Finite Group Databasebeta

← #120022

Subgroups — #120022

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14347C211yesmaximal, characteristic
2C2×#1444C211yesmaximal, characteristic
3#10730C212yesmaximal
4C2×#1294C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C23×C4⋊C4—21no—
7C23×C4⋊C4C2211yescharacteristic
8C22×C22⋊C8C2212yes—
9C2×#380C2212yes—
10C22×#95—21no—
11C22×#95C2211yescharacteristic
12C22×#95C2211yescharacteristic
13C2×#340—24no—
14#1444—22no—
15#1444C414yescomplemented
16#1442—24no—
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—24no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—44no—
32C22×C4⋊C4—44no—
33C22×C4⋊C4—44no—
34C22×C4⋊C4—44no—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4D414yes—
37C22×C22⋊C4—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8Q814yes—
44#380—22no—
45#380—22no—
46#380—22no—
47#380—22no—
48C2×#95—22no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—22no—
53C2×#95—24no—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—44no—
60C2×#95C2×C412yes—
61C2×#95C2×C412yes—
62C2×#95D412yes—
63C2×#95D412yes—
64C2×#95Q812yes—
65#340—48no—
66C25C2×D411yescharacteristic
67C23×C4—24no—
68C23×C4—24no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—24no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—24no—
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—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q811yescharacteristic
107C23×C4C2×Q812yes—
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4#4112yes—
111C23×C4#4112yes—
112C22×C8—24no—
113C22×C8—44no—
114C22×C8—44no—
115C22×C8—42no—
116C22×C8—42no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4—432no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142#95—22no—
143#95—24no—
144#95—22no—
145#95—24no—
146#95—22no—
147#95—48no—
148#95—42no—
149#95—42no—
150#95—82no—
151#95—82no—
152#95—82no—
153#95—82no—
154#95C4⋊C414yes—
155C24—22no—
156C24—22no—
157C24—22no—
158C24—22no—
159C24C2×C22⋊C411yescharacteristic
160C24#12011yescharacteristic
161C24#12111yescharacteristic
162C24C22⋊Q811yescharacteristic
163C24#12311yescharacteristic
164C24#12411yescharacteristic
165C24C4⋊Q811yescharacteristic
166C24#13512yes—
167C24#13512yes—
168C24AGL(1,ℤ/8)12yes—
169C24#13712yes—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—48no—
200C22×C4—48no—
201C22×C4—48no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4—84no—
244C22×C4—84no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4#11711yescharacteristic
253C22×C4C4×D412yes—
254C22×C4C4×Q812yes—
255C22×C4#12112yes—
256C22×C4#12112yes—
257C22×C4#12111yescharacteristic
258C22×C4C22⋊Q812yes—
259C22×C4C22⋊Q812yes—
260C22×C4C22⋊Q812yes—
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12312yes—
263C22×C4#12512yes—
264C22×C4#12512yes—
265C22×C4#12612yes—
266C22×C4C4⋊Q812yes—
267C2×C8—48no—
268C2×C8—44no—
269C2×C8—82no—
270C2×C8—82no—
271C2×C8—84no—
272C4⋊C4—28no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—432no—
277C4⋊C4—84no—
278C22⋊C4—82no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C22⋊C4—82no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—24no—
291C23—24no—
292C23—24no—
293C23—24no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23#38011yescharacteristic
307C23#38611yescharacteristic
308C23#39111yescharacteristic
309C23#39411yescharacteristic
310C23#39511yescharacteristic
311C23#39811yescharacteristic
312C23#40011yescharacteristic
313C23#41611yescharacteristic
314C23#41611yescharacteristic
315C23#41811yescharacteristic
316C23#41911yescharacteristic
317C23#44912yes—
318C23#45212yes—
319C23#45312yes—
320C23#45612yes—
321C23B(4,𝔽2)12yes—
322C23#45812yes—
323C23#46312yes—
324C23#46412yes—
325C23#47812yes—
326C23#47912yes—
327C23#48212yes—
328C23#48512yes—
329C23#48812yes—
330C23#48912yes—
331C23#49912yes—
332C23#50112yes—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—416no—
357C2×C4—416no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—88no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4#38712yes—
391C2×C4#38912yes—
392C2×C4#39612yes—
393C2×C4#39812yes—
394C8—84no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—24no—
408C22—24no—
409C22—24no—
410C22—24no—
411C22—24no—
412C22—24no—
413C22—24no—
414C22—24no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22#144111yescharacteristic
424C22#160014yes—
425C22#160314yes—
426C22#160914yes—
427C22#161814yes—
428C22#180411yescharacteristic
429C22#180511yescharacteristic
430C22#187311yescharacteristic
431C22#187511yescharacteristic
432C22#188611yescharacteristic
433C22#188711yescharacteristic
434C22#188911yescharacteristic
435C22#189011yescharacteristic
436C22#199111yescharacteristic
437C22#199311yescharacteristic
438C22#203912yes—
439C22#204212yes—
440C22#204412yes—
441C22#205912yes—
442C22#206012yes—
443C22#206212yes—
444C22#208512yes—
445C22#210112yes—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—48no—
451C4—82no—
452C4—84no—
453C4—82no—
454C4—164no—
455C2—21no—
456C2—21no—
457C2—22no—
458C2—22no—
459C2—21no—
460C2—21no—
461C2#901411yescharacteristic
462C2#902011yescharacteristic
463C2#902411yescharacteristic
464C2#902611yescharacteristic
465C2#1040514yes—
466C2#1057214yes—
467C2#1120811yescharacteristic
468C2#1121011yescharacteristic
469C2#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
15. #1444C4semidirect

Maximal subgroups

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

subgroupindex
#143472
C2×#14442
#107302
C2×#12942
C2×#12802