Finite Group Databasebeta

← #4388248

Subgroups — #4388248

Order 512.

2084 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2845C211yesmaximal, characteristic, complemented
2C2×#2314C211yesmaximal, characteristic
3#16521C214yesmaximal
4#15773C214yesmaximal
5#15765C214yesmaximal
6#15438C214yesmaximal, complemented
7#15435C214yesmaximal, complemented
8#14804C214yesmaximal, complemented
9#14354C212yesmaximal, complemented
10#14168C211yesmaximal, characteristic, complemented
11#13957C212yesmaximal
12C22×#117C2211yescharacteristic
13#2845—28no—
14C2×#399C2214yes—
15C2×#398C2214yes—
16C2×#398C2214yes—
17C2×#398C2214yes—
18C2×#396C2214yes—
19#2314—28no—
20C2×#389C2214yes—
21C2×#388C2214yes—
22C2×#387C2214yes—
23C2×#387C2214yes—
24C2×#384C2211yescharacteristic
25C2×#384C2214yes—
26C2×#382C2212yes—
27C2×#382C2214yes—
28C2×#382C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×#376C2212yes—
32#1445C2214yes—
33#1444C2214yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1444C2214yes—
38#1444C2214yes—
39#1442C2214yes—
40#1441C2212yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2214yes—
46#1441C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1439C2214yes—
50#1439C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1434C2214yes—
58C22⋊#95C2214yes—
59C24×C4C2311yescharacteristic
60C2×#117—24no—
61C2×#117—24no—
62C2×#117Q818yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C42C2314yes—
82C22×C42C2311yescharacteristic
83C22×C42C2314yes—
84C22×C42C2311yescharacteristic
85#399—48no—
86#398—216no—
87#398—216no—
88#398—216no—
89#396—48no—
90#389—48no—
91#388—216no—
92#387—216no—
93#387—48no—
94#384—24no—
95#384—48no—
96#382—28no—
97#382—48no—
98#382D418yes—
99#380—28no—
100C4×C4⋊C4—28no—
101#376—28no—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2312yes—
112C2×#95C2312yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C25C2411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C2×Q814yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2414yes—
133C23×C4C2414yes—
134C23×C4C2412yes—
135C23×C4C2411yescharacteristic
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2412yes—
139C23×C4C2411yescharacteristic
140#117—232no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C2×D418yes—
180C2×C4⋊C4C2×Q818yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4C2×Q818yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—28no—
196C2×C42—28no—
197C2×C42—28no—
198C2×C42—28no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42C2×D414yes—
202C2×C42C2×Q814yes—
203C2×C42C2×Q814yes—
204C2×C42#4114yes—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—28no—
210#95—28no—
211#95—24no—
212#95—24no—
213#95—28no—
214#95—28no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95C2×D4116yes—
227C24—21no—
228C24—21no—
229C24—24no—
230C24—24no—
231C24—22no—
232C24—22no—
233C24C22×Q812yes—
234C24C2511yescharacteristic
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
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—48no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4C22×D418yes—
292C22×C4C22×D412yes—
293C22×C4C22×D414yes—
294C22×C4C22×Q814yes—
295C22×C4C22×Q812yes—
296C22×C4C22×Q812yes—
297C22×C4C22×Q814yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4114yes—
300C22×C4H(5,𝔽2)18yes—
301C22×C42−1+418yes—
302C4⋊C4—216no—
303C4⋊C4—48no—
304C4⋊C4—416no—
305C4⋊C4—48no—
306C4⋊C4—416no—
307C4⋊C4—88no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—88no—
311C4⋊C4—88no—
312C4⋊C4—88no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C22⋊C4—216no—
317C22⋊C4—88no—
318C22⋊C4—88no—
319C22⋊C4—88no—
320C42—232no—
321C42—28no—
322C42—88no—
323C42C22⋊Q8116yes—
324C23—22no—
325C23—24no—
326C23—28no—
327C23—41no—
328C23—41no—
329C23—44no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23C23×D411yescharacteristic
338C23C23×Q811yescharacteristic
339C23C22×#4111yescharacteristic
340C23C2×H(5,𝔽2)14yes—
341C23C2×2−1+414yes—
342C23#58512yes—
343C23#58512yes—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—416no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4C2×C22⋊Q814yes—
406C2×C4C2×C22⋊Q818yes—
407C22—48no—
408C22—42no—
409C22—44no—
410C22—82no—
411C22—81no—
412C22—84no—
413C22—81no—
414C22C22×C22⋊Q811yescharacteristic
415C22#345812yes—
416C22#348511yescharacteristic
417C22#348711yescharacteristic
418C22#349214yes—
419C22#349312yes—
420C22#349812yes—
421C22#349912yes—
422C22#350314yes—
423C22#353514yes—
424C22#354214yes—
425C22#354714yes—
426C22#356314yes—
427C4—24no—
428C4—28no—
429C4—42no—
430C4—48no—
431C4—48no—
432C4—48no—
433C4—44no—
434C4—84no—
435C4—84no—
436C4—84no—
437C2—82no—
438C2#3515111yescharacteristic
439C2#3553212yes—
440C2#4200614yes—
441C2#4780314yes—
442C2#5154612yes—
443C2#5154711yescharacteristic
444C2#5190511yescharacteristic
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. C2×#2845C2semidirect
6. #15438C2semidirect
7. #15435C2semidirect
8. #14804C2semidirect
9. #14354C2semidirect
10. #14168C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28452
C2×#23142
#165212
#157732
#157652
#154382
#154352
#148042
#143542
#141682
#139572