no—
516C8—44no—
517C22—28no—
518C22—24no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—42no—
523C22—42no—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—44no—
528C22C2×#39111yescharacteristic
529C22#229711yescharacteristic
530C22#230811yescharacteristic
531C22#256111yescharacteristic
532C22#256111yescharacteristic
533C22#258311yescharacteristic
534C22#258311yescharacteristic
535C22#289712yes—
536C22#289812yes—
537C22#311812yes—
538C22#312112yes—
539C22#312412yes—
540C22#312512yes—
541C22#313612yes—
542C22#313712yes—
543C22#316312yes—
544C22#316412yes—
545C22#319812yes—
546C22#319911yescharacteristic
547C22#320011yescharacteristic
548C22#320011yescharacteristic
549C22#320112yes—
550C22#320211yescharacteristic
551C4—24no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—84no—
557C4—84no—
558C4—84no—
559C4—84no—
560C4#236812yes—
561C4#236812yes—
562C4#256412yes—
563C4#256412yes—
564C2—44no—
565C2#1415911yescharacteristic
566C2#2132612yes—
567C2#2139712yes—
568C2#2177011yescharacteristic
569C2#2177111yescharacteristic
570C2#2911611yescharacteristic
571C2#2911811yescharacteristic
572C2#2912012yes—
573C2#2930812yes—
574C2#2931011yescharacteristic
575C2#2931111yescharacteristic
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
4. #11384C2semidirect
5. #11358C2semidirect
6. #10881C2semidirect
7. #10880C2semidirect
18. #2628C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
85. C2×C22⋊C8Q8semidirect
97. #389C2×C4semidirect
99. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17502
#137952
#113842
#113582
#108812
#108802
#107312
#107302