Finite Group Databasebeta

← #414790

Subgroups — #414790

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#14161C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11323C212yesmaximal
9#11124C212yesmaximal
10#11122C212yesmaximal
11#10835C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#501—24no—
16C2×#501—24no—
17C2×#501—24no—
18C2×#475—24no—
19C2×#475—24no—
20C2×#475—24no—
21C22×#107C2212yes—
22C22×#106C2212yes—
23C22×#106C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#97C2211yescharacteristic
28#2565—24no—
29#2565—24no—
30C2×#398—22no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#387—22no—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1989—24no—
39#1763—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1446—22no—
46C4⋊#95—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125D414yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8D414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#501—232no—
88#475—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106—24no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8D414yes—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103Q818yescomplemented
107C2×#97—22no—
108C2×#97—22no—
109C2×#97D414yes—
110#398—216no—
111#391—22no—
112#391—22no—
113#391—22no—
114#391—22no—
115#387—216no—
116#384—22no—
117#384—22no—
118#382—22no—
119#382—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95D412yes—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336—24no—
136C22×Q8—24no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—84no—
148C22×Q8C2×D412yes—
149C22×Q8C2×Q814yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C2×D414yes—
172C22×C8C2×D414yes—
173C22×C8C2×Q814yes—
174C22×C8#4114yes—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—44no—
178C4⋊Q8—44no—
179#125—28no—
180#125—28no—
181#125—44no—
182#125—44no—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—44no—
186C4×Q8—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×Q814yes—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—44no—
247C2×C42—42no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254#107—216no—
255#106—216no—
256#106—216no—
257C4⋊C8—28no—
258#103—216no—
259#103—216no—
260#97—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—42no—
266#95—84no—
267#95—84no—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—48no—
287C2×Q8—88no—
288C2×Q8—88no—
289C2×Q8—88no—
290C2×Q8C4⋊Q814yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
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—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4#12012yes—
333C22×C4#12012yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#12712yes—
337C22×C4#12712yes—
338C22×C4#13714yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q812yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C42−1+412yes—
348C22×C42−1+412yes—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8C22⋊Q818yes—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—48no—
374C4⋊C4—416no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—42no—
394C42—42no—
395C42—48no—
396C42—84no—
397C42#12014yes—
398C42#12014yes—
399C42#12114yes—
400C42#12114yes—
401C23C2×#12011yescharacteristic
402C23C2×#12111yescharacteristic
403C23C2×#12711yescharacteristic
404C23#54712yes—
405C23D4×Q812yes—
406C23C2×#13712yes—
407C23#57512yes—
408C23#57714yes—
409Q8—24no—
410Q8—24no—
411Q8—42no—
412Q8—44no—
413Q8—44no—
414Q8—42no—
415Q8—816no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—84no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39214yes—
463C2×C4#39214yes—
464C2×C4C2×#12011yescharacteristic
465C2×C4C2×#12011yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×C22⋊Q812yes—
469C2×C4C2×C4⋊Q811yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4#53312yes—
472C2×C4#53612yes—
473C2×C4#53611yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#54112yes—
476C2×C4#54712yes—
477C2×C4#54812yes—
478C2×C4#54812yes—
479C2×C4D4×Q812yes—
480C2×C4D4×Q812yes—
481C2×C4D4×Q812yes—
482C8—28no—
483C8—44no—
484C22C2×#39211yescharacteristic
485C22#242912yes—
486C22#243211yescharacteristic
487C22#243411yescharacteristic
488C22#254212yes—
489C22#300712yes—
490C22#301012yes—
491C22#303712yes—
492C22#304212yes—
493C22#315211yescharacteristic
494C22#315311yescharacteristic
495C22#315412yes—
496C22#334414yes—
497C22#334814yes—
498C22#338714yes—
499C22#338814yes—
500C4—24no—
501C4—24no—
502C4—44no—
503C4—44no—
504C4—82no—
505C4—82no—
506C4—84no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4#240912yes—
511C4#242912yes—
512C4#243211yescharacteristic
513C4#243411yescharacteristic
514C4#262111yescharacteristic
515C4#285711yescharacteristic
516C2#1517811yescharacteristic
517C2#2342411yescharacteristic
518C2#2342911yescharacteristic
519C2#2546414yes—
520C2#2571512yes—
521C2#2584812yes—
522C2#2854912yes—
523C2#2855012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
106. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4752
C2×#25652
C2×#19892
C2×#17632
#141612
#113252
#113232
#111242
#111222
#108352