Finite Group Databasebeta

← #559857

Subgroups — #559857

Order 512.

3486 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#2381C212yesmaximal, complemented
3#14683C211yesmaximal, characteristic
4#14683C211yesmaximal, characteristic
5#14663C212yesmaximal, complemented
6#14474C214yesmaximal, complemented
7#14449C214yesmaximal
8#14448C214yesmaximal
9#13944C214yesmaximal
10#13931C214yesmaximal, complemented
11#13927C214yesmaximal, complemented
12C2×#533—216no—
13C2×#533C4116yescomplemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#125C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2212yes—
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2381—216no—
24C2×#391C2218yes—
25C2×#387C2218yes—
26C2×#387C2218yes—
27C2×#383C2212yes—
28C2×#383C2212yes—
29#1446C2218yes—
30#1445C2218yes—
31#1445C2218yes—
32#1445C2218yes—
33#1441C2218yes—
34#1439C2218yes—
35#1437C2218yes—
36#1436C2218yes—
37C4⋊#95C2214yes—
38C4⋊#95C2218yes—
39#1434C2212yes—
40#1434C2212yes—
41#1434C2212yes—
42#1434C2212yes—
43#1434C2212yes—
44#1434C2212yes—
45#1434C2214yes—
46#1434C2214yes—
47C22⋊#95C2212yes—
48C22⋊#95C2212yes—
49C22⋊#95C2212yes—
50C22⋊#95C2212yes—
51C22⋊#95C2212yes—
52C22⋊#95C2212yes—
53C4×#95C2214yes—
54C23×Q8C2312yes—
55C23×Q8C2312yes—
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2311yescharacteristic
59#533—2128no—
60C2×C4⋊Q8—28no—
61C2×C4⋊Q8C2×C418yes—
62C2×C4⋊Q8C2×C418yes—
63C2×C4⋊Q8D4116yes—
64C2×#125—216no—
65C2×#125C2×C4116yes—
66C2×C22⋊Q8—216no—
67C2×C22⋊Q8—216no—
68C2×C22⋊Q8C2×C4116yes—
69C2×C22⋊Q8C2×C4116yes—
70C2×C4×Q8—216no—
71C2×C4×Q8—216no—
72C2×C4×Q8C2×C4116yes—
73C2×C4×Q8C2×C4116yes—
74C2×#117—216no—
75C2×#117C2×C4116yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2318yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C42—28no—
87C22×C42C2×C418yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2314yes—
92C22×C42C2318yes—
93C22×C42C2311yescharacteristic
94#391—232no—
95#387—232no—
96#387—232no—
97#383—28no—
98#383—28no—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2318yes—
108C2×#95C2318yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C25C2411yescharacteristic
118C22×Q8—28no—
119C22×Q8—42no—
120C22×Q8—44no—
121C22×Q8—42no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C22×C418yes—
126C22×Q8C22×C418yes—
127C22×Q8C2×D4116yes—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C22×C414yes—
138C23×C4C22×C414yes—
139C23×C4C22×C414yes—
140C23×C4C2418yes—
141C23×C4C2418yes—
142C23×C4C2414yes—
143C23×C4C2411yescharacteristic
144C23×C4C2411yescharacteristic
145C23×C4C2414yes—
146C23×C4C2411yescharacteristic
147C23×C4C2411yescharacteristic
148C23×C4C2411yescharacteristic
149C23×C4C2411yescharacteristic
150C4⋊Q8—232no—
151C4⋊Q8C22⋊C4164yescomplemented
152#125—264no—
153C22⋊Q8—264no—
154C22⋊Q8—264no—
155C4×Q8—264no—
156C4×Q8—264no—
157#117—264no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—216no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—48no—
165C2×C4⋊C4C22×C418yes—
166C2×C4⋊C4C22×C4116yes—
167C2×C4⋊C4C22×C4116yes—
168C2×C4⋊C4C22×C418yes—
169C2×C4⋊C4C22×C418yes—
170C2×C4⋊C4C22×C418yes—
171C2×C4⋊C4C2×D4116yes—
172C2×C4⋊C4C2×D4116yes—
173C2×C22⋊C4—216no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4C22×C4116yes—
181C2×C42—216no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—232no—
186C2×C42—216no—
187C2×C42—48no—
188C2×C42—48no—
189C2×C42C22×C4116yes—
190C2×C42C22×C414yes—
191C2×C42C22×C414yes—
192C2×C42C22×C414yes—
193C2×C42C22×C414yes—
194C2×C42C2×D418yes—
195#95—28no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—216no—
202#95—216no—
203#95—28no—
204#95—28no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—48no—
210#95—48no—
211#95—48no—
212#95—48no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24C23×C412yes—
220C242−1+414yes—
221C24C2511yescharacteristic
222C2×Q8—216no—
223C2×Q8—48no—
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—48no—
227C2×Q8—416no—
228C2×Q8—416no—
229C2×Q8—416no—
230C2×Q8—416no—
231C2×Q8C2×C22⋊C4132yes—
232C22×C4—232no—
233C22×C4—28no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—416no—
278C22×C4—416no—
279C22×C4C23×C412yes—
280C22×C4C23×C412yes—
281C22×C4C23×C418yes—
282C22×C4C23×C412yes—
283C22×C4C23×C412yes—
284C22×C4C23×C412yes—
285C22×C4C23×C418yes—
286C22×C4C23×C412yes—
287C22×C4C22×D414yes—
288C22×C4C22×D414yes—
289C22×C4C22×D414yes—
290C22×C4C22×D4116yes—
291C22×C42−1+414yes—
292C4⋊C4—216no—
293C4⋊C4—232no—
294C4⋊C4—232no—
295C4⋊C4—216no—
296C4⋊C4—816no—
297C4⋊C4C2×C22⋊C4132yes—
298C4⋊C4C2×C22⋊C4132yes—
299C22⋊C4—232no—
300C22⋊C4—88no—
301C22⋊C4—88no—
302C42—28no—
303C42—28no—
304C42—28no—
305C42—232no—
306C42—816no—
307C42C2×C22⋊C4116yes—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—28no—
315C23—28no—
316C23—28no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23#51914yes—
322C23C24×C411yescharacteristic
323C23C23×D412yes—
324C23C2×2−1+412yes—
325C23C2×2−1+412yes—
326C23#58512yes—
327C23#58512yes—
328C23#58512yes—
329C23#58512yes—
330Q8—432no—
331Q8—432no—
332C2×C4—232no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—216no—
344C2×C4—216no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—28no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—416no—
359C2×C4—416no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4C22×C22⋊C414yes—
379C2×C4C22×C22⋊C414yes—
380C2×C4C22×C22⋊C414yes—
381C2×C4C22×C22⋊C4116yes—
382C2×C4#51914yes—
383C22—24no—
384C22—24no—
385C22—24no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—416no—
391C22C23×C22⋊C411yescharacteristic
392C22C2×#51911yescharacteristic
393C22C2×#51911yescharacteristic
394C22#342811yescharacteristic
395C22#342811yescharacteristic
396C22#342811yescharacteristic
397C22#342811yescharacteristic
398C22#348514yes—
399C22#348614yes—
400C22#348714yes—
401C22#353114yes—
402C22#355714yes—
403C22#356414yes—
404C22#356414yes—
405C4—24no—
406C4—216no—
407C4—24no—
408C4—24no—
409C4—24no—
410C4—42no—
411C4—42no—
412C4—48no—
413C4—88no—
414C4—88no—
415C2—44no—
416C2#3417211yescharacteristic
417C2#3417311yescharacteristic
418C2#3417411yescharacteristic
419C2#3434811yescharacteristic
420C2#3437511yescharacteristic
421C2#3438911yescharacteristic
422C2#3438911yescharacteristic
423C2#4627718yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2381C2semidirect
5. #14663C2semidirect
6. #14474C2semidirect
10. #13931C2semidirect
11. #13927C2semidirect
13. C2×#533C4semidirect
151. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#23812
#146832
#146632
#144742
#144492
#144482
#139442
#139312
#139272