Finite Group Databasebeta

← #1494824

Subgroups — #1494824

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3#15762C212yesmaximal
4#15625C214yesmaximal
5#15625C212yesmaximal
6#15510C214yesmaximal
7#15199C212yesmaximal
8#15143C214yesmaximal
9#14467C212yesmaximal
10#14241C212yesmaximal
11#14236C214yesmaximal
12#13976C212yesmaximal
13#13826C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17#2811—28no—
18#2567—28no—
19C2×#395C2214yes—
20C2×#395C2212yes—
21C2×#395C2214yes—
22C2×#395C2214yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#391C2211yescharacteristic
30C2×#391C2214yes—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#384C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#384C2214yes—
39C2×#384C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41#1446C2212yes—
42#1446C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2212yes—
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1436C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4×#95C2214yes—
67C4×#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C23×Q8C2311yescharacteristic
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—48no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—44no—
78C2×C4×Q8—24no—
79C2×C4×Q8—42no—
80C2×C4×Q8—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2314yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107#395—28no—
108#395—216no—
109#395—216no—
110#395—44no—
111#395—48no—
112#395—48no—
113#395Q8116yescomplemented
114#391—24no—
115#391—28no—
116#391—28no—
117#391—216no—
118#391—48no—
119#391D418yes—
120#391Q8116yescomplemented
121#389—216no—
122#389—48no—
123#389—44no—
124#384—24no—
125#384—24no—
126#384—48no—
127#384—44no—
128C4×C4⋊C4—24no—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8C2×D414yes—
170C22×Q8C2×Q818yes—
171C22×Q8C2×Q818yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2414yes—
185C4⋊Q8—416no—
186C4⋊Q8—432no—
187C4×Q8—416no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×Q818yes—
239C2×C4⋊C4C2×Q818yes—
240C2×C4⋊C4C2×Q8116yes—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—28no—
248C2×C42—28no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C2×D414yes—
261C2×C42C2×Q818yes—
262#95—24no—
263#95—28no—
264#95—24no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95C2×D4116yes—
281#95C2×Q8116yes—
282#95C2×Q8116yes—
283#95C2×Q8116yes—
284#95C2×Q818yes—
285C24C2511yescharacteristic
286C2×Q8—216no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—416no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—88no—
298C2×Q8—84no—
299C2×Q8—88no—
300C2×Q8—88no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×Q8—88no—
305C2×Q8C4⋊Q8116yescomplemented
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4C22×D414yes—
363C22×C4C22×D414yes—
364C22×C4C22×D412yes—
365C22×C4C22×D414yes—
366C22×C4C22×Q818yes—
367C22×C4C22×Q818yes—
368C22×C4C22×Q814yes—
369C22×C4C22×Q814yes—
370C22×C4C22×Q814yes—
371C22×C4C22×Q814yes—
372C22×C4C22×Q814yes—
373C22×C4C22×Q818yes—
374C22×C4C22×Q814yes—
375C22×C4C22×Q818yes—
376C22×C4H(5,𝔽2)12yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C4H(5,𝔽2)18yes—
379C22×C42−1+414yes—
380C4⋊C4—216no—
381C4⋊C4—44no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—416no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—416no—
390C4⋊C4—84no—
391C4⋊C4—88no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C42—28no—
399C42—44no—
400C42—48no—
401C42—416no—
402C42—48no—
403C42—48no—
404C42—44no—
405C42—84no—
406C23C23×D411yescharacteristic
407C23C23×Q812yes—
408C23C23×Q812yes—
409C23C2×H(5,𝔽2)11yescharacteristic
410C23C2×H(5,𝔽2)14yes—
411C23C2×H(5,𝔽2)11yescharacteristic
412C23C2×2−1+412yes—
413C23#58512yes—
414Q8—416no—
415Q8—816no—
416Q8—816no—
417Q8—816no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
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—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4C2×C4⋊Q818yes—
488C2×C4C2×C4⋊Q814yes—
489C2×C4D4×Q818yes—
490C2×C4Q8214yes—
491C2×C4Q82116yes—
492C22C22×C4⋊Q811yescharacteristic
493C22C2×D4×Q812yes—
494C22C2×Q8211yescharacteristic
495C22C2×Q8214yes—
496C22#348411yescharacteristic
497C22#348511yescharacteristic
498C22#349812yes—
499C22#350114yes—
500C22#350112yes—
501C22#350112yes—
502C22#350312yes—
503C22#350314yes—
504C22#353212yes—
505C22#353514yes—
506C22#354411yescharacteristic
507C22#354912yes—
508C4—24no—
509C4—216no—
510C4—24no—
511C4—28no—
512C4—28no—
513C4—48no—
514C4—42no—
515C4—44no—
516C4—44no—
517C4—48no—
518C4—82no—
519C4—82no—
520C4—84no—
521C2#3539111yescharacteristic
522C2Q8×C4⋊Q812yes—
523C2#4349412yes—
524C2#4355314yes—
525C2#5160112yes—
526C2#5190211yescharacteristic
527C2#5915111yescharacteristic
528C2#5992412yes—
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
113. #395Q8semidirect
120. #391Q8semidirect
305. C2×Q8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#25672
#157622
#156252
#155102
#151992
#151432
#144672
#142412
#142362
#139762
#138262