Finite Group Databasebeta

← #4474592

Subgroups — #4474592

Order 512.

2038 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17230C214yesmaximal, complemented
2#16903C211yesmaximal, characteristic
3#16769C214yesmaximal, complemented
4#16608C214yesmaximal, complemented
5#16254C211yesmaximal, characteristic
6#15996C214yesmaximal
7#15849C214yesmaximal, complemented
8#15484C214yesmaximal
9#14720C212yesmaximal
10#14449C212yesmaximal
11#14339C211yesmaximal, characteristic
12C22×#125C2214yes—
13C22×C22⋊Q8C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C2×#401C2214yes—
16C2×#395C2214yes—
17C2×#395C2212yes—
18C2×#395C2212yes—
19C2×#391C2214yes—
20C2×#385C2214yes—
21C2×#381C2212yes—
22C2×#380C2212yes—
23#1445C2214yes—
24#1445C2214yes—
25#1445C2214yes—
26#1444C2214yes—
27#1444C2214yes—
28#1444C2214yes—
29#1443C2214yes—
30#1442C2214yes—
31#1442C2214yes—
32#1442C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1441C2214yes—
36#1441C2214yes—
37#1441C2212yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1434C2214yes—
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2214yes—
57C23×Q8C2312yes—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C24×C4C2311yescharacteristic
61C2×#125—48no—
62C2×#125—48no—
63C2×#125—48no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—48no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2312yes—
82C22×C42C2314yes—
83#401—48no—
84#395—44no—
85#395—44no—
86#395—48no—
87#391—48no—
88#385—48no—
89#381—44no—
90#380—44no—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2312yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2312yes—
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—
117C25C2411yescharacteristic
118C22×Q8—24no—
119C22×Q8—28no—
120C22×Q8—42no—
121C22×Q8—44no—
122C22×Q8—42no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4#4114yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2412yes—
139C23×C4C2414yes—
140C23×C4C2412yes—
141C23×C4C2411yescharacteristic
142C23×C4C2414yes—
143C23×C4C2411yescharacteristic
144#125—464no—
145C22⋊Q8—432no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
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—416no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C42—28no—
189C2×C42—44no—
190C2×C42—44no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—28no—
197#95—28no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—42no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—42no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217C24—24no—
218C24—21no—
219C24—22no—
220C24—24no—
221C24—21no—
222C24C2×#4112yes—
223C242−1+414yes—
224C24C2511yescharacteristic
225C2×Q8—216no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8—88no—
229C2×Q8—88no—
230C22×C4—28no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
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—44no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—88no—
295C22×C4—84no—
296C22×C4—88no—
297C22×C4C2×#4114yes—
298C22×C4H(5,𝔽2)12yes—
299C22×C4H(5,𝔽2)18yes—
300C22×C4H(5,𝔽2)14yes—
301C22×C42−1+418yes—
302C22×C42−1+412yes—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—416no—
306C4⋊C4—416no—
307C4⋊C4—416no—
308C4⋊C4—416no—
309C4⋊C4—48no—
310C4⋊C4—416no—
311C4⋊C4—416no—
312C4⋊C4—416no—
313C22⋊C4—416no—
314C22⋊C4—416no—
315C22⋊C4—84no—
316C22⋊C4—84no—
317C42—416no—
318C23—24no—
319C23—22no—
320C23—22no—
321C23—24no—
322C23—24no—
323C23—28no—
324C23—24no—
325C23—28no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23—44no—
330C23#54114yes—
331C23C22×#4111yescharacteristic
332C23C2×H(5,𝔽2)12yes—
333C23C2×H(5,𝔽2)14yes—
334C23C2×H(5,𝔽2)11yescharacteristic
335C23C2×2−1+412yes—
336C23C2×2−1+411yescharacteristic
337C23C2×2−1+414yes—
338Q8—816no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—416no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
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—88no—
403C2×C4—88no—
404C22—28no—
405C22—24no—
406C22—48no—
407C22—48no—
408C22—42no—
409C22—42no—
410C22—44no—
411C22C2×#54111yescharacteristic
412C22#349012yes—
413C22#349011yescharacteristic
414C22#349211yescharacteristic
415C22#350912yes—
416C22#353214yes—
417C22#353612yes—
418C22#353812yes—
419C22#353814yes—
420C22#354814yes—
421C22#355214yes—
422C22#355214yes—
423C22#355214yes—
424C4—48no—
425C4—42no—
426C4—42no—
427C4—44no—
428C4—44no—
429C4—48no—
430C4—84no—
431C4—84no—
432C4—84no—
433C4—84no—
434C2—44no—
435C2#3672811yescharacteristic
436C2#4054712yes—
437C2#4645311yescharacteristic
438C2#4699611yescharacteristic
439C2#4959412yes—
440C2#5512214yes—
441C2#5638714yes—
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. #17230C2semidirect
3. #16769C2semidirect
4. #16608C2semidirect
7. #15849C2semidirect

Maximal subgroups

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

subgroupindex
#172302
#169032
#167692
#166082
#162542
#159962
#158492
#154842
#147202
#144492
#143392