Finite Group Databasebeta

← #127016

Subgroups — #127016

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1465C211yesmaximal, characteristic
2C2×#1460C211yesmaximal, characteristic
3#13872C211yesmaximal, characteristic
4#7187C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C2×#431—21no—
8C2×#407—21no—
9C2×#386C412yes—
10C2×#380—21no—
11C2×#380C411yescharacteristic
12#1465—22no—
13#1465C414yescomplemented
14#1460—22no—
15#1460C414yescomplemented
16C2×#324—21no—
17C2×#323—21no—
18C2×#323C2212yes—
19C2×#323C2212yes—
20#1441—21no—
21#1439—21no—
22C4⋊#95—21no—
23C22⋊#95C2212yes—
24C23×D4C2×C411yescharacteristic
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C412yes—
27C24×C4D411yescharacteristic
28C22×M16—41no—
29C2×C4×D4—41no—
30C2×C4×D4—41no—
31C2×C4×D4—42no—
32C2×C4×D4C2×C412yes—
33C2×C4×D4C2×C412yes—
34C2×C4×D4D412yes—
35C2×C4×D4Q812yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4—41no—
42C22×C4⋊C4D411yescharacteristic
43C22×C22⋊C4—22no—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4D411yescharacteristic
46C22×C22⋊C4C2311yescharacteristic
47C22×C42D411yescharacteristic
48#431—24no—
49#431—42no—
50C2×C4⋊C8—41no—
51#407—24no—
52#407—42no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—41no—
56C2×C4×C8—41no—
57#386—44no—
58#380—44no—
59#380—42no—
60C2×#95—22no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64#324—24no—
65#323—24no—
66#323—22no—
67#323—22no—
68#323C2×C414yes—
69#323C2×C414yes—
70C25C22⋊C411yescharacteristic
71C25C22×C411yescharacteristic
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4C4212yes—
79C22×D4C4⋊C412yes—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—44no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—82no—
97C23×C4C4212yes—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—44no—
107C2×M16—82no—
108C2×M16—82no—
109C22×C8—42no—
110C22×C8—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—42no—
118C4×D4—42no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—48no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C4⋊C412yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—82no—
145C2×C22⋊C4C22⋊C412yes—
146C2×C22⋊C4C4⋊C412yes—
147C2×C22⋊C4C22×C411yescharacteristic
148C2×C22⋊C4C22×C411yescharacteristic
149C2×C22⋊C4C2×D411yescharacteristic
150C2×C22⋊C4C2×Q811yescharacteristic
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42C22⋊C412yes—
155C2×C42C4⋊C412yes—
156C4⋊C8—44no—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—42no—
161C22⋊C8—44no—
162C4×C8—44no—
163#95—42no—
164#95—44no—
165#95—44no—
166#95—82no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24—21no—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24#9512yes—
186C24C2×C4211yescharacteristic
187C24C2×C22⋊C411yescharacteristic
188C24C2×C4⋊C411yescharacteristic
189C24#13112yes—
190C24#13112yes—
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—48no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—84no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—88no—
258C22×C4—88no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4#9512yes—
262C22×C4#9512yes—
263C22×C4#9512yes—
264C22×C4#9912yes—
265C22×C4#9911yescharacteristic
266C22×C4#9911yescharacteristic
267C22×C4#10012yes—
268C22×C4#10112yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C4⋊C411yescharacteristic
272C22×C4C2×C4⋊C411yescharacteristic
273M16—88no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—82no—
277C2×C8—82no—
278C2×C8—82no—
279C2×C8—82no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—84no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—88no—
293C22⋊C4—84no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23C2×#9511yescharacteristic
352C23#40512yes—
353C23#40811yescharacteristic
354C23#40811yescharacteristic
355C23C2×#9911yescharacteristic
356C23C2×#9911yescharacteristic
357C23C2×#10011yescharacteristic
358C23C2×#10111yescharacteristic
359C23#42412yes—
360D4—48no—
361D4—48no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—88no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—88no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#34212yes—
416C2×C4#34212yes—
417C2×C4#34314yes—
418C2×C4AΣL(1,𝔽16)12yes—
419C2×C4#35212yes—
420C8—84no—
421C8—84no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—22no—
436C22—42no—
437C22—42no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—44no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#147111yescharacteristic
462C22#147111yescharacteristic
463C22#147211yescharacteristic
464C22#147211yescharacteristic
465C22#172911yescharacteristic
466C22C2×#34211yescharacteristic
467C22C2×#34311yescharacteristic
468C22C2×AΣL(1,𝔽16)11yescharacteristic
469C22C2×#35211yescharacteristic
470C22#212211yescharacteristic
471C22#212311yescharacteristic
472C4—22no—
473C4—24no—
474C4—41no—
475C4—42no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—84no—
480C4—162no—
481C4#139014yes—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—82no—
489C2#791511yescharacteristic
490C2#1156111yescharacteristic
491C2#1156211yescharacteristic
492C2#1157911yescharacteristic
493C2#1158011yescharacteristic
494C2C2×#139011yescharacteristic
495C2#1257511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7187C2semidirect
13. #1465C4semidirect
15. #1460C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14652
C2×#14602
#138722
#71872