Finite Group Databasebeta

← #410197

Subgroups — #410197

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic
7#11257C214yesmaximal, complemented
8#11231C214yesmaximal, complemented
9#10721C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#106—24no—
14C22×#106—24no—
15C22×#106—24no—
16C22×#106C2214yes—
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#381—21no—
20C2×#381—21no—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1850—24no—
25#1824—24no—
26#1823—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2214yes—
32C2×#336C2212yes—
33#1445—22no—
34#1445—22no—
35C4⋊#95C2214yes—
36C4⋊#95C2214yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C23×C8—22no—
44C23×C8—28no—
45C23×C8D414yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D414yes—
62C22×C4⋊C4D414yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2312yes—
72C2×#106—232no—
73C2×#106—28no—
74C2×#106—28no—
75C2×#106—28no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—28no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81#381—44no—
82#381—44no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—24no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95D412yes—
96#340—28no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#336—24no—
102C25C2×D411yescharacteristic
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—84no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4#4112yes—
129C23×C4C2411yescharacteristic
130C22×C8—216no—
131C22×C8—216no—
132C22×C8—28no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—24no—
136C22×C8—216no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—24no—
144C22×C8—24no—
145C22×C8C2×D414yes—
146C22×C8#4118yes—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4SD1614yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197#106—232no—
198C4⋊C8—48no—
199C22⋊C8—44no—
200#95—24no—
201#95—48no—
202#95—42no—
203#95—42no—
204#95—48no—
205#95—84no—
206#95—84no—
207C24—24no—
208C24—24no—
209C24#12112yes—
210C24#12112yes—
211C24#12112yes—
212C24C2×SD1612yes—
213C24#13512yes—
214C24AGL(1,ℤ/8)12yes—
215C24#13712yes—
216C24C22×D411yescharacteristic
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—28no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—88no—
263C22×C4—88no—
264C22×C4—88no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4#12014yes—
270C22×C4#12112yes—
271C22×C4#12112yes—
272C22×C4#12112yes—
273C22×C4#12314yes—
274C22×C4#12314yes—
275C22×C4#12414yes—
276C22×C4C2×SD1614yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4114yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C42−1+414yes—
285C2×C8—232no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—28no—
289C2×C8—28no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8#12318yes—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C22⋊C4—28no—
316C22⋊C4—44no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C42—42no—
320C42—42no—
321C42—44no—
322C23—28no—
323C23—28no—
324C23—28no—
325C23—28no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—24no—
332C23#39414yes—
333C23#46014yes—
334C23#46114yes—
335C23#46314yes—
336C23#46414yes—
337C23#46514yes—
338C23#46814yes—
339C23C2×#12111yescharacteristic
340C23C2×#12111yescharacteristic
341C23C2×#12111yescharacteristic
342C23#54714yes—
343C23C22×SD1611yescharacteristic
344C23C2×#13511yescharacteristic
345C23C2×AGL(1,ℤ/8)11yescharacteristic
346C23C2×#13711yescharacteristic
347C23#57714yes—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—24no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—816no—
373C2×C4—816no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#39414yes—
391C2×C4C2×#12011yescharacteristic
392C2×C4C2×#12312yes—
393C2×C4C2×#12311yescharacteristic
394C2×C4C2×#12311yescharacteristic
395C2×C4#52512yes—
396C2×C4#52512yes—
397C2×C4C2×#12411yescharacteristic
398C2×C4#53612yes—
399C2×C4#53612yes—
400C2×C4#53612yes—
401C2×C4#55214yes—
402C2×C4#55214yes—
403C2×C4#55214yes—
404C8—28no—
405C8—81no—
406C8—81no—
407C22—28no—
408C22—28no—
409C22—28no—
410C22—28no—
411C22—28no—
412C22—28no—
413C22—24no—
414C22—24no—
415C22#205518yes—
416C22C2×#39411yescharacteristic
417C22#241312yes—
418C22#241312yes—
419C22#248612yes—
420C22C2×#46111yescharacteristic
421C22C2×#46311yescharacteristic
422C22C2×#46011yescharacteristic
423C22C2×#46411yescharacteristic
424C22#304212yes—
425C22#304312yes—
426C22C2×#46511yescharacteristic
427C22C2×#46811yescharacteristic
428C22#305712yes—
429C22#333514yes—
430C22#333714yes—
431C22#334414yes—
432C22#334514yes—
433C4—24no—
434C4—44no—
435C4—82no—
436C4—84no—
437C4—84no—
438C4—162no—
439C4—162no—
440C4#248412yes—
441C4#248412yes—
442C4#250312yes—
443C4#261411yescharacteristic
444C4#284911yescharacteristic
445C2—28no—
446C2#1507311yescharacteristic
447C2C2×#205511yescharacteristic
448C2#2354111yescharacteristic
449C2#2457812yes—
450C2#2457912yes—
451C2#2458012yes—
452C2#2458112yes—
453C2#2765812yes—
454C2#2766312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11257C2semidirect
8. #11231C2semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#18502
C2×#18242
C2×#18232
#139242
#139042
#112572
#112312
#107212