Finite Group Databasebeta

← #414732

Subgroups — #414732

Order 512.

1784 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
1C22×#474C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×#2580C211yesmaximal, characteristic
4C2×#1993C211yesmaximal, characteristic, complemented
5C2×#1985C211yesmaximal, characteristic, complemented
6C2×#1762C211yesmaximal, characteristic
7#11328C214yesmaximal, complemented
8#11066C214yesmaximal
9#10850C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#474—24no—
14C2×#474—24no—
15C2×#474—24no—
16C22×#107C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#97C2211yescharacteristic
21#2580—24no—
22#2580—24no—
23#2567—24no—
24#2567—24no—
25C2×#399—21no—
26C2×#395—22no—
27C2×#393—21no—
28C2×#391—21no—
29C2×#390C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#385—22no—
33C2×#384—21no—
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1993—24no—
37#1985—24no—
38#1762—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D414yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4D414yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#474—232no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8D414yes—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102Q818yescomplemented
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98#399—28no—
99#395—216no—
100#393—28no—
101#391—28no—
102#390—24no—
103#390—24no—
104#389—28no—
105#389—28no—
106#389—24no—
107#389—24no—
108#385—216no—
109#384—28no—
110#382—22no—
111#382D414yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—24no—
118C2×#95—21no—
119#340—28no—
120#340—24no—
121#340—28no—
122#336—28no—
123C25—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—84no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C2×D412yes—
142C22×D4C2×Q814yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—84no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—24no—
162C22×C8—22no—
163C22×C8—24no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C22×C8C2×Q814yes—
168C4⋊Q8—28no—
169C4⋊Q8—28no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×Q8—44no—
175C4×Q8—44no—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—44no—
179C4×D4—44no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—216no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4—84no—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—84no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—44no—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238#107—216no—
239C4⋊C8—28no—
240#102—216no—
241#102—216no—
242#97—44no—
243#95—28no—
244#95—216no—
245#95—24no—
246#95—24no—
247#95—24no—
248C24—22no—
249C24—24no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—44no—
256C24C22×D411yescharacteristic
257C2×Q8—28no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—48no—
263C2×Q8—88no—
264C2×Q8—88no—
265C2×Q8—88no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4C4⋊Q814yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—48no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4#12012yes—
324C22×C4#12012yes—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#13714yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×Q812yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C42−1+412yes—
337C22×C42−1+412yes—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—28no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—44no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—816no—
368C4⋊C4—88no—
369C4⋊C4C22⋊Q818yes—
370C22⋊C4—416no—
371C22⋊C4—48no—
372C22⋊C4—88no—
373C42—48no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42#12014yes—
379C42#12114yes—
380C42#12114yes—
381C42#12714yes—
382C23—24no—
383C23—48no—
384C23—48no—
385C23—42no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—84no—
392C23—84no—
393C23—81no—
394C23—81no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23C2×#12011yescharacteristic
399C23C2×#12011yescharacteristic
400C23C2×#12111yescharacteristic
401C23#54812yes—
402C23D4×Q812yes—
403C23C2×#13712yes—
404C23#57512yes—
405C23#57614yes—
406Q8—416no—
407Q8—816no—
408D4—24no—
409D4—24no—
410D4—42no—
411D4—44no—
412D4—42no—
413D4—44no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—816no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4#39214yes—
454C2×C4#39214yes—
455C2×C4C2×#12011yescharacteristic
456C2×C4C2×#12111yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×C22⋊Q812yes—
459C2×C4C2×#12711yescharacteristic
460C2×C4C2×C4⋊Q811yescharacteristic
461C2×C4#53211yescharacteristic
462C2×C4#53312yes—
463C2×C4#53611yescharacteristic
464C2×C4#53611yescharacteristic
465C2×C4#53612yes—
466C2×C4#54112yes—
467C2×C4#54712yes—
468C2×C4#54712yes—
469C2×C4#54812yes—
470C2×C4D4×Q812yes—
471C2×C4D4×Q812yes—
472C2×C4D4×Q812yes—
473C8—44no—
474C8—44no—
475C22—44no—
476C22—48no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—82no—
481C22—82no—
482C22—88no—
483C22C2×#39211yescharacteristic
484C22#240912yes—
485C22#242912yes—
486C22#243211yescharacteristic
487C22#243411yescharacteristic
488C22#300411yescharacteristic
489C22#300511yescharacteristic
490C22#300712yes—
491C22#300812yes—
492C22#300812yes—
493C22#305612yes—
494C22#305812yes—
495C22#337014yes—
496C22#337514yes—
497C22#338514yes—
498C22#338914yes—
499C4—24no—
500C4—24no—
501C4—48no—
502C4—44no—
503C4—84no—
504C4—82no—
505C4—82no—
506C4—84no—
507C4#242912yes—
508C4#243211yescharacteristic
509C4#243411yescharacteristic
510C4#254212yes—
511C4#262111yescharacteristic
512C4#285711yescharacteristic
513C2—84no—
514C2#1517811yescharacteristic
515C2#2338711yescharacteristic
516C2#2339211yescharacteristic
517C2#2433912yes—
518C2#2434112yes—
519C2#2548114yes—
520C2#2566012yes—
521C2#2585012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
4. C2×#1993C2semidirect
5. C2×#1985C2semidirect
7. #11328C2semidirect
9. #10850C2semidirect
94. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#25672
C2×#25802
C2×#19932
C2×#19852
C2×#17622
#113282
#110662
#108502