Finite Group Databasebeta

← #382780

Subgroups — #382780

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic, complemented
3#14684C211yesmaximal, characteristic, complemented
4#11203C212yesmaximal, complemented
5#11197C212yesmaximal, complemented
6#11043C212yesmaximal
7#10966C212yesmaximal
8#10801C212yesmaximal, complemented
9#10761C212yesmaximal
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×#99C2214yes—
14C4×#123—22no—
15C4×#123—22no—
16C4×#123—22no—
17C4×#123—22no—
18C2×#388C412yescomplemented
19C2×#387C412yescomplemented
20C2×#386C2211yescharacteristic
21C2×#385C2211yescharacteristic
22C2×#381C2212yes—
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C4⋊C4C412yescomplemented
27C2×C4×C22⋊C4C412yescomplemented
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×#342C2214yes—
30C2×#342C2214yes—
31C2×#342C2214yes—
32C4⋊#95—22no—
33#1434—22no—
34#1434—22no—
35#1434—22no—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C22⋊#95C2212yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C2×#123—24no—
45C2×#123—22no—
46C2×#123—21no—
47C2×#123—21no—
48C2×#123—24no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—44no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2314yes—
60C22×C22⋊C4C2314yes—
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2311yescharacteristic
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42C2×C412yes—
69C22×C42D412yes—
70C22×C42D412yes—
71C22×C42D412yes—
72C22×C42D412yes—
73C2×#99—28no—
74C2×#99—44no—
75C2×#99—44no—
76#388—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#386—21no—
85#386—22no—
86#386—21no—
87#385—21no—
88#385—22no—
89#385—21no—
90#381—22no—
91#381—21no—
92#381—21no—
93#381—44no—
94#381—44no—
95#380—44no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118#342—28no—
119#342—44no—
120#342—44no—
121C25C22×C411yescharacteristic
122C25C2411yescharacteristic
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4#4114yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—44no—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161#123—28no—
162#123—28no—
163#123—216no—
164C4×D4—48no—
165C4×D4—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—44no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4#4114yes—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42C22⋊C412yes—
221C2×C42C22⋊C412yes—
222C2×C42C22⋊C412yes—
223C2×C42C22⋊C412yes—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22⋊C412yes—
227C2×C42C22⋊C412yes—
228#99—416no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—48no—
240#95—48no—
241#95—48no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—48no—
246C24—21no—
247C24—21no—
248C24—22no—
249C24—22no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24#11712yes—
255C24C23×C411yescharacteristic
256C24C22×D411yescharacteristic
257C24C22×D411yescharacteristic
258C24C2×#4112yes—
259C24C2×#4112yes—
260C24H(5,𝔽2)14yes—
261C24H(5,𝔽2)12yes—
262C242−1+412yes—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—44no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—48no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—88no—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4#11712yes—
353C22×C4#11714yes—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—88no—
360C22⋊C4—28no—
361C22⋊C4—28no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—44no—
389C42—44no—
390C42—44no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—82no—
413C23—84no—
414C23—82no—
415C23C22×C22⋊C411yescharacteristic
416C23C2×#11711yescharacteristic
417C23C2×#11711yescharacteristic
418C23#51812yes—
419C23#52012yes—
420C23#52512yes—
421C23#52512yes—
422C23#53412yes—
423C23#53412yes—
424C23#53511yescharacteristic
425C23#53611yescharacteristic
426C23#53712yes—
427C23#53814yes—
428C23#53914yes—
429C23#54014yes—
430C23#54214yes—
431D4—48no—
432D4—44no—
433D4—44no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—88no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—88no—
487C2×C4—88no—
488C2×C4—88no—
489C2×C4—88no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4#41014yes—
496C2×C4#41014yes—
497C22—22no—
498C22—24no—
499C22—24no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—24no—
504C22—24no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—44no—
517C22—84no—
518C22—84no—
519C22—82no—
520C22—81no—
521C22—81no—
522C22#228011yescharacteristic
523C22#230811yescharacteristic
524C22#231111yescharacteristic
525C22#231812yes—
526C22#231912yes—
527C22#263014yes—
528C22#263512yes—
529C22#263812yes—
530C22C2×#41012yes—
531C22C2×#41012yes—
532C4—24no—
533C4—24no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—44no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—81no—
542C4—81no—
543C4—164no—
544C4—164no—
545C2—22no—
546C2—24no—
547C2—22no—
548C2—22no—
549C2—22no—
550C2—82no—
551C2#1434411yescharacteristic
552C2#2072312yes—
553C2#2174711yescharacteristic
554C2#2174811yescharacteristic
555C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14685C2semidirect
3. #14684C2semidirect
4. #11203C2semidirect
5. #11197C2semidirect
8. #10801C2semidirect
18. C2×#388C4semidirect
19. C2×#387C4semidirect
26. C2×C4×C4⋊C4C4semidirect
27. C2×C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
#146852
#146842
#112032
#111972
#110432
#109662
#108012
#107612