Finite Group Databasebeta

← #4320112

Subgroups — #4320112

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2764C212yesmaximal
2#17059C211yesmaximal, characteristic
3#16823C211yesmaximal, characteristic
4#16675C212yesmaximal
5#16409C214yesmaximal, complemented
6#15933C214yesmaximal, complemented
7#15511C214yesmaximal, complemented
8#15050C212yesmaximal
9#14939C214yesmaximal, complemented
10#14821C212yesmaximal
11#14422C212yesmaximal
12#14229C212yesmaximal
13#13776C211yesmaximal, characteristic
14C23×C4⋊C4C2211yescharacteristic
15#2764—216no—
16C2×#400C2214yes—
17C2×#398C2214yes—
18C2×#398C2212yes—
19C2×#398C2212yes—
20C2×#397C2212yes—
21C2×#397C2212yes—
22C2×#393C2212yes—
23C2×#391C2214yes—
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#385C2214yes—
28C2×#383C2214yes—
29C2×#381C2211yescharacteristic
30C2×#381C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1446C2212yes—
35#1446C2214yes—
36#1445C2212yes—
37#1445C2214yes—
38#1445C2212yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2212yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1440C2214yes—
54#1439C2212yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1434C2214yes—
60#1434C2212yes—
61#1434C2214yes—
62#1434C2214yes—
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2211yescharacteristic
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C22×C4⋊C4—24no—
73C22×C4⋊C4—42no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2311yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2311yescharacteristic
90C22×C22⋊C4C2311yescharacteristic
91C22×C42C2314yes—
92C22×C42C2312yes—
93C22×C42C2312yes—
94#400—216no—
95#398—216no—
96#398—44no—
97#398—44no—
98#397—28no—
99#397—28no—
100#393—28no—
101#391—216no—
102#391—44no—
103#389—44no—
104#388—216no—
105#385—216no—
106#383—216no—
107#381—24no—
108#381D418yes—
109#380—28no—
110C4×C22⋊C4—42no—
111C4×C22⋊C4—42no—
112C2×#95C2314yes—
113C2×#95C2312yes—
114C2×#95C2314yes—
115C2×#95C2312yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
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×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C25C2411yescharacteristic
143C22×Q8—24no—
144C22×Q8—28no—
145C22×Q8—44no—
146C22×Q8—42no—
147C22×Q8—42no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4C2×D414yes—
162C23×C4#4114yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2414yes—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—416no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4C2×D414yes—
220C2×C22⋊C4C2×D414yes—
221C2×C22⋊C4#4114yes—
222C2×C42—28no—
223C2×C42—28no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42—44no—
229#95—24no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—28no—
238#95—28no—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—44no—
256#95—42no—
257#95—42no—
258#95C2×D4116yes—
259C24—21no—
260C24—21no—
261C24—24no—
262C24—21no—
263C24—21no—
264C24—24no—
265C24C22×D412yes—
266C24C2×#4112yes—
267C24C2×#4112yes—
268C24C2511yescharacteristic
269C2×Q8—416no—
270C2×Q8—48no—
271C2×Q8—84no—
272C2×Q8—88no—
273C2×Q8—84no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
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—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4C22×D414yes—
350C22×C4C22×D414yes—
351C22×C4C22×D414yes—
352C22×C4C2×#4114yes—
353C22×C4C2×#4114yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+418yes—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C4⋊C4—88no—
363C4⋊C4—84no—
364C4⋊C4—88no—
365C4⋊C4—88no—
366C4⋊C4—84no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—416no—
371C22⋊C4—44no—
372C22⋊C4—416no—
373C42—416no—
374C42—84no—
375C42—84no—
376C23—22no—
377C23—22no—
378C23—28no—
379C23—22no—
380C23—24no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—28no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—44no—
392C23#53214yes—
393C23C23×D411yescharacteristic
394C23C22×#4111yescharacteristic
395C23C22×#4111yescharacteristic
396C23C2×H(5,𝔽2)11yescharacteristic
397C23C2×H(5,𝔽2)11yescharacteristic
398C23C2×2−1+414yes—
399C23#58511yescharacteristic
400C23#58511yescharacteristic
401C23#58514yes—
402Q8—816no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—24no—
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—28no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—416no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
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—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—88no—
473C2×C4—88no—
474C22—28no—
475C22—24no—
476C22—44no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22C2×#53211yescharacteristic
484C22#345911yescharacteristic
485C22#346011yescharacteristic
486C22#348914yes—
487C22#349011yescharacteristic
488C22#349112yes—
489C22#349311yescharacteristic
490C22#349611yescharacteristic
491C22#349611yescharacteristic
492C22#350912yes—
493C22#351514yes—
494C22#353812yes—
495C22#354512yes—
496C22#354512yes—
497C22#355012yes—
498C22#356414yes—
499C22#356514yes—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—48no—
504C4—44no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—84no—
509C4—84no—
510C4—82no—
511C4—84no—
512C2—44no—
513C2#3500311yescharacteristic
514C2#3602412yes—
515C2#3716212yes—
516C2#4191412yes—
517C2#4683711yescharacteristic
518C2#4703411yescharacteristic
519C2#4745912yes—
520C2#4879912yes—
521C2#5088712yes—
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. #16409C2semidirect
6. #15933C2semidirect
7. #15511C2semidirect
9. #14939C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27642
#170592
#168232
#166752
#164092
#159332
#155112
#150502
#149392
#148212
#144222
#142292
#137762