Finite Group Databasebeta

← #4456469

Subgroups — #4456469

Order 512.

2114 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic
3#16697C214yesmaximal
4#16430C214yesmaximal, complemented
5#16393C214yesmaximal
6#16279C214yesmaximal, complemented
7#14948C214yesmaximal, complemented
8#14779C214yesmaximal, complemented
9#14292C211yesmaximal, characteristic
10#14165C211yesmaximal, characteristic
11#14138C211yesmaximal, characteristic
12#13870C211yesmaximal, characteristic
13#13852C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2583—28no—
18C2×#398C2212yes—
19C2×#398C2214yes—
20C2×#395C2212yes—
21#2313—28no—
22C2×#391C2212yes—
23C2×#389C2214yes—
24C2×#387C2214yes—
25C2×#387C2214yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#381C2211yescharacteristic
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1446C2212yes—
37#1445C2214yes—
38#1445C2214yes—
39#1445C2214yes—
40#1445C2212yes—
41#1445C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1436C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C2×C22⋊Q8—24no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—24no—
76C2×#117—24no—
77C2×#117—44no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4D418yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2312yes—
102#398—28no—
103#398—48no—
104#395—28no—
105#391—44no—
106#389—216no—
107#387—216no—
108#387—216no—
109#385—28no—
110#385—28no—
111#381—28no—
112#381—24no—
113#381—44no—
114#380—28no—
115C4×C4⋊C4—44no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—24no—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
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—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—42no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D414yes—
163C23×C4C2×D414yes—
164C23×C4C2×D414yes—
165C23×C4#4114yes—
166C23×C4#4114yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C22⋊Q8—416no—
180#117—416no—
181C2×C4⋊C4—232no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
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⋊C4C2×D418yes—
211C2×C4⋊C4C2×D418yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
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—44no—
266#95—44no—
267#95—44no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24C22×D412yes—
273C24C2×#4112yes—
274C24C2×#4112yes—
275C24C2511yescharacteristic
276C2×Q8—48no—
277C2×Q8—44no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×Q8—84no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—28no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
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—44no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—84no—
352C22×C4#123116yes—
353C22×C4C22×D414yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)14yes—
362C22×C42−1+418yes—
363C4⋊C4—216no—
364C4⋊C4—216no—
365C4⋊C4—416no—
366C4⋊C4—48no—
367C4⋊C4—416no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4—88no—
373C22⋊C4—216no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—84no—
379C42—28no—
380C42—44no—
381C42—84no—
382C42—84no—
383C42—84no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23C2×#12314yes—
399C23C23×D411yescharacteristic
400C23C22×#4111yescharacteristic
401C23C22×#4111yescharacteristic
402C23C2×H(5,𝔽2)12yes—
403C23C2×2−1+414yes—
404C23#58514yes—
405C23#58512yes—
406Q8—88no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—416no—
426C2×C4—416no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
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—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—88no—
476C2×C4C2×#12314yes—
477C2×C4C2×#12314yes—
478C22—24no—
479C22—24no—
480C22—24no—
481C22—48no—
482C22—44no—
483C22—44no—
484C22—48no—
485C22C22×#12311yescharacteristic
486C22#344912yes—
487C22#345512yes—
488C22#345512yes—
489C22#348612yes—
490C22#349011yescharacteristic
491C22#349111yescharacteristic
492C22#349612yes—
493C22#350512yes—
494C22#350912yes—
495C22#350912yes—
496C22#353514yes—
497C22#355314yes—
498C22#355414yes—
499C22#356514yes—
500C4—24no—
501C4—24no—
502C4—44no—
503C4—44no—
504C4—48no—
505C4—44no—
506C4—82no—
507C4—82no—
508C4—84no—
509C4—84no—
510C4—82no—
511C4—84no—
512C2—44no—
513C2#3463511yescharacteristic
514C2#3536011yescharacteristic
515C2#3567411yescharacteristic
516C2#3681312yes—
517C2#3701112yes—
518C2#4052614yes—
519C2#4068414yes—
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
4. #16430C2semidirect
6. #16279C2semidirect
7. #14948C2semidirect
8. #14779C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#23132
#166972
#164302
#163932
#162792
#149482
#147792
#142922
#141652
#141382
#138702
#138522