Finite Group Databasebeta

← #4929397

Subgroups — #4929397

Order 512.

2514 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#14965C214yesmaximal
4#14906C214yesmaximal
5#14844C214yesmaximal, complemented
6#14787C214yesmaximal, complemented
7#14292C211yesmaximal, characteristic
8#14192C214yesmaximal, complemented
9#14191C214yesmaximal, complemented
10#14165C211yesmaximal, characteristic
11#14138C211yesmaximal, characteristic
12#13870C211yesmaximal, characteristic
13#13852C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2212yes—
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#2583—28no—
19C2×#400C2214yes—
20C2×#398C2212yes—
21C2×#398C2214yes—
22C2×#395C2212yes—
23C2×#391C2212yes—
24C2×#389C2214yes—
25C2×#387C2214yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#381C2211yescharacteristic
29C2×#381C2212yes—
30#2274—28no—
31C2×C4×C4⋊C4C2214yes—
32C2×C4×C4⋊C4C2214yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38#1446C2212yes—
39#1445C2214yes—
40#1445C2214yes—
41#1445C2212yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1436C2212yes—
53#1436C2212yes—
54#1436C2212yes—
55#1436C2212yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
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—28no—
77C2×#117—24no—
78C2×#117—216no—
79C2×#117—44no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4D418yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C42C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2314yes—
105C22×C42C2312yes—
106#400—216no—
107#398—28no—
108#398—216no—
109#395—28no—
110#391—44no—
111#389—216no—
112#387—48no—
113#385—28no—
114#385—28no—
115#381—24no—
116#381—44no—
117C4×C4⋊C4—216no—
118C4×C4⋊C4—216no—
119C4×C4⋊C4—44no—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—28no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—24no—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2411yescharacteristic
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—42no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2×D414yes—
166C23×C4C2×D414yes—
167C23×C4C2×D414yes—
168C23×C4#4114yes—
169C23×C4#4114yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C22⋊Q8—416no—
183#117—264no—
184#117—416no—
185C2×C4⋊C4—232no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×D418yes—
214C2×C4⋊C4C2×D418yes—
215C2×C4⋊C4#41116yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C42—28no—
232C2×C42—28no—
233C2×C42—28no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—28no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—44no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—44no—
250#95—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276C24—24no—
277C24—22no—
278C24—24no—
279C24—22no—
280C24C22×D412yes—
281C24C2×#4112yes—
282C24C2×#4112yes—
283C24C2511yescharacteristic
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—84no—
289C22×C4—216no—
290C22×C4—216no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—216no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4—44no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—84no—
357C22×C4#123116yes—
358C22×C4C22×D414yes—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D414yes—
362C22×C4C2×#4118yes—
363C22×C4C2×#4118yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4118yes—
369C22×C4H(5,𝔽2)14yes—
370C22×C42−1+418yes—
371C4⋊C4—216no—
372C4⋊C4—216no—
373C4⋊C4—232no—
374C4⋊C4—416no—
375C4⋊C4—416no—
376C4⋊C4—48no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—416no—
380C4⋊C4—88no—
381C22⋊C4—216no—
382C22⋊C4—216no—
383C22⋊C4—216no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—84no—
387C42—232no—
388C42—28no—
389C42—44no—
390C42—84no—
391C42—84no—
392C42—84no—
393C42—88no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—22no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23C2×#12314yes—
409C23C23×D411yescharacteristic
410C23C22×#4111yescharacteristic
411C23C22×#4111yescharacteristic
412C23C22×#4114yes—
413C23C2×H(5,𝔽2)12yes—
414C23C2×2−1+414yes—
415C23#58512yes—
416Q8—88no—
417C2×C4—232no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—28no—
434C2×C4—24no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—416no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—88no—
485C2×C4C2×#12314yes—
486C2×C4C2×#12314yes—
487C2×C4#52518yes—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—48no—
492C22—44no—
493C22—48no—
494C22—44no—
495C22C22×#12311yescharacteristic
496C22C2×#52512yes—
497C22#344412yes—
498C22#344412yes—
499C22#348612yes—
500C22#349011yescharacteristic
501C22#349111yescharacteristic
502C22#349612yes—
503C22#350512yes—
504C22#350912yes—
505C22#350912yes—
506C22#351514yes—
507C22#351614yes—
508C22#352014yes—
509C22#352214yes—
510C4—28no—
511C4—24no—
512C4—24no—
513C4—44no—
514C4—44no—
515C4—48no—
516C4—48no—
517C4—48no—
518C4—82no—
519C4—82no—
520C4—84no—
521C4—82no—
522C2—44no—
523C2#3536011yescharacteristic
524C2#3543512yes—
525C2#3549612yes—
526C2#3567411yescharacteristic
527C2#3572414yes—
528C2#3587214yes—
529C2#3662611yescharacteristic
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
5. #14844C2semidirect
6. #14787C2semidirect
8. #14192C2semidirect
9. #14191C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#22742
#149652
#149062
#148442
#147872
#142922
#141922
#141912
#141652
#141382
#138702
#138522