Finite Group Databasebeta

← #196971

Subgroups — #196971

Order 512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35250C211yesmaximal, characteristic
2C2×#2339C211yesmaximal, characteristic, complemented
3#20581C211yesmaximal, characteristic, complemented
4C2×#1519C211yesmaximal, characteristic, complemented
5C2×#1507C211yesmaximal, characteristic, complemented
6#9158C214yesmaximal, complemented
7#9147C214yesmaximal, complemented
8#8016C211yesmaximal, characteristic, complemented
9#8012C211yesmaximal, characteristic, complemented
10C23×#41C2211yescharacteristic
11C2×#535C412yescomplemented
12C22×#121C2211yescharacteristic
13C22×C4×D4—21no—
14C2×#408—22no—
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2847C2214yes—
18#2615C414yescomplemented
19#2436—21no—
20#2433—21no—
21#2431—22no—
22#2404—22no—
23#2403—22no—
24#2339—24no—
25#2339—24no—
26#2311C412yescomplemented
27C2×#391—21no—
28C2×#386—22no—
29#2280C2211yescharacteristic
30C2×#381C2212yescomplemented
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2212yescomplemented
34#1780—22no—
35#1779C2214yescomplemented
36#1519—24no—
37#1507—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1294C2212yescomplemented
42#1294C2212yescomplemented
43#1294C2214yescomplemented
44C22×#41—22no—
45C22×#41—22no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—48no—
49C22×#41C2×C412yes—
50C22×#41C2×C412yes—
51C22×#41D412yes—
52C22×#41D412yes—
53C23×Q8D411yescharacteristic
54C23×D4—21no—
55C23×D4C2×C411yescharacteristic, complemented
56C23×D4D411yescharacteristic
57C23×D4D412yes—
58C24×C4—22no—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C22×M16—21no—
62C23×C8—21no—
63#535—48no—
64C2×#127—22no—
65C2×#124—22no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2×C412yescomplemented
73C2×#121C2×C418yescomplemented
74C2×#121C2312yes—
75C2×#120—24no—
76C2×#120—24no—
77C2×C4×D4—28no—
78C2×C4×D4—48no—
79C2×#117C2×C412yescomplemented
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic, complemented
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C42—21no—
87#408—48no—
88#407—22no—
89#407—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—28no—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97#393—24no—
98#392—28no—
99#392—24no—
100#391—28no—
101#391—22no—
102#390—22no—
103#386—216no—
104#386—24no—
105#386—24no—
106#386—28no—
107#381—21no—
108#381—24no—
109#381—24no—
110#380—22no—
111#380C2×C418yescomplemented
112#380C2×C412yescomplemented
113#380D414yescomplemented
114#380C2312yes—
115#379—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4D414yescomplemented
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95C2311yescharacteristic
125C2×#95C2312yes—
126C2×#95C2311yescharacteristic
127#327—24no—
128#327—24no—
129#327—28no—
130C25—21no—
131C25C2×D412yes—
132C25C2×D411yescharacteristic
133C2×#41—28no—
134C2×#41—28no—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—24no—
138C2×#41—24no—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—44no—
146C2×#41—44no—
147C2×#41—416no—
148C2×#41—416no—
149C2×#41—416no—
150C2×#41—416no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—44no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22⋊C412yes—
156C22×D4—28no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—22no—
160C22×D4—28no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—44no—
168C22×D4—48no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—82no—
172C22×D4—82no—
173C22×D4C22⋊C412yes—
174C22×D4C22⋊C414yes—
175C22×D4C22⋊C414yes—
176C22×D4C22⋊C412yes—
177C22×D4C22×C412yes—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—28no—
186C23×C4—21no—
187C23×C4—44no—
188C23×C4—48no—
189C23×C4—48no—
190C23×C4—48no—
191C23×C4—84no—
192C23×C4C22×C414yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2×D414yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D412yes—
201C23×C4C2×D412yes—
202C23×C4#4112yes—
203C23×C4#4112yes—
204C23×C4C2411yescharacteristic
205C2×M16—44no—
206C2×M16—42no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—44no—
210C22×C8—42no—
211#127—48no—
212#124—48no—
213#121—24no—
214#121—24no—
215#121—22no—
216#121—22no—
217#121—22no—
218#121—22no—
219#121—416no—
220#121—42no—
221#121—42no—
222#121—42no—
223#121—42no—
224#120—416no—
225#120—84no—
226#120—84no—
227C4×D4—432no—
228#117—44no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C414yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—82no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—44no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—416no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261#95—28no—
262#95—28no—
263#95—22no—
264#95—28no—
265#95—21no—
266#95—22no—
267#95—21no—
268#95—22no—
269#95C2×D412yes—
270#95C2×D414yes—
271#95C2×D412yes—
272C24—22no—
273C24—22no—
274C24—28no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—44no—
281C24—42no—
282C24—42no—
283C24—44no—
284C24—42no—
285C24—82no—
286C24—82no—
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C414yes—
290C24#12012yes—
291C24#12312yes—
292C24C22×D411yescharacteristic
293#41—416no—
294#41—416no—
295#41—416no—
296#41—416no—
297#41—416no—
298#41—416no—
299#41—416no—
300#41—416no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—44no—
306C2×Q8—48no—
307C2×Q8—48no—
308C2×D4—216no—
309C2×D4—216no—
310C2×D4—28no—
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—416no—
324C2×D4—416no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—416no—
331C2×D4—42no—
332C2×D4—44no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—84no—
340C2×D4—84no—
341C2×D4#9918yes—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—416no—
370C22×C4—42no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—416no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—82no—
385C22×C4—88no—
386C22×C4—832no—
387C22×C4C2×C22⋊C414yes—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4C4×D412yes—
393C22×C4C4×D412yes—
394C22×C4#12012yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4#12112yes—
398C22×C4#12112yes—
399C22×C4#12312yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406M16—84no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—84no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—84no—
428C22⋊C4—88no—
429C42—48no—
430C42—44no—
431C23—22no—
432C23—22no—
433C23—28no—
434C23—28no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—28no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—28no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—416no—
453C23—416no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—168no—
469C23#38612yes—
470C23#38612yes—
471C23C2×#9918yes—
472C23C22×C22⋊C411yescharacteristic
473C23C2×#12011yescharacteristic
474C23C2×#12311yescharacteristic
475C23#57612yes—
476C23#57712yes—
477Q8—48no—
478Q8—48no—
479D4—216no—
480D4—216no—
481D4—416no—
482D4—416no—
483D4—82no—
484D4—82no—
485D4—88no—
486D4—84no—
487D4—82no—
488D4—82no—
489D4—84no—
490D4—1616no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—48no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—832no—
533C2×C4—84no—
534C2×C4—88no—
535C2×C4—84no—
536C2×C4#38612yes—
537C2×C4#38612yes—
538C2×C4C2×#9914yes—
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C8—82no—
544C8—82no—
545C22—24no—
546C22—24no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—22no—
553C22—216no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—28no—
559C22—48no—
560C22—48no—
561C22—416no—
562C22—44no—
563C22—48no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—84no—
570C22—82no—
571C22—88no—
572C22—82no—
573C22—82no—
574C22—168no—
575C22C2×#38611yescharacteristic
576C22C22×#9912yes—
577C22#289712yes—
578C22#290112yes—
579C22#300811yescharacteristic
580C22#301011yescharacteristic
581C22#309612yes—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—22no—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—84no—
591C4—82no—
592C4—84no—
593C4#189814yes—
594C2—21no—
595C2—21no—
596C2—22no—
597C2—48no—
598C2—42no—
599C2—42no—
600C2—162no—
601C2#1760314yes—
602C2#2091411yescharacteristic
603C2#2098011yescharacteristic
604C2C2×#189811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35250C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2339C2semidirect
3. #20581C2semidirect
4. C2×#1519C2semidirect
5. C2×#1507C2semidirect
6. #9158C2semidirect
7. #9147C2semidirect
8. #8016C2semidirect
9. #8012C2semidirect
11. C2×#535C4semidirect
18. #2615C4semidirect
26. #2311C4semidirect
30. C2×#381C22semidirect
33. C22×#95C22semidirect
35. #1779C22semidirect
41. #1294C22semidirect
42. #1294C22semidirect
43. #1294C22semidirect
55. C23×D4C2×C4semidirect
72. C2×#121C2×C4semidirect
73. C2×#121C2×C4semidirect
79. C2×#117C2×C4semidirect
81. C22×C4⋊C4C2×C4semidirect
111. #380C2×C4semidirect
112. #380C2×C4semidirect
113. #380D4semidirect
119. C4×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#352502
C2×#23392
#205812
C2×#15192
C2×#15072
#91582
#91472
#80162
#80122