Finite Group Databasebeta

← #539113

Subgroups — #539113

Order 512.

2394 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C212yesmaximal
2C2×#2351C211yesmaximal, characteristic
3#14426C214yesmaximal
4#14388C214yesmaximal
5#14089C214yesmaximal
6#14041C212yesmaximal
7#13982C212yesmaximal
8#13931C212yesmaximal
9#13926C211yesmaximal, characteristic
10#13911C211yesmaximal, characteristic
11#13900C214yesmaximal
12#13848C211yesmaximal, characteristic
13#13804C211yesmaximal, characteristic
14#13786C211yesmaximal, characteristic
15#13779C211yesmaximal, characteristic
16C22×C4×Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18#2356—216no—
19#2351C4116yescomplemented
20C2×#391C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#384C2211yescharacteristic
28C2×#383C2214yes—
29C2×#382C2212yes—
30C2×#382C2212yes—
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×#376C2214yes—
40C2×#376C2211yescharacteristic
41#1445C2214yes—
42#1445C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1444C2212yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2212yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2212yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2211yescharacteristic
70#1434C2214yes—
71#1434C2214yes—
72#1434C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2311yescharacteristic
78C2×C4×Q8—28no—
79C2×C4×Q8—28no—
80C2×C4×Q8—24no—
81C2×C4×Q8—24no—
82C2×C4×Q8C2×C418yes—
83C2×C4×Q8C2×C4116yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2314yes—
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2311yescharacteristic
108#391—28no—
109#391—24no—
110#391D418yes—
111#389C2×C4132yes—
112#389C2×C4116yes—
113#387—216no—
114#387—216no—
115#384—24no—
116#383—216no—
117#382—28no—
118#382C2×C4116yes—
119#382C2×C418yes—
120C4×C4⋊C4—28no—
121C4×C4⋊C4—28no—
122C4×C4⋊C4—28no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4C2×C4116yes—
126C4×C4⋊C4C2×C418yes—
127#376—24no—
128#376—216no—
129C2×#95C2311yescharacteristic
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8C22×C414yes—
168C22×Q8C2×D414yes—
169C22×Q8#4114yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C4×Q8—264no—
186C4×Q8—232no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C22×C4116yes—
221C2×C4⋊C4C22×C418yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C22×C418yes—
224C2×C4⋊C4C22×C4116yes—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4C2×D414yes—
227C2×C42—28no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—28no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—28no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—28no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42C22×C418yes—
250C2×C42C22×C4116yes—
251C2×C42C22×C418yes—
252C2×C42C22×C414yes—
253C2×C42C2×D414yes—
254C2×C42#4114yes—
255C2×C42#4114yes—
256#95—24no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—22no—
267#95—24no—
268#95—22no—
269#95—28no—
270#95—24no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95C22×C414yes—
285#95C22×C418yes—
286#95C22×C414yes—
287#95C22×C4116yes—
288#95C22×C418yes—
289#95C2×D4116yes—
290C24C2511yescharacteristic
291C2×Q8—28no—
292C2×Q8—28no—
293C2×Q8—28no—
294C2×Q8—28no—
295C2×Q8—28no—
296C2×Q8—28no—
297C2×Q8C4×D4116yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4C23×C414yes—
371C22×C4C23×C418yes—
372C22×C4C23×C414yes—
373C22×C4C23×C414yes—
374C22×C4C23×C412yes—
375C22×C4C23×C412yes—
376C22×C4C23×C414yes—
377C22×C4C23×C412yes—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×D414yes—
381C22×C4C22×D412yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C42−1+414yes—
391C22×C42−1+412yes—
392C22×C42−1+412yes—
393C4⋊C4—216no—
394C4⋊C4—28no—
395C4⋊C4—232no—
396C4⋊C4—216no—
397C4⋊C4—28no—
398C4⋊C4—48no—
399C4⋊C4—416no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C42—216no—
408C42—232no—
409C42—216no—
410C42—28no—
411C42—416no—
412C42—416no—
413C42—48no—
414C42—44no—
415C42—84no—
416C42#117116yes—
417C23C24×C411yescharacteristic
418C23C23×D411yescharacteristic
419C23C22×#4111yescharacteristic
420C23C22×#4111yescharacteristic
421C23C22×#4111yescharacteristic
422C23C2×2−1+411yescharacteristic
423C23C2×2−1+411yescharacteristic
424C23C2×2−1+412yes—
425C23#58512yes—
426C23#58512yes—
427C23#58512yes—
428Q8—232no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4C2×#11714yes—
515C2×C4C2×#11714yes—
516C2×C4C2×#11714yes—
517C2×C4C2×C4×D418yes—
518C2×C4C2×C4×D414yes—
519C2×C4#51914yes—
520C2×C4#54714yes—
521C2×C4#54714yes—
522C2×C4#55214yes—
523C2×C4#55214yes—
524C22C22×#11711yescharacteristic
525C22C22×C4×D411yescharacteristic
526C22C2×#51911yescharacteristic
527C22#342812yes—
528C22C4×2−1+412yes—
529C22C2×#54711yescharacteristic
530C22C2×#54711yescharacteristic
531C22C2×#55211yescharacteristic
532C22C2×#55211yescharacteristic
533C22#348712yes—
534C22#348912yes—
535C22#349512yes—
536C22#350911yescharacteristic
537C22#351011yescharacteristic
538C22#351011yescharacteristic
539C22#351011yescharacteristic
540C22#351212yes—
541C22#351812yes—
542C22#352412yes—
543C22#355512yes—
544C22#355712yes—
545C22#356312yes—
546C22#356512yes—
547C4—28no—
548C4—24no—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—44no—
555C4—48no—
556C4—44no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4—82no—
561C4#234418yes—
562C2C2×#234411yescharacteristic
563C2#3418911yescharacteristic
564C2#3425411yescharacteristic
565C2#3429312yes—
566C2#3449012yes—
567C2#4526412yes—
568C2#4526611yescharacteristic
569C2#4530412yes—
570C2#4530611yescharacteristic
571C2#4531811yescharacteristic
572C2#4542911yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2351C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23562
C2×#23512
#144262
#143882
#140892
#140412
#139822
#139312
#139262
#139112
#139002
#138482
#138042
#137862
#137792