Finite Group Databasebeta

← #620256

Subgroups — #620256

Order 512.

3550 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#2625C212yesmaximal
3C2×#2285C211yesmaximal, characteristic
4#15911C218yesmaximal
5#15881C214yesmaximal, complemented
6#15850C214yesmaximal, complemented
7#15218C212yesmaximal, complemented
8#15180C212yesmaximal, complemented
9#14702C212yesmaximal
10#14343C211yesmaximal, characteristic
11#14017C214yesmaximal, complemented
12C2×#541—216no—
13C2×#541—28no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×#126C2214yes—
16C22×#125C2214yes—
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2625—216no—
24C2×#400C2214yes—
25C2×#398C2214yes—
26C2×#397C2218yes—
27C2×#391C2214yes—
28C2×#391C2214yes—
29C2×#389C2214yes—
30#2285—28no—
31C2×#384C2212yes—
32C2×#382C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37#1445C2214yes—
38#1444C2218yes—
39#1444C2218yes—
40#1442C2218yes—
41#1442C2218yes—
42#1441C2218yes—
43#1441C2218yes—
44#1440C2214yes—
45#1436C2214yes—
46#1436C2218yes—
47#1436C2218yes—
48#1436C2214yes—
49C4⋊#95C2212yes—
50C4⋊#95C2212yes—
51C4⋊#95C2214yes—
52C22⋊#95C2218yes—
53C23×Q8C2312yes—
54C24×C4C2312yes—
55C24×C4C2311yescharacteristic
56#541—2128no—
57C2×C4⋊Q8—28no—
58C2×C4⋊Q8—24no—
59C2×#126—216no—
60C2×#126—216no—
61C2×#126—216no—
62C2×#125—28no—
63C2×#125—216no—
64C2×#125—216no—
65C2×#125—216no—
66C2×#125—24no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8D4116yes—
70C2×C4×Q8—28no—
71C2×C4×Q8—28no—
72C2×C4×Q8—28no—
73C2×#117—28no—
74C2×#117—28no—
75C2×#117—24no—
76C2×#117D418yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2318yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C42—24no—
93C22×C42—28no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2312yes—
97C22×C42C2314yes—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101#400—216no—
102#398—216no—
103#397—232no—
104#391—216no—
105#391—216no—
106#389—216no—
107#384—28no—
108#382—28no—
109#380—28no—
110#380—216no—
111C4×C4⋊C4—28no—
112C4×C4⋊C4—28no—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2314yes—
117C2×#95C2318yes—
118C2×#95C2318yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2318yes—
123C2×#95C2318yes—
124C2×#95C2318yes—
125C25C2411yescharacteristic
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—28no—
129C22×Q8—28no—
130C22×Q8C2×D418yes—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—28no—
137C23×C4—44no—
138C23×C4C2×D418yes—
139C23×C4C2×D414yes—
140C23×C4C2414yes—
141C23×C4C2418yes—
142C23×C4C2414yes—
143C23×C4C2412yes—
144C23×C4C2412yes—
145C23×C4C2414yes—
146C23×C4C2412yes—
147C23×C4C2411yescharacteristic
148C23×C4C2412yes—
149C23×C4C2411yescharacteristic
150C4⋊Q8—232no—
151#126—2128no—
152#125—2128no—
153#125—232no—
154C22⋊Q8—264no—
155C4×Q8—264no—
156#117—232no—
157#117—416no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4C2×D418yes—
189C2×C4⋊C4C2×D418yes—
190C2×C4⋊C4C2×D4116yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4C2×D418yes—
200C2×C22⋊C4C2×D4116yes—
201C2×C42—232no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—28no—
205C2×C42—28no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—28no—
215C2×C42C2×D414yes—
216C2×C42C2×D414yes—
217C2×C42#4118yes—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—216no—
222#95—216no—
223#95—216no—
224#95—216no—
225#95—216no—
226#95—216no—
227#95—48no—
228#95—48no—
229#95—48no—
230C24—22no—
231C24—22no—
232C24—24no—
233C24—24no—
234C24C22×D414yes—
235C24C22×D412yes—
236C24C2511yescharacteristic
237C2×Q8—216no—
238C2×Q8—232no—
239C2×Q8—48no—
240C2×Q8—48no—
241C2×Q8—416no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—232no—
250C22×C4—216no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—416no—
286C22×C4—416no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4#120116yes—
290C22×C4C22×D418yes—
291C22×C4C22×D414yes—
292C22×C4C22×D418yes—
293C22×C4C22×D414yes—
294C22×C4C22×D414yes—
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D414yes—
298C22×C4C2×#4114yes—
299C22×C4C2×#4114yes—
300C22×C4C2×#4114yes—
301C22×C42−1+418yes—
302C22×C42−1+414yes—
303C4⋊C4—232no—
304C4⋊C4—232no—
305C4⋊C4—232no—
306C4⋊C4—216no—
307C4⋊C4—216no—
308C4⋊C4—232no—
309C4⋊C4—416no—
310C4⋊C4—416no—
311C4⋊C4—48no—
312C4⋊C4—416no—
313C4⋊C4—816no—
314C22⋊C4—216no—
315C22⋊C4—232no—
316C22⋊C4—48no—
317C22⋊C4—88no—
318C42—28no—
319C42—28no—
320C42—232no—
321C42—28no—
322C42—28no—
323C42—216no—
324C42—48no—
325C23—22no—
326C23—22no—
327C23—28no—
328C23—28no—
329C23—28no—
330C23—24no—
331C23—24no—
332C23—42no—
333C23—42no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23C2×#12014yes—
339C23C23×D411yescharacteristic
340C23C23×D412yes—
341C23C22×#4112yes—
342C23C2×2−1+412yes—
343C23C2×2−1+414yes—
344C23#58514yes—
345Q8—432no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—232no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—216no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—432no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4C2×#12014yes—
395C2×C4C2×#12014yes—
396C2×C4#52514yes—
397C2×C4#53614yes—
398C2×C4#54118yescomplemented
399C2×C4#547116yes—
400C22—28no—
401C22—24no—
402C22—44no—
403C22—48no—
404C22—48no—
405C22—44no—
406C22C22×#12011yescharacteristic
407C22C2×#52511yescharacteristic
408C22C2×#53611yescharacteristic
409C22C2×#54112yes—
410C22#345512yes—
411C22C2×#54714yes—
412C22#348914yes—
413C22#349514yes—
414C22#349714yes—
415C22#349912yes—
416C22#350012yes—
417C22#355514yes—
418C22#355612yes—
419C22#355712yes—
420C4—24no—
421C4—28no—
422C4—44no—
423C4—44no—
424C4—48no—
425C4—48no—
426C4—44no—
427C4—84no—
428C4—88no—
429C4#240618yes—
430C2—44no—
431C2C2×#240611yescharacteristic
432C2#3531012yes—
433C2#3566012yes—
434C2#3737712yes—
435C2#3934714yes—
436C2#3935314yes—
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
5. #15881C2semidirect
6. #15850C2semidirect
7. #15218C2semidirect
8. #15180C2semidirect
11. #14017C2semidirect
398. C2×C4#541semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#26252
C2×#22852
#159112
#158812
#158502
#152182
#151802
#147022
#143432
#140172