Finite Group Databasebeta

← #761280

Subgroups — #761280

Order 512.

3214 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C214yesmaximal
2C22×#389C211yesmaximal, characteristic
3C2×#2285C211yesmaximal, characteristic
4#16964C214yesmaximal, complemented
5#15153C212yesmaximal, complemented
6#15145C212yesmaximal, complemented
7#15082C214yesmaximal, complemented
8#14936C214yesmaximal
9#14049C214yesmaximal, complemented
10#13904C212yesmaximal
11#13880C212yesmaximal
12C4×#381C211yesmaximal, characteristic
13C22×#117C2212yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17C23×C42C2211yescharacteristic
18#2484—232no—
19C2×#400C2218yes—
20C2×#398C2218yes—
21C2×#389—28no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C22116yes—
27C2×#385C2218yes—
28#2285—28no—
29C2×#384C2211yescharacteristic
30C2×#384C2214yes—
31C2×#384C2211yescharacteristic
32C2×#382C2212yes—
33C2×#381C2218yes—
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2212yes—
41#1445C2218yes—
42#1445C2218yes—
43#1445C22116yes—
44#1444C2218yes—
45#1444C2218yes—
46C4⋊#95C2214yes—
47C4⋊#95C2214yes—
48C4⋊#95C2214yes—
49#1434C2212yes—
50#1434C2212yes—
51C22⋊#95C2212yes—
52C4×#95C2212yes—
53C4×#95C2212yes—
54C24×C4C2312yes—
55C24×C4C2312yes—
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59C2×#117—216no—
60C2×#117—28no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—28no—
63C22×C4⋊C4—216no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C23116yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C42—28no—
83C22×C42D418yes—
84C22×C42C2314yes—
85C22×C42C2314yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#400—232no—
91#398—232no—
92#389—28no—
93#389—28no—
94#389—28no—
95#389—28no—
96#387—264no—
97#385—232no—
98#384—216no—
99#384—24no—
100#384—24no—
101#382—28no—
102#381—232no—
103#381D4116yes—
104#380—28no—
105#380—28no—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—28no—
108C4×C22⋊C4—28no—
109C2×#95—28no—
110C2×#95C23116yes—
111C2×#95C23116yes—
112C2×#95C23116yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C25C2411yescharacteristic
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—28no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—28no—
133C23×C4—28no—
134C23×C4—28no—
135C23×C4—28no—
136C23×C4—42no—
137C23×C4—44no—
138C23×C4C2×D414yes—
139C23×C4C2×D414yes—
140C23×C4C2×D418yes—
141C23×C4C2×D414yes—
142C23×C4#4118yes—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C23×C4C2414yes—
146C23×C4C2418yes—
147C23×C4C2411yescharacteristic
148C23×C4C2414yes—
149C23×C4C2411yescharacteristic
150C23×C4C2411yescharacteristic
151C23×C4C2411yescharacteristic
152C23×C4C2411yescharacteristic
153C23×C4C2411yescharacteristic
154#117—432no—
155C2×C4⋊C4—232no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—216no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—232no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—232no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—416no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—416no—
184C2×C4⋊C4—432no—
185C2×C4⋊C4#4118yes—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—216no—
190C2×C22⋊C4C2×D4116yes—
191C2×C42—232no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—28no—
197C2×C42—216no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42—28no—
201C2×C42C2×D414yes—
202C2×C42C2×D414yes—
203C2×C42C2×D414yes—
204C2×C42C2×D414yes—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—232no—
210#95—232no—
211#95—232no—
212#95—28no—
213#95—28no—
214#95—44no—
215#95C2×D4116yes—
216#95C2×D4116yes—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24C22×D412yes—
222C24C22×D414yes—
223C24C2×#4114yes—
224C242−1+414yes—
225C24C2511yescharacteristic
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—232no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—232no—
240C22×C4—216no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—22no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—44no—
269C22×C4—416no—
270C22×C4—432no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—432no—
277C22×C4#123116yes—
278C22×C4#127116yes—
279C22×C4C22×D412yes—
280C22×C4C22×D4116yes—
281C22×C4C22×D412yes—
282C22×C4C22×D414yes—
283C22×C4C22×D414yes—
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C2×#4118yes—
289C22×C4C2×#4114yes—
290C22×C4C2×#4114yes—
291C22×C4C2×#4114yes—
292C22×C42−1+414yes—
293C22×C42−1+414yes—
294C4⋊C4—216no—
295C4⋊C4—216no—
296C4⋊C4—464no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—416no—
302C4⋊C4—416no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C22⋊C4—232no—
308C22⋊C4—416no—
309C42—28no—
310C42—28no—
311C42—28no—
312C42—28no—
313C42—416no—
314C42—416no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—28no—
320C23—28no—
321C23—28no—
322C23—28no—
323C23—28no—
324C23—28no—
325C23C2×#12314yes—
326C23C2×#12714yes—
327C23#53614yes—
328C23#547116yes—
329C23C23×D411yescharacteristic
330C23C23×D412yes—
331C23C22×#4112yes—
332C23C22×#4112yes—
333C23C2×2−1+412yes—
334C23C2×2−1+412yes—
335C23C2×2−1+412yes—
336C23#58512yes—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—232no—
350C2×C4—232no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—216no—
356C2×C4—216no—
357C2×C4—216no—
358C2×C4—216no—
359C2×C4—416no—
360C2×C4—432no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—416no—
379C2×C4—416no—
380C2×C4—416no—
381C2×C4—816no—
382C2×C4C2×#12314yes—
383C2×C4C2×#12314yes—
384C2×C4#52514yes—
385C2×C4C2×#12714yes—
386C2×C4C2×#12714yes—
387C2×C4#53614yes—
388C2×C4#53614yes—
389C2×C4#552116yes—
390C22—24no—
391C22—232no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—24no—
396C22—24no—
397C22#250118yes—
398C22C22×#12311yescharacteristic
399C22C2×#52511yescharacteristic
400C22C22×#12711yescharacteristic
401C22C2×#53611yescharacteristic
402C22C2×#53611yescharacteristic
403C22C2×#53611yescharacteristic
404C22#344911yescharacteristic
405C22C2×#54714yes—
406C22D4×#4114yes—
407C22C2×#55214yes—
408C22#349514yes—
409C22#349914yes—
410C22#352314yes—
411C22#356314yes—
412C4—24no—
413C4—24no—
414C4—24no—
415C4—24no—
416C4—48no—
417C4—416no—
418C4—42no—
419C4—84no—
420C4—84no—
421C4—84no—
422C4#248418yescomplemented
423C2—28no—
424C2C2×#250111yescharacteristic
425C2C2×#248411yescharacteristic
426C2#3493611yescharacteristic
427C2#3524811yescharacteristic
428C2#3531111yescharacteristic
429C2#3531711yescharacteristic
430C2#3754011yescharacteristic
431C2#5929018yes—
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
4. #16964C2semidirect
5. #15153C2semidirect
6. #15145C2semidirect
7. #15082C2semidirect
9. #14049C2semidirect
422. C4#2484semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C22×#3892
C2×#22852
#169642
#151532
#151452
#150822
#149362
#140492
#139042
#138802
C4×#3812