Finite Group Databasebeta

← #127019

Subgroups — #127019

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1465C211yesmaximal, characteristic
2C2×#1459C211yesmaximal, characteristic
3#13906C211yesmaximal, characteristic
4#7189C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#407—21no—
7C22×C22⋊C8—21no—
8C2×#390C412yes—
9C2×#384C411yescharacteristic
10C2×#380—21no—
11C2×#380—21no—
12C2×C4×C4⋊C4C411yescharacteristic
13#1465—22no—
14#1465C414yescomplemented
15#1459—22no—
16#1459C414yescomplemented
17C2×#324C2212yes—
18C2×#324C2212yes—
19C2×#323—21no—
20C2×#323—21no—
21#1439—21no—
22#1439—21no—
23C4⋊#95C2212yes—
24C23×D4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—41no—
28C23×C8—41no—
29C2×C4×D4—42no—
30C2×C4×D4—41no—
31C2×C4×D4—41no—
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⋊C4C2×C412yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—22no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4D411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C412yes—
47#407—24no—
48#407—42no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—42no—
53C2×C22⋊C8—41no—
54C2×C22⋊C8—41no—
55#390—44no—
56#384—41no—
57#384—41no—
58#380—44no—
59#380—44no—
60C4×C4⋊C4—42no—
61C2×#95—21no—
62C2×#95—22no—
63C2×#95—22no—
64#324—22no—
65#324—22no—
66#324C2×C414yes—
67#324C2×C414yes—
68#323—24no—
69#323—24no—
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—21no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C412yes—
97C23×C4C4⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—44no—
102C2×M16—82no—
103C2×M16—82no—
104C22×C8—42no—
105C22×C8—44no—
106C22×C8—42no—
107C22×C8—82no—
108C22×C8—82no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C22×C411yescharacteristic
132C2×C4⋊C4C22×C411yescharacteristic
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×Q811yescharacteristic
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—82no—
144C2×C22⋊C4—82no—
145C2×C22⋊C4C22⋊C412yes—
146C2×C22⋊C4C22⋊C412yes—
147C2×C22⋊C4C4⋊C412yes—
148C2×C22⋊C4C4⋊C412yes—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—24no—
153C2×C42—42no—
154C2×C42—42no—
155C2×C42C4212yes—
156C2×C42C4⋊C412yes—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—44no—
161C22⋊C8—42no—
162C22⋊C8—44no—
163#95—42no—
164#95—44no—
165#95—44no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24—41no—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24#9512yes—
185C24#9512yes—
186C24#10012yes—
187C24#10112yes—
188C24C2×C22⋊C411yescharacteristic
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
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—42no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4#9512yes—
254C22×C4#9512yes—
255C22×C4#9912yes—
256C22×C4#9911yescharacteristic
257C22×C4#9911yescharacteristic
258C22×C4C2×C4211yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C2×C4⋊C411yescharacteristic
264C22×C4#13112yes—
265C22×C4#13112yes—
266M16—88no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—82no—
270C2×C8—82no—
271C2×C8—88no—
272C2×C8—82no—
273C2×C8—82no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—48no—
279C4⋊C4—84no—
280C4⋊C4—82no—
281C4⋊C4—84no—
282C4⋊C4—82no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—88no—
288C22⋊C4—88no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—84no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23#34314yes—
344C23C2×#9511yescharacteristic
345C23#40811yescharacteristic
346C23#40811yescharacteristic
347C23C2×#9911yescharacteristic
348C23C2×#9911yescharacteristic
349C23C2×#10011yescharacteristic
350C23C2×#10111yescharacteristic
351D4—48no—
352D4—48no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#34212yes—
407C2×C4#34212yes—
408C2×C4#35312yes—
409C2×C4#35412yes—
410C2×C4#40512yes—
411C2×C4#42412yes—
412C8—84no—
413C8—84no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—24no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#147111yescharacteristic
449C22#147111yescharacteristic
450C22#147211yescharacteristic
451C22#147211yescharacteristic
452C22#172911yescharacteristic
453C22C2×#34211yescharacteristic
454C22C2×#34311yescharacteristic
455C22C2×#35311yescharacteristic
456C22C2×#35411yescharacteristic
457C22#213411yescharacteristic
458C22#213511yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—42no—
467C4—84no—
468C4—162no—
469C4#139314yes—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—44no—
475C2—82no—
476C2#791511yescharacteristic
477C2#1156311yescharacteristic
478C2#1156411yescharacteristic
479C2#1156911yescharacteristic
480C2#1157011yescharacteristic
481C2C2×#139311yescharacteristic
482C2#1258711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7189C2semidirect
14. #1465C4semidirect
16. #1459C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14652
C2×#14592
#139062
#71892