Finite Group Databasebeta

← #4386460

Subgroups — #4386460

Order 512.

2002 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16358C212yesmaximal
2#16339C212yesmaximal
3#16230C214yesmaximal, complemented
4#15850C214yesmaximal, complemented
5#15720C214yesmaximal, complemented
6#15438C214yesmaximal, complemented
7#14969C212yesmaximal
8#14916C212yesmaximal
9#14674C211yesmaximal, characteristic
10#14297C212yesmaximal
11#14165C211yesmaximal, characteristic
12#13880C212yesmaximal
13#13776C211yesmaximal, characteristic
14C22×#125C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C2×#400C2214yes—
17C2×#398C2214yes—
18C2×#389C2212yes—
19C2×#387C2214yes—
20C2×#385C2214yes—
21C2×#385C2214yes—
22C2×#384C2212yes—
23C2×#384C2212yes—
24C2×#381C2211yescharacteristic
25C2×#381C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1445C2214yes—
31#1444C2214yes—
32#1444C2214yes—
33#1444C2212yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2212yes—
37#1444C2212yes—
38#1443C2214yes—
39#1442C2212yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1439C2212yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2212yes—
53#1439C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C2×#125—44no—
71C2×#125—48no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2311yescharacteristic
91C22×C22⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2311yescharacteristic
94C22×C22⋊C4C2311yescharacteristic
95C22×C42C2312yes—
96C22×C42C2312yes—
97C22×C42C2312yes—
98C22×C42C2312yes—
99#400—48no—
100#398—48no—
101#389—44no—
102#387—48no—
103#385—48no—
104#385—48no—
105#384—44no—
106#384—44no—
107#381—24no—
108#381Q818yes—
109#380—28no—
110C4×C22⋊C4—28no—
111C4×C22⋊C4—42no—
112C4×C22⋊C4—42no—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C25C2411yescharacteristic
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4C2×Q814yes—
153C23×C4#4114yes—
154C23×C4#4114yes—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2412yes—
158C23×C4C2412yes—
159C23×C4C2412yes—
160C23×C4C2412yes—
161C23×C4C2412yes—
162C23×C4C2414yes—
163C23×C4C2411yescharacteristic
164C23×C4C2411yescharacteristic
165C23×C4C2414yes—
166C23×C4C2412yes—
167#125—432no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—416no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
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×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4C2×Q814yes—
224C2×C22⋊C4C2×Q814yes—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—44no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—44no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—42no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—42no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95C2×Q8116yes—
261C24—24no—
262C24—21no—
263C24—21no—
264C24—24no—
265C24—21no—
266C24—21no—
267C24C22×Q812yes—
268C24C2×#4112yes—
269C24C2×#4112yes—
270C24C2511yescharacteristic
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
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—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4—84no—
344C22×C4#125116yescomplemented
345C22×C4C22×Q814yes—
346C22×C4C22×Q814yes—
347C22×C4C22×Q814yes—
348C22×C4C2×#4114yes—
349C22×C4C2×#4114yes—
350C22×C4H(5,𝔽2)18yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+418yes—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—416no—
358C4⋊C4—48no—
359C4⋊C4—416no—
360C4⋊C4—416no—
361C4⋊C4—48no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C4⋊C4—88no—
365C4⋊C4—88no—
366C4⋊C4—84no—
367C4⋊C4—88no—
368C22⋊C4—232no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—88no—
374C42—48no—
375C42—48no—
376C42—48no—
377C42—84no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—28no—
383C23—22no—
384C23—22no—
385C23—28no—
386C23—24no—
387C23—24no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23C2×#12514yes—
395C23C23×Q811yescharacteristic
396C23C22×#4111yescharacteristic
397C23C22×#4111yescharacteristic
398C23C2×H(5,𝔽2)11yescharacteristic
399C23C2×H(5,𝔽2)11yescharacteristic
400C23C2×H(5,𝔽2)14yes—
401C23C2×2−1+414yes—
402C23#58511yescharacteristic
403C23#58511yescharacteristic
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—416no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—88no—
473C2×C4—88no—
474C2×C4C2×#12518yes—
475C22—24no—
476C22—28no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22C22×#12511yescharacteristic
485C22#345911yescharacteristic
486C22#345911yescharacteristic
487C22#349011yescharacteristic
488C22#349111yescharacteristic
489C22#349214yes—
490C22#350111yescharacteristic
491C22#350111yescharacteristic
492C22#350314yes—
493C22#350812yes—
494C22#350912yes—
495C22#353212yes—
496C22#353812yes—
497C22#354114yes—
498C22#354312yes—
499C22#355012yes—
500C22#355914yes—
501C4—28no—
502C4—44no—
503C4—48no—
504C4—48no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—82no—
510C4—84no—
511C4—84no—
512C4—82no—
513C2—44no—
514C2#3499411yescharacteristic
515C2#3555512yes—
516C2#4097412yes—
517C2#4099012yes—
518C2#4703111yescharacteristic
519C2#4703411yescharacteristic
520C2#4722412yes—
521C2#5070012yes—
522C2#5107412yes—
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
3. #16230C2semidirect
4. #15850C2semidirect
5. #15720C2semidirect
6. #15438C2semidirect
344. C22×C4#125semidirect

Maximal subgroups

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

subgroupindex
#163582
#163392
#162302
#158502
#157202
#154382
#149692
#149162
#146742
#142972
#141652
#138802
#137762