Finite Group Databasebeta

← #204235

Subgroups — #204235

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3#20534C211yesmaximal, characteristic, complemented
4C2×#1467C212yesmaximal, complemented
5C2×#1464C212yesmaximal
6#9654C212yesmaximal, complemented
7#9652C212yesmaximal
8#9651C212yesmaximal
9#9650C212yesmaximal, complemented
10C22×C22⋊Q8C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C2×#432C2212yes—
14C2×#432C2212yes—
15#2323—24no—
16#2323—24no—
17C2×#391C411yescharacteristic
18C2×#389C411yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#388C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#387C411yescharacteristic
26C2×#387C411yescharacteristic
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1940—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1771—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1467—28no—
36#1464—28no—
37C2×#324C2214yes—
38C2×#324C2214yes—
39C2×#324C2214yes—
40C2×#323C2214yes—
41C23×Q8C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—24no—
49C2×C4×Q8—24no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117—24no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#432—24no—
78#432—24no—
79#432—24no—
80#432D418yes—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8C2314yes—
86C2×C22⋊C8C2314yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97—22no—
90C2×C4×C8—22no—
91#391—24no—
92#389—24no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—24no—
102#388—24no—
103#387—24no—
104#387—24no—
105#380—24no—
106C4×C4⋊C4D414yes—
107C4×C4⋊C4D414yes—
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2×C411yescharacteristic
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#324—28no—
123#324—28no—
124#324—28no—
125#323—28no—
126C25C22×C411yescharacteristic
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—42no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—42no—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—24no—
152C22×C8—24no—
153C22⋊Q8—232no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×Q8—44no—
157C4×Q8—44no—
158#117—44no—
159#117—44no—
160#117—44no—
161#117—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—81no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4C22⋊C414yes—
191C2×C4⋊C4C22⋊C414yes—
192C2×C4⋊C4M1614yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—81no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4M1614yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—81no—
216C2×C42—81no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D414yes—
223C2×C42C2×D412yes—
224C2×C42#4114yes—
225C4⋊C8—28no—
226C4⋊C8—44no—
227C4⋊C8—82no—
228C4⋊C8—82no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8C2×D418yes—
233#97—28no—
234#97—44no—
235#97—82no—
236C4×C8—82no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95C22⋊C414yes—
246#95C22⋊C414yes—
247C24—21no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—41no—
255C24C2×M1612yes—
256C24C23×C411yescharacteristic
257C2×Q8—28no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—44no—
261C2×Q8—44no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—28no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—48no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—48no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C4×D414yes—
321C22×C4C4×D414yes—
322C22×C4C2×M1612yes—
323C22×C4C2×M1612yes—
324C22×C4C2×M1612yes—
325C22×C4C2×M1612yes—
326C22×C4C2×M1612yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C2×#4112yes—
331C2×C8—28no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C22⋊C4—28no—
355C22⋊C4—28no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—44no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—44no—
367C42—44no—
368C42—44no—
369C42#12114yes—
370C42#12114yes—
371C42#12414yes—
372C42#12714yes—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—24no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—41no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23C22×C22⋊C411yescharacteristic
395C23C2×C4×D412yes—
396C23C22×M1611yescharacteristic
397C23C22×M1611yescharacteristic
398C23#56812yes—
399Q8—416no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—416no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39014yes—
463C2×C4#39014yes—
464C2×C4#40714yes—
465C2×C4C2×#12112yes—
466C2×C4C2×#12411yescharacteristic
467C2×C4C2×#12711yescharacteristic
468C8—48no—
469C8—84no—
470C22—22no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—84no—
482C22—84no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22C2×#39011yescharacteristic
488C22C2×#40711yescharacteristic
489C22#287911yescharacteristic
490C22#288311yescharacteristic
491C22#288411yescharacteristic
492C22#288911yescharacteristic
493C22#288911yescharacteristic
494C22D4×M1612yes—
495C22D4×M1612yes—
496C4—24no—
497C4—24no—
498C4—22no—
499C4—21no—
500C4—21no—
501C4—24no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C2—22no—
513C2—22no—
514C2—82no—
515C2#1736811yescharacteristic
516C2#1737111yescharacteristic
517C2#1740711yescharacteristic
518C2#1741011yescharacteristic
519C2#2119011yescharacteristic
520C2#2121411yescharacteristic
521C2#2122011yescharacteristic
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. #21955C2semidirect
3. #20534C2semidirect
4. C2×#1467C2semidirect
6. #9654C2semidirect
9. #9650C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
#219552
#205342
C2×#14672
C2×#14642
#96542
#96522
#96512
#96502