Finite Group Databasebeta

← #224294

Subgroups — #224294

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2#22629C212yesmaximal, complemented
3#9883C214yesmaximal, complemented
4#9878C212yesmaximal, complemented
5#8557C212yesmaximal
6#8557C211yesmaximal, characteristic
7#8152C212yesmaximal
8C4×#323C211yesmaximal, characteristic
9C22×C22⋊Q8C411yescharacteristic
10C22×C4×Q8C411yescharacteristic
11C23×C4⋊C4C411yescharacteristic
12#2360—24no—
13#2360—22no—
14#2360—22no—
15C2×#391C2211yescharacteristic
16C2×#389C411yescharacteristic
17C2×#389C2212yes—
18C2×#387C412yes—
19C2×#385C412yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2212yes—
23#1939—22no—
24#1917—24no—
25#1797C2212yes—
26#1797C2214yes—
27#1770—22no—
28#1749C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#323C2211yescharacteristic
31C2×#323C2212yes—
32C2×#323C2211yescharacteristic
33#1280C2214yes—
34#1280C2212yes—
35#1280C2214yes—
36#1280C2212yes—
37C23×Q8C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2×C412yes—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—22no—
46C2×C4×Q8—21no—
47C2×C4×Q8—44no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—44no—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42C2×C412yes—
63C22×C42C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2312yes—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C22⋊C8C2312yes—
69C2×C22⋊C8C2314yes—
70C2×C22⋊C8C2312yes—
71C2×#97—24no—
72C2×C4×C8—22no—
73#391—22no—
74#391—22no—
75#389—22no—
76#389—22no—
77#389—24no—
78#389—24no—
79#387—24no—
80#387—24no—
81#385—24no—
82#385—24no—
83C4×C4⋊C4D414yes—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—24no—
90C2×#95C2×C411yescharacteristic
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2×C412yes—
94C2×#95C2311yescharacteristic
95C2×#95C2312yes—
96#336—22no—
97#336—24no—
98#336—22no—
99#336—24no—
100#323—22no—
101#323—24no—
102#323Q814yescomplemented
103C25C22×C411yescharacteristic
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8M1614yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—44no—
120C23×C4—82no—
121C23×C4M1614yes—
122C23×C4C22×C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—24no—
135C22⋊Q8—216no—
136C22⋊Q8—216no—
137C4×Q8—44no—
138C4×Q8—48no—
139C4×Q8—44no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4C2×Q812yes—
174C2×C22⋊C4#4112yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—44no—
182C2×C42—82no—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D414yes—
186C2×C42C2×D414yes—
187C4⋊C8—44no—
188C4⋊C8—84no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—24no—
192C22⋊C8—22no—
193C22⋊C8—24no—
194C22⋊C8C2×Q814yes—
195#97—84no—
196C4×C8—44no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—22no—
201#95—24no—
202#95C22⋊C414yes—
203#95C22⋊C414yes—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24C4×Q812yes—
213C24C2×M1612yes—
214C24#13112yes—
215C24C23×C411yescharacteristic
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—48no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—48no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—48no—
257C22×C4—48no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—88no—
277C22×C4—88no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C414yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C4×Q812yes—
285C22×C4C2×M1612yes—
286C22×C4C2×M1612yes—
287C22×C4C2×M1614yes—
288C22×C4C2×M1612yes—
289C22×C4C22×D412yes—
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C42−1+412yes—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—44no—
302C4⋊C4—216no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—84no—
314C4⋊C4#10014yes—
315C22⋊C4—216no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—44no—
319C42—24no—
320C42—44no—
321C42—44no—
322C42—42no—
323C42—42no—
324C42—44no—
325C42—44no—
326C42—42no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23#44614yes—
354C23C22×C22⋊C411yescharacteristic
355C23C2×C4×Q811yescharacteristic
356C23#51911yescharacteristic
357C23C22×M1611yescharacteristic
358C23C22×M1611yescharacteristic
359C23C2×#13111yescharacteristic
360C23#56811yescharacteristic
361Q8—48no—
362Q8—48no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—88no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—88no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4#40712yes—
419C2×C4#40712yes—
420C2×C4#41012yes—
421C2×C4C2×#10014yes—
422C2×C4C2×#10012yes—
423C2×C4#54812yes—
424C2×C4D4×Q812yes—
425C8—44no—
426C8—82no—
427C8—84no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—24no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22Q8×C22⋊C411yescharacteristic
452C22C2×#40711yescharacteristic
453C22#288011yescharacteristic
454C22C2×#41011yescharacteristic
455C22#288311yescharacteristic
456C22C22×#10011yescharacteristic
457C22#288911yescharacteristic
458C22C2×#44611yescharacteristic
459C22Q8×M1611yescharacteristic
460C22#298311yescharacteristic
461C22#298411yescharacteristic
462C4—24no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—44no—
471C4—42no—
472C4—81no—
473C4—82no—
474C4#146614yes—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—44no—
480C2C2×#146611yescharacteristic
481C2#1737011yescharacteristic
482C2#1741311yescharacteristic
483C2#1742111yescharacteristic
484C2#2251011yescharacteristic
485C2#2251311yescharacteristic
486C2Q8×#10011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22629C2semidirect
3. #9883C2semidirect
4. #9878C2semidirect
102. #323Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
#226292
#98832
#98782
#85572
#81522
C4×#3232