Finite Group Databasebeta

← #443530

Subgroups — #443530

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic
2C2×#2551C211yesmaximal, characteristic
3C2×#2058C212yesmaximal
4C2×#2058C211yesmaximal, characteristic
5#15128C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11327C212yesmaximal
8#11323C212yesmaximal
9#10908C212yesmaximal
10#10880C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#498—28no—
17C2×#498—24no—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×#106C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2551—24no—
23#2551—24no—
24C2×#395—21no—
25C2×#395—21no—
26C2×#395—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2058—28no—
35#2058—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8—21no—
51C23×Q8—21no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8—42no—
65C2×C4⋊Q8—42no—
66C2×C4⋊Q8—44no—
67C2×C4⋊Q8—44no—
68C2×C4⋊Q8D414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#498—232no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106—42no—
98C2×#106D418yes—
99C2×C4×C8—24no—
100C2×C4×C8—22no—
101#395—28no—
102#395—28no—
103#395—44no—
104#389—28no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—24no—
109#389—42no—
110#389D414yes—
111#384—22no—
112#384—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340Q818yescomplemented
132#336—24no—
133#336—22no—
134#336Q818yescomplemented
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×Q8—44no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8—82no—
148C22×Q8—82no—
149C22×Q8—82no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
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×C8—22no—
172C22×C8—22no—
173C22×C8C2×D414yes—
174C22×C8C2×Q814yes—
175C22×C8C2×Q814yes—
176C22×C8C2×Q814yes—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179C4⋊Q8—44no—
180C4⋊Q8—48no—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—48no—
184C4⋊Q8—432no—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
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—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
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—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4SD1614yes—
232C2×C4⋊C4SD1614yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C4⋊C4C2×Q814yes—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×Q814yes—
256#106—216no—
257#106—216no—
258#106—48no—
259C4×C8—28no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265#95—44no—
266#95—42no—
267#95—44no—
268#95—42no—
269#95—82no—
270C24C22×D411yescharacteristic
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—84no—
277C2×Q8—88no—
278C2×Q8—88no—
279C2×Q8—88no—
280C2×Q8—88no—
281C2×Q8—88no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—88no—
285C2×Q8—84no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4#12012yes—
341C22×C4#12012yes—
342C22×C4C4⋊Q812yes—
343C22×C4C4⋊Q812yes—
344C22×C4C2×SD1612yes—
345C22×C4C2×SD1612yes—
346C22×C4C2×SD1612yes—
347C22×C4C2×SD1612yes—
348C22×C4C2×SD1612yes—
349C22×C4C2×SD1612yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4#13712yes—
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q812yes—
356C22×C4C22×Q812yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C42−1+412yes—
359C22×C42−1+412yes—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8C4⋊Q818yescomplemented
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—48no—
400C4⋊C4—88no—
401C4⋊C4—84no—
402C4⋊C4#12014yes—
403C4⋊C4C4⋊Q814yes—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—48no—
410C42—41no—
411C42—41no—
412C42—42no—
413C23C2×#12011yescharacteristic
414C23C2×C4⋊Q811yescharacteristic
415C23#53611yescharacteristic
416C23D4212yes—
417C23D4×Q812yes—
418C23C22×SD1611yescharacteristic
419C23C22×SD1611yescharacteristic
420C23C2×AGL(1,ℤ/8)11yescharacteristic
421C23C2×#13711yescharacteristic
422C23#57712yes—
423C23#57712yes—
424Q8—816no—
425Q8—816no—
426Q8—816no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4#44814yes—
491C2×C4#45014yes—
492C2×C4#49814yescomplemented
493C2×C4C2×#12011yescharacteristic
494C2×C4C2×C4⋊Q812yes—
495C2×C4C2×C4⋊Q811yescharacteristic
496C2×C4#53411yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53612yes—
499C2×C4#54412yes—
500C2×C4#54412yes—
501C2×C4D4212yes—
502C2×C4D4×Q812yes—
503C2×C4D4×Q812yes—
504C2×C4D4×Q812yes—
505C2×C4D4×Q812yes—
506C2×C4D4×Q812yes—
507C2×C4Q8212yes—
508C2×C4#56312yes—
509C8—28no—
510C8—44no—
511C22#240311yescharacteristic
512C22#243411yescharacteristic
513C22#243411yescharacteristic
514C22#255112yes—
515C22#267712yes—
516C22C2×#45011yescharacteristic
517C22C2×#44811yescharacteristic
518C22#300911yescharacteristic
519C22#301011yescharacteristic
520C22C2×#49811yescharacteristic
521C22#316312yes—
522C22#316511yescharacteristic
523C22#322711yescharacteristic
524C22#322911yescharacteristic
525C22#324412yes—
526C22#326512yes—
527C22#326612yes—
528C22D4×SD1612yes—
529C22D4×SD1612yes—
530C22Q8×SD1612yes—
531C22Q8×SD1612yes—
532C22#338412yes—
533C22#338712yes—
534C4—24no—
535C4—24no—
536C4—24no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—42no—
541C4—44no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—84no—
546C4#243412yes—
547C4#255112yes—
548C4#267712yes—
549C4#273412yes—
550C2#1576211yescharacteristic
551C2#2384911yescharacteristic
552C2#2385511yescharacteristic
553C2#2581111yescharacteristic
554C2#2582011yescharacteristic
555C2#2583411yescharacteristic
556C2#2583711yescharacteristic
557C2#2880912yes—
558C2#2881912yes—
559C2#3094912yes—
560C2#3095412yes—
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
131. #340Q8semidirect
134. #336Q8semidirect
373. C2×C8C4⋊Q8semidirect
492. C2×C4#498semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#25512
C2×#20582
#151282
#113842
#113272
#113232
#109082
#108802
#108742