Finite Group Databasebeta

← #4387778

Subgroups — #4387778

Order 512.

2130 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2844C211yesmaximal, characteristic, complemented
2C2×#2317C211yesmaximal, characteristic
3#16686C212yesmaximal, complemented
4#16570C212yesmaximal, complemented
5#15686C214yesmaximal, complemented
6#15624C214yesmaximal, complemented
7#15594C214yesmaximal
8#15475C214yesmaximal
9#14887C212yesmaximal
10#14822C212yesmaximal
11#14348C212yesmaximal, complemented
12#14168C211yesmaximal, characteristic, complemented
13#13957C212yesmaximal
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2844—28no—
18C2×#400C2214yes—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#395C2214yes—
22C2×#395C2212yes—
23C2×#393C2212yes—
24#2317—28no—
25C2×#391C2212yes—
26C2×#389C2214yes—
27C2×#385C2214yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2212yes—
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#376C2212yes—
35#1446C2214yes—
36#1443C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2212yes—
56#1439C2212yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64#1434C2212yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×#117—24no—
73C2×#117—24no—
74C2×#117—24no—
75C2×#117—42no—
76C2×#117—42no—
77C2×#117Q818yescomplemented
78C22×C4⋊C4—24no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2314yes—
102C22×C42C2311yescharacteristic
103#400—28no—
104#400—216no—
105#400—44no—
106#395—28no—
107#395—216no—
108#393—44no—
109#391—44no—
110#389—216no—
111#385—216no—
112#384—24no—
113#384—24no—
114#382—24no—
115#382—28no—
116#382D418yes—
117#380—44no—
118#376—28no—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2411yescharacteristic
148C22×Q8—24no—
149C22×Q8—44no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—42no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4C2×Q814yes—
165C23×C4C2412yes—
166C23×C4C2414yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177#117—232no—
178#117—416no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—416no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×Q818yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4C2×Q818yes—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42C2×D414yes—
239C2×C42C2×Q814yes—
240C2×C42C2×Q814yes—
241C2×C42#4114yes—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—24no—
246#95—28no—
247#95—24no—
248#95—28no—
249#95—24no—
250#95—24no—
251#95—28no—
252#95—28no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95—42no—
268#95—44no—
269#95C2×D4116yes—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—24no—
275C24—24no—
276C24C22×Q812yes—
277C24H(5,𝔽2)12yes—
278C242−1+412yes—
279C24C2511yescharacteristic
280C2×Q8—48no—
281C2×Q8—84no—
282C2×Q8—88no—
283C2×Q8—84no—
284C2×Q8—88no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—88no—
355C22×C4—84no—
356C22×C4—88no—
357C22×C4C22×D414yes—
358C22×C4C22×D414yes—
359C22×C4C22×D412yes—
360C22×C4C22×D414yes—
361C22×C4C22×Q814yes—
362C22×C4C22×Q812yes—
363C22×C4C22×Q814yes—
364C22×C4C22×Q812yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4114yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C42−1+412yes—
369C4⋊C4—216no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—84no—
376C4⋊C4—88no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—88no—
380C4⋊C4—88no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C22⋊C4—216no—
384C22⋊C4—48no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—28no—
388C42—48no—
389C42—416no—
390C42C22⋊Q8116yes—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—28no—
398C23—24no—
399C23—24no—
400C23—28no—
401C23—44no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23#54414yes—
408C23C23×D411yescharacteristic
409C23C23×Q811yescharacteristic
410C23C22×#4111yescharacteristic
411C23C2×H(5,𝔽2)11yescharacteristic
412C23C2×H(5,𝔽2)11yescharacteristic
413C23C2×2−1+411yescharacteristic
414C23C2×2−1+411yescharacteristic
415C23#58514yes—
416C23#58514yes—
417Q8—816no—
418C2×C4—232no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
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—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—816no—
489C2×C4C2×C22⋊Q814yes—
490C2×C4C2×C22⋊Q818yes—
491C2×C4#54414yes—
492C22—24no—
493C22—28no—
494C22—42no—
495C22—44no—
496C22—42no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22C22×C22⋊Q811yescharacteristic
502C22C2×#54411yescharacteristic
503C22C2×#54411yescharacteristic
504C22#348411yescharacteristic
505C22#348611yescharacteristic
506C22#349211yescharacteristic
507C22#349211yescharacteristic
508C22#349314yes—
509C22#350012yes—
510C22#350012yes—
511C22#350414yes—
512C22#353312yes—
513C22#354012yes—
514C22#354512yes—
515C22#355112yes—
516C22#356414yes—
517C22#356514yes—
518C4—28no—
519C4—24no—
520C4—24no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—82no—
525C4—82no—
526C4—84no—
527C4—84no—
528C4—84no—
529C4—84no—
530C2—44no—
531C2#3522411yescharacteristic
532C2#3562012yes—
533C2#4218112yes—
534C2#4298112yes—
535C2#4780512yes—
536C2#4780612yes—
537C2#5174911yescharacteristic
538C2#5175012yes—
539C2#5197811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2844C2semidirect
3. #16686C2semidirect
4. #16570C2semidirect
5. #15686C2semidirect
6. #15624C2semidirect
11. #14348C2semidirect
12. #14168C2semidirect
77. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
C2×#23172
#166862
#165702
#156862
#156242
#155942
#154752
#148872
#148222
#143482
#141682
#139572