Finite Group Databasebeta

← #4414817

Subgroups — #4414817

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17081C211yesmaximal, characteristic, complemented
2#17073C212yesmaximal, complemented
3#17073C211yesmaximal, characteristic, complemented
4#16571C214yesmaximal, complemented
5#16525C214yesmaximal, complemented
6#14766C214yesmaximal
7#14736C214yesmaximal
8#14728C214yesmaximal
9#14682C212yesmaximal
10#14640C214yesmaximal, complemented
11C2×C4⋊#95C211yesmaximal, characteristic
12C23×C4⋊C4C2212yes—
13C23×C42C2211yescharacteristic
14C2×#398C2214yes—
15C2×#398C2214yes—
16C2×#398C2214yes—
17C2×#398C2214yes—
18C2×#385C2214yes—
19C2×#385C2214yes—
20C2×#382C2212yes—
21C2×#382C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2214yes—
25C22×#95C2214yes—
26#1445C2214yes—
27#1445C2214yes—
28#1444C2214yes—
29#1444C2214yes—
30#1444C2214yes—
31#1444C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C4⋊#95C2214yes—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
52#1434C2214yes—
53#1434C2214yes—
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C24×C4C2311yescharacteristic
58C24×C4C2314yes—
59C24×C4C2312yes—
60C22×C4⋊C4—28no—
61C22×C4⋊C4—44no—
62C22×C4⋊C4—44no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C42—44no—
79C22×C42Q818yes—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#398—216no—
84#398—48no—
85#398—48no—
86#398—48no—
87#385—48no—
88#385—48no—
89#382—44no—
90#382—44no—
91#380—44no—
92#380—48no—
93#380—44no—
94C2×#95—48no—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C25C2411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—28no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—28no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4C2×Q814yes—
138C23×C4C2×Q818yes—
139C23×C4C2414yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2412yes—
145C23×C4C2414yes—
146C23×C4C2411yescharacteristic
147C23×C4C2411yescharacteristic
148C23×C4C2412yes—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—432no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4#4118yes—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—416no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42C2×Q818yes—
201C2×C42C2×Q814yes—
202C2×C42C2×Q814yes—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24C22×Q812yes—
232C24H(5,𝔽2)14yes—
233C24H(5,𝔽2)12yes—
234C242−1+412yes—
235C242−1+414yes—
236C24C2511yescharacteristic
237C22×C4—216no—
238C22×C4—216no—
239C22×C4—28no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—216no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—216no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
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—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—48no—
305C22×C4—48no—
306C22×C4—416no—
307C22×C4—416no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—88no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4C22×Q814yes—
314C22×C4C22×Q812yes—
315C22×C4C22×Q812yes—
316C22×C4C22×Q814yes—
317C22×C4C2×#4118yes—
318C22×C4C2×#4114yes—
319C22×C4H(5,𝔽2)14yes—
320C22×C4H(5,𝔽2)14yes—
321C22×C42−1+414yes—
322C4⋊C4—48no—
323C4⋊C4—416no—
324C4⋊C4—48no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—88no—
330C4⋊C4—88no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C4⋊C4—88no—
334C4⋊C4—88no—
335C22⋊C4—416no—
336C22⋊C4—416no—
337C42—216no—
338C42—44no—
339C42—44no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23#54418yes—
357C23#54414yes—
358C23#56214yes—
359C23#56218yes—
360C23#56414yes—
361C23C23×Q811yescharacteristic
362C23C22×#4112yes—
363C23C2×H(5,𝔽2)12yes—
364C23C2×H(5,𝔽2)11yescharacteristic
365C23C2×H(5,𝔽2)12yes—
366C23C2×H(5,𝔽2)12yes—
367C23C2×2−1+411yescharacteristic
368C23C2×2−1+412yes—
369C23C2×2−1+412yes—
370C2×C4—232no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—432no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—832no—
436C2×C4#53314yes—
437C2×C4#54314yes—
438C2×C4#54418yes—
439C22—28no—
440C22—28no—
441C22—28no—
442C22—28no—
443C22—28no—
444C22—28no—
445C22—24no—
446C22—24no—
447C22#284518yes—
448C22C2×#53311yescharacteristic
449C22C2×#54311yescharacteristic
450C22C2×#54412yes—
451C22C2×#54411yescharacteristic
452C22C2×#54412yes—
453C22#349012yes—
454C22#349212yes—
455C22#351912yes—
456C22#352012yes—
457C22#352212yes—
458C22#352312yes—
459C22C2×#56212yes—
460C22C2×#56211yescharacteristic
461C22C2×#56411yescharacteristic
462C22#353612yes—
463C22#353612yes—
464C22#353712yes—
465C22#353812yes—
466C22#354311yescharacteristic
467C22#354411yescharacteristic
468C22#355212yes—
469C4—28no—
470C4—24no—
471C4—24no—
472C4—48no—
473C4—44no—
474C4—84no—
475C4—84no—
476C4—84no—
477C4—84no—
478C4—84no—
479C2—28no—
480C2#3505211yescharacteristic
481C2#3523912yes—
482C2C2×#284511yescharacteristic
483C2#3747712yes—
484C2#3762311yescharacteristic
485C2#4608812yes—
486C2#4609012yes—
487C2#4609812yes—
488C2#4620012yes—
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. #17081C2semidirect
2. #17073C2semidirect
3. #17073C2semidirect
4. #16571C2semidirect
5. #16525C2semidirect
10. #14640C2semidirect

Maximal subgroups

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

subgroupindex
#170812
#170732
#165712
#165252
#147662
#147362
#147282
#146822
#146402
C2×C4⋊#952