Finite Group Databasebeta

← #825996

Subgroups — #825996

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C212yesmaximal
2#17080C212yesmaximal
3#15964C212yesmaximal
4#15363C212yesmaximal
5#15333C214yesmaximal
6#15196C211yesmaximal, characteristic
7#15184C214yesmaximal
8#15133C212yesmaximal
9#14238C212yesmaximal
10#14053C212yesmaximal
11#14053C212yesmaximal
12#14045C212yesmaximal
13#14005C212yesmaximal
14#13910C211yesmaximal, characteristic
15C4×#398C211yesmaximal, characteristic
16C22×C4⋊Q8C2212yes—
17C2×#398C2214yes—
18C2×#398C2211yescharacteristic
19#2369—216no—
20C2×#391C2214yes—
21C2×#391C2214yes—
22C2×#391C2214yes—
23C2×#391C2214yes—
24C2×#389C2214yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#384C2212yes—
33C2×#384C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2214yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1446C2214yes—
42#1446C2214yes—
43#1445C2211yescharacteristic
44#1444C2211yescharacteristic
45#1444C2212yes—
46#1444C2212yes—
47#1444C2214yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2214yes—
51#1442C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2212yes—
57#1441C2211yescharacteristic
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2314yes—
77C2×C4⋊Q8—28no—
78C2×C4⋊Q8—28no—
79C2×C4⋊Q8Q8116yescomplemented
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2312yes—
108#398—48no—
109#398Q818yescomplemented
110#391—216no—
111#391—216no—
112#391—48no—
113#391—48no—
114#389—28no—
115#389—28no—
116#389—28no—
117#389—28no—
118#389—216no—
119#389—48no—
120#389—44no—
121#384—42no—
122#384—42no—
123#384—44no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—216no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4D418yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C22×Q8—28no—
162C22×Q8—28no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8C2×Q8116yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2414yes—
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2414yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C4⋊Q8—264no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D418yes—
233C2×C4⋊C4C2×Q818yes—
234C2×C4⋊C4C2×Q8116yes—
235C2×C4⋊C4C2×Q818yes—
236C2×C4⋊C4C2×Q814yes—
237C2×C4⋊C4C2×Q814yes—
238C2×C4⋊C4C2×Q818yes—
239C2×C4⋊C4#4118yes—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—28no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—28no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42—44no—
259C2×C42C2×D414yes—
260C2×C42C2×D418yes—
261C2×C42C2×Q818yes—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—28no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—42no—
277#95—44no—
278#95—41no—
279#95—42no—
280#95—42no—
281#95—41no—
282#95—41no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95C2×D418yes—
290#95C2×Q818yes—
291#95C2×Q814yes—
292C24C2511yescharacteristic
293C2×Q8—232no—
294C2×Q8—416no—
295C2×Q8—416no—
296C2×Q8—88no—
297C2×Q8—88no—
298C2×Q8—88no—
299C2×Q8—88no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4C22×D412yes—
373C22×C4C22×D418yes—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×Q812yes—
377C22×C4C22×Q814yes—
378C22×C4C22×Q812yes—
379C22×C4C22×Q814yes—
380C22×C4C22×Q818yes—
381C22×C4C22×Q818yes—
382C22×C4C22×Q812yes—
383C22×C4C22×Q814yes—
384C22×C4C22×Q814yes—
385C22×C4C22×Q814yes—
386C22×C4C2×#4118yes—
387C22×C4C2×#4114yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)14yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392C22×C42−1+412yes—
393C4⋊C4—232no—
394C4⋊C4—216no—
395C4⋊C4—216no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—416no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—84no—
407C4⋊C4—88no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—88no—
412C4⋊C4C22⋊Q8132yes—
413C42—216no—
414C42—28no—
415C42—48no—
416C42—416no—
417C42—48no—
418C42—44no—
419C42—44no—
420C42—88no—
421C23C23×D411yescharacteristic
422C23C23×Q812yes—
423C23C23×Q811yescharacteristic
424C23C22×#4112yes—
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×H(5,𝔽2)12yes—
427C23C2×H(5,𝔽2)11yescharacteristic
428C23C2×2−1+411yescharacteristic
429C23C2×2−1+411yescharacteristic
430C23#58512yes—
431C23#58511yescharacteristic
432Q8—832no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—28no—
441C2×C4—22no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4C2×C22⋊Q8116yes—
519C2×C4C2×C22⋊Q818yes—
520C2×C4#53714yes—
521C2×C4D4×Q814yes—
522C2×C4#55718yes—
523C2×C4Q8218yes—
524C2×C4Q8214yes—
525C2×C4#55914yes—
526C22C22×C22⋊Q812yes—
527C22C2×#53711yescharacteristic
528C22C2×D4×Q811yescharacteristic
529C22C2×#55712yes—
530C22C2×Q8212yes—
531C22C2×Q8211yescharacteristic
532C22Q8×#4112yes—
533C22C2×#55911yescharacteristic
534C22#348812yes—
535C22#349611yescharacteristic
536C22#350112yes—
537C22#350112yes—
538C22#350111yescharacteristic
539C22#350212yes—
540C22#350311yescharacteristic
541C22#351912yes—
542C22#352112yes—
543C22#352212yes—
544C22#353611yescharacteristic
545C22#354011yescharacteristic
546C22#354712yes—
547C22#354911yescharacteristic
548C22#356311yescharacteristic
549C4—28no—
550C4—24no—
551C4—216no—
552C4—44no—
553C4—48no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—84no—
561C4—81no—
562C4—84no—
563C4#258218yes—
564C2C2×#258211yescharacteristic
565C2#3577612yes—
566C2Q8×C22⋊Q812yes—
567C2#3961411yescharacteristic
568C2#4297111yescharacteristic
569C2#4495312yes—
570C2#4504711yescharacteristic
571C2#4536411yescharacteristic
572C2#5962911yescharacteristic
573C2#5966812yes—
574C2#5992411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
79. C2×C4⋊Q8Q8semidirect
109. #398Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
#170802
#159642
#153632
#153332
#151962
#151842
#151332
#142382
#140532
#140452
#140052
#139102
C4×#3982