Finite Group Databasebeta

← #110034

Subgroups — #110034

Order 512.

812 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20524C211yesmaximal, characteristic
2#14149C211yesmaximal, characteristic
3C2×#1305C212yesmaximal
4#7911C211yesmaximal, characteristic
5#7909C212yesmaximal
6C2×#407C2212yes—
7C2×#407C2212yes—
8C2×#385C411yescharacteristic
9C2×#385C411yescharacteristic
10C2×#380C2211yescharacteristic
11C2×C4×C22⋊C4C2211yescharacteristic
12C22×#95C2211yescharacteristic
13#1753—24no—
14#1749—21no—
15C4×C22⋊C8—21no—
16#1441—21no—
17#1441—21no—
18#1436—21no—
19#1436—21no—
20C22⋊#95C411yescharacteristic
21C22⋊#95C411yescharacteristic
22#1305C418yescomplemented
23#1280—24no—
24#1280—21no—
25#1280—21no—
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4C2311yescharacteristic
29C22×M16—22no—
30C22×M16—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C411yescharacteristic
36C22×C22⋊C4C2×C411yescharacteristic
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C42D411yescharacteristic
39C22×C42D411yescharacteristic
40#407—22no—
41#407—22no—
42#407—22no—
43#407—22no—
44#407—22no—
45#407C2×C414yes—
46#407C2×C414yes—
47#407Q814yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8D414yes—
52C2×#97—22no—
53C2×C4×C8—22no—
54#385—42no—
55#385—42no—
56#380—22no—
57#380C2×C414yes—
58C4×C22⋊C4—22no—
59C4×C22⋊C4—22no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95D411yescharacteristic
72C2×#95D411yescharacteristic
73#336—22no—
74#336—22no—
75C25C22×C411yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4—41no—
87C23×C4—41no—
88C23×C4—81no—
89C23×C4—81no—
90C23×C4—81no—
91C23×C4—81no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22×C412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×Q812yes—
101C23×C4#4112yes—
102C23×C4#4112yes—
103C23×C4#4112yes—
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—42no—
108C2×M16—82no—
109C2×M16—82no—
110C2×M16—82no—
111C2×M16—82no—
112C22×C8—24no—
113C22×C8—42no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—81no—
120C2×C4⋊C4—81no—
121C2×C4⋊C4—81no—
122C2×C4⋊C4—81no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4C4212yes—
136C2×C22⋊C4C4212yes—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—81no—
142C2×C42—81no—
143C22⋊C8—24no—
144C22⋊C8—24no—
145C22⋊C8—24no—
146C22⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—42no—
149C22⋊C8—42no—
150C22⋊C8C4⋊C418yescomplemented
151#97—82no—
152C4×C8—82no—
153#95—24no—
154#95—22no—
155#95—22no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95—41no—
161#95—42no—
162#95—41no—
163#95—81no—
164#95—81no—
165C24—21no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24C2×C4211yescharacteristic
174C24C2×C22⋊C411yescharacteristic
175C24#11711yescharacteristic
176C24C4×D412yes—
177C24C4×Q812yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4#10012yes—
249C22×C4#10112yes—
250C22×C4C2×C4⋊C412yes—
251C22×C4#11712yes—
252C22×C4C4×D412yes—
253C22×C4C4×D412yes—
254C22×C4C4×D412yes—
255C22×C4C4×Q812yes—
256C22×C4#12011yescharacteristic
257C22×C4#12112yes—
258C22×C4C22⋊Q812yes—
259C22×C4C22⋊Q812yes—
260C22×C4#12311yescharacteristic
261C22×C4#12312yes—
262C22×C4#12411yescharacteristic
263C22×C4#12512yes—
264C22×C4#12612yes—
265C22×C4C4⋊Q811yescharacteristic
266M16—84no—
267M16—84no—
268M16—84no—
269M16—84no—
270C2×C8—28no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—82no—
275C2×C8—82no—
276C4⋊C4—84no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C42—42no—
286C42—42no—
287C42—81no—
288C42—81no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23C4×C22⋊C411yescharacteristic
320C23C4×C4⋊C412yes—
321C23#38611yescharacteristic
322C23#38712yes—
323C23#39111yescharacteristic
324C23C2×#10011yescharacteristic
325C23C2×#10111yescharacteristic
326C23#41311yescharacteristic
327C23#41311yescharacteristic
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4#38512yes—
400C2×C4#38712yes—
401C2×C4#38812yes—
402C2×C4#38912yes—
403C8—48no—
404C8—84no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22—44no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22#143611yescharacteristic
434C22C4×#10011yescharacteristic
435C22C4×#10111yescharacteristic
436C22#189111yescharacteristic
437C22#189111yescharacteristic
438C22#190011yescharacteristic
439C22#190111yescharacteristic
440C22#191311yescharacteristic
441C22#191311yescharacteristic
442C22#199711yescharacteristic
443C22#199811yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4—82no—
453C4—81no—
454C4—82no—
455C4—81no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—42no—
461C2—42no—
462C2#951511yescharacteristic
463C2#951611yescharacteristic
464C2#951711yescharacteristic
465C2#951811yescharacteristic
466C2#1091311yescharacteristic
467C2#1091511yescharacteristic
468C2#1091911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1305C4semidirect
150. C22⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#205242
#141492
C2×#13052
#79112
#79092