Finite Group Databasebeta

← #120021

Subgroups — #120021

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14347C211yesmaximal, characteristic
2C2×#1442C211yesmaximal, characteristic
3#10730C212yesmaximal
4C2×#1294C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C23×C4⋊C4C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2212yes—
9C22×#95—21no—
10C22×#95—21no—
11C22×#95C2211yescharacteristic
12C22×#95C2211yescharacteristic
13C2×#340—24no—
14#1442—22no—
15#1442—24no—
16#1442C414yescomplemented
17C22⋊#95—21no—
18C22⋊#95—21no—
19#1294—24no—
20#1280—22no—
21C24×C4—21no—
22C24×C4—21no—
23C24×C4D411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4C2311yescharacteristic
26C23×C8—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—22no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—44no—
31C22×C4⋊C4C2×C412yes—
32C22×C4⋊C4D414yes—
33C22×C22⋊C4—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8Q814yes—
40#380—22no—
41#380—22no—
42#380—22no—
43#380—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—22no—
49C2×#95—24no—
50C2×#95—24no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—44no—
57C2×#95—44no—
58C2×#95C2×C412yes—
59C2×#95C2×C412yes—
60C2×#95D412yes—
61C2×#95D412yes—
62C2×#95Q812yes—
63#340—48no—
64C25C2×D411yescharacteristic
65C23×C4—24no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—24no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—41no—
93C23×C4—41no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—42no—
98C23×C4C22⋊C412yes—
99C23×C4C22⋊C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×Q811yescharacteristic
105C23×C4C2×Q812yes—
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C23×C4#4112yes—
109C23×C4#4112yes—
110C22×C8—24no—
111C22×C8—44no—
112C22×C8—44no—
113C22×C8—42no—
114C22×C8—42no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—44no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C22⋊C8—24no—
133C22⋊C8—24no—
134C22⋊C8—42no—
135C22⋊C8—42no—
136#95—22no—
137#95—24no—
138#95—22no—
139#95—24no—
140#95—22no—
141#95—48no—
142#95—48no—
143#95—42no—
144#95—42no—
145#95—82no—
146#95—82no—
147#95—82no—
148#95—82no—
149#95C4⋊C414yes—
150C24—22no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24C2×C22⋊C411yescharacteristic
155C24#12011yescharacteristic
156C24#12111yescharacteristic
157C24C22⋊Q811yescharacteristic
158C24#12311yescharacteristic
159C24#12411yescharacteristic
160C24C4⋊Q811yescharacteristic
161C24#13512yes—
162C24#13512yes—
163C24AGL(1,ℤ/8)12yes—
164C24#13712yes—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—44no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—48no—
202C22×C4—48no—
203C22×C4—48no—
204C22×C4—48no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—84no—
239C22×C4—84no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4#11711yescharacteristic
248C22×C4C4×D412yes—
249C22×C4C4×Q812yes—
250C22×C4#12112yes—
251C22×C4#12112yes—
252C22×C4#12111yescharacteristic
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12312yes—
258C22×C4#12512yes—
259C22×C4#12512yes—
260C22×C4#12612yes—
261C22×C4C4⋊Q812yes—
262C2×C8—48no—
263C2×C8—44no—
264C2×C8—82no—
265C2×C8—82no—
266C2×C8—84no—
267C4⋊C4—28no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—84no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—24no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—24no—
286C23—24no—
287C23—24no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23#38011yescharacteristic
301C23#38611yescharacteristic
302C23#39111yescharacteristic
303C23#39411yescharacteristic
304C23#39511yescharacteristic
305C23#39811yescharacteristic
306C23#40011yescharacteristic
307C23#41611yescharacteristic
308C23#41611yescharacteristic
309C23#41811yescharacteristic
310C23#41911yescharacteristic
311C23#44912yes—
312C23#45212yes—
313C23#45412yes—
314C23#45512yes—
315C23B(4,𝔽2)12yes—
316C23#45812yes—
317C23#46312yes—
318C23#46412yes—
319C23#47812yes—
320C23#47912yes—
321C23#48212yes—
322C23#48512yes—
323C23#48812yes—
324C23#48912yes—
325C23#49912yes—
326C23#50112yes—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—88no—
364C2×C4—88no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—88no—
369C2×C4—84no—
370C2×C4—88no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4#38712yes—
385C2×C4#38912yes—
386C2×C4#39612yes—
387C2×C4#39812yes—
388C8—84no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—24no—
402C22—24no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—24no—
407C22—24no—
408C22—24no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22#144111yescharacteristic
418C22#159714yes—
419C22#160614yes—
420C22#161214yes—
421C22#161514yes—
422C22#180411yescharacteristic
423C22#180511yescharacteristic
424C22#187311yescharacteristic
425C22#187511yescharacteristic
426C22#188511yescharacteristic
427C22#188811yescharacteristic
428C22#188911yescharacteristic
429C22#189011yescharacteristic
430C22#199111yescharacteristic
431C22#199311yescharacteristic
432C22#204012yes—
433C22#204212yes—
434C22#204412yes—
435C22#205912yes—
436C22#206112yes—
437C22#206212yes—
438C22#208512yes—
439C22#210112yes—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—84no—
445C4—82no—
446C4—84no—
447C4—82no—
448C4—164no—
449C2—21no—
450C2—21no—
451C2—22no—
452C2—22no—
453C2—21no—
454C2—21no—
455C2#901611yescharacteristic
456C2#901811yescharacteristic
457C2#902211yescharacteristic
458C2#902811yescharacteristic
459C2#1040414yes—
460C2#1057114yes—
461C2#1120811yescharacteristic
462C2#1121111yescharacteristic
463C2#1122211yescharacteristic
subgroupG/H
center, socleC24#120
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1442C4semidirect

Maximal subgroups

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

subgroupindex
#143472
C2×#14422
#107302
C2×#12942
C2×#12802