Finite Group Databasebeta

← #480267

Subgroups — #480267

Order 512.

1616 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34044C211yesmaximal, characteristic
2#17357C212yesmaximal, complemented
3#11508C214yesmaximal, complemented
4#11507C214yesmaximal, complemented
5#11434C212yesmaximal, complemented
6#11433C212yesmaximal, complemented
7C23×#41C411yescharacteristic
8C22×#117C411yescharacteristic
9C22×#117C2211yescharacteristic
10#2920—22no—
11#2311—22no—
12#2311C412yes—
13#2311C2212yes—
14#2310C411yescharacteristic
15#2310C2211yescharacteristic
16#2309C411yescharacteristic
17#2309C2211yescharacteristic
18#2308—22no—
19#2297—22no—
20#2297C412yes—
21#2297C2212yes—
22#2280—22no—
23#1464—22no—
24#1464C2214yescomplemented
25#1463C2214yescomplemented
26#1462—22no—
27#1458—22no—
28#1458C2214yes—
29#1319C2218yescomplemented
30#1318C2218yescomplemented
31C22×#41—24no—
32C22×#41—24no—
33C22×#41—21no—
34C22×#41—21no—
35C22×#41—42no—
36C22×#41—42no—
37C22×#41—44no—
38C22×#41C2×C412yes—
39C22×#41C2×C412yes—
40C23×Q8C2×C411yescharacteristic
41C23×D4—21no—
42C23×D4C2×C411yescharacteristic
43C24×C4—21no—
44C24×C4C2×C411yescharacteristic
45C2×#117—21no—
46C2×#117—21no—
47C2×#117—42no—
48C2×#117—42no—
49C2×#117—44no—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C412yes—
55C2×#117D412yes—
56C2×#117D412yes—
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4D412yes—
64C22×C42C2×C412yes—
65C22×C42D412yes—
66#432—24no—
67#432—42no—
68#431—24no—
69#431—42no—
70#423—42no—
71#407—24no—
72C4×M16—42no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2312yes—
79#390—22no—
80#390—22no—
81#390C2×C412yes—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2312yes—
85#386—24no—
86#386—24no—
87#386—24no—
88#386—24no—
89#381—22no—
90#381—22no—
91#380—24no—
92#380—24no—
93#379—22no—
94#379—22no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—24no—
97#324—24no—
98#324C2318yes—
99#323—24no—
100#323—24no—
101C25—21no—
102C25C22⋊C411yescharacteristic
103C2×#41—28no—
104C2×#41—28no—
105C2×#41—24no—
106C2×#41—24no—
107C2×#41—24no—
108C2×#41—24no—
109C2×#41—416no—
110C2×#41—416no—
111C2×#41—416no—
112C2×#41—416no—
113C2×#41—44no—
114C2×#41—44no—
115C2×#41—44no—
116C2×#41—44no—
117C2×#41—42no—
118C2×#41—42no—
119C2×#41—44no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—44no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8M1614yescomplemented
128C22×Q8C22×C412yes—
129C22×D4—28no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4M1614yescomplemented
139C22×D4C22×C412yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—28no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C2×M16—44no—
162#117—42no—
163#117—41no—
164#117—41no—
165#117—42no—
166#117—42no—
167#117—42no—
168#117—42no—
169#117—42no—
170#117—41no—
171#117—41no—
172#117—44no—
173#117—44no—
174#117—42no—
175#117—42no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C2×D412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42C22⋊C412yes—
210C2×C42C22⋊C412yes—
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C4⋊C8—44no—
216C4⋊C8—44no—
217C22⋊C8—28no—
218C22⋊C8—44no—
219#97—44no—
220C4×C8—44no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—24no—
230C24—42no—
231C24—42no—
232C24—81no—
233C24—81no—
234C24—82no—
235C24—82no—
236C24C2×C22⋊C411yescharacteristic
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C411yescharacteristic
239#41—48no—
240#41—48no—
241#41—416no—
242#41—48no—
243#41—48no—
244#41—416no—
245#41—416no—
246#41—416no—
247#41—432no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8C2×M1614yes—
257C2×D4—28no—
258C2×D4—28no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4—432no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4C2×M1614yes—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—416no—
307C22×C4—416no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4#9912yes—
321C22×C4#9912yes—
322C22×C4#10014yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×M1612yes—
330C22×C4C2×M1612yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D412yes—
333M16—88no—
334C2×C8—48no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—84no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C22⋊C4—84no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—42no—
364C42—42no—
365C42—44no—
366C23—28no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—24no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—44no—
381C23—44no—
382C23—48no—
383C23—42no—
384C23—42no—
385C23—82no—
386C23—82no—
387C23—84no—
388C23—84no—
389C23—88no—
390C23—88no—
391C23—84no—
392C23—84no—
393C23#40712yes—
394C23C2×#9912yes—
395C23C2×#10012yes—
396C23C22×C22⋊C411yescharacteristic
397Q8—44no—
398Q8—44no—
399Q8—48no—
400D4—44no—
401D4—44no—
402D4—416no—
403D4—416no—
404D4—48no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—416no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—88no—
450C2×C4#40712yes—
451C2×C4C2×#9911yescharacteristic
452C2×C4C2×#9911yescharacteristic
453C2×C4C2×#9912yes—
454C2×C4C2×#10012yes—
455C2×C4C2×#10012yes—
456C2×C4C22×M1611yescharacteristic
457C2×C4C22×M1611yescharacteristic
458C8—88no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—48no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—44no—
478C22—84no—
479C22—88no—
480C22—816no—
481C22—82no—
482C22—82no—
483C22C2×#40711yescharacteristic
484C22C22×#9911yescharacteristic
485C22C22×#10011yescharacteristic
486C4—21no—
487C4—21no—
488C4—22no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—44no—
496C4—82no—
497C4—82no—
498C4#146214yes—
499C2—21no—
500C2—21no—
501C2—42no—
502C2—42no—
503C2—42no—
504C2—84no—
505C2C2×#146211yescharacteristic
506C2#3335811yescharacteristic
507C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17357C2semidirect
3. #11508C2semidirect
4. #11507C2semidirect
5. #11434C2semidirect
6. #11433C2semidirect
24. #1464C22semidirect
25. #1463C22semidirect
29. #1319C22semidirect
30. #1318C22semidirect
127. C22×Q8M16semidirect
138. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
#340442
#173572
#115082
#115072
#114342
#114332