Finite Group Databasebeta

← #195338

Subgroups — #195338

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#14159C211yesmaximal, characteristic
6#9046C214yesmaximal, complemented
7#9040C214yesmaximal, complemented
8C2×#1294C211yesmaximal, characteristic
9#7907C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#3189—24no—
13#3189C418yescomplemented
14C2×#484—21no—
15C2×#484C2212yes—
16C2×#467—22no—
17C2×#451—21no—
18C2×#451C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2368—22no—
22#2360—22no—
23#2345—22no—
24#2345—21no—
25Q8×C22⋊C4—21no—
26#2329C2214yes—
27#2315—21no—
28C4×C22⋊Q8—22no—
29C2×#380C2211yescharacteristic
30#2273—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1892—22no—
34#1872—22no—
35#1824C2212yes—
36#1787C2212yes—
37C2×#328—24no—
38C2×#328C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1436—22no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44#1294C2214yes—
45#1280C2214yes—
46C23×Q8D411yescharacteristic
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×Q16—22no—
51C23×C8—21no—
52C23×C8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×C22⋊Q8C2×C414yes—
66C2×C22⋊Q8D414yes—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8C2312yes—
69C2×C4×Q8—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42D412yes—
81#484—24no—
82#484—28no—
83#484C2×C418yescomplemented
84#467—216no—
85#451—28no—
86#451—24no—
87#451C2×C418yescomplemented
88C2×#107—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8C2×C414yes—
98C2×C22⋊C8C2312yes—
99C2×C22⋊C8C2312yes—
100#391—22no—
101#391—24no—
102#389—22no—
103#389—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#385—24no—
108#385—24no—
109#380—22no—
110#380—22no—
111#380—21no—
112#380—21no—
113#380D412yes—
114#380C2312yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—24no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2312yes—
134C2×#95C2312yes—
135#340—24no—
136#328—28no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×Q8C22⋊C414yes—
148C22×Q8C22×C412yes—
149C22×Q8C2×D412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C22×C412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C2×Q16—28no—
175C2×Q16—84no—
176C22×C8—22no—
177C22×C8—24no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—82no—
181C22×C8—82no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8C22⋊C418yes—
198C22⋊Q8C22×C414yes—
199C22⋊Q8C2×D414yes—
200C4×Q8—84no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22⋊C414yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4Q1614yescomplemented
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—82no—
249#107—28no—
250#103—28no—
251#103—28no—
252#103—84no—
253C22⋊C8—24no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—42no—
258C22⋊C8C22×C414yes—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—82no—
273#95—82no—
274C24—42no—
275C24—42no—
276C24C4×D412yes—
277C24#12011yescharacteristic
278C24#12011yescharacteristic
279C24#12112yes—
280C24C2×Q1612yes—
281C24#13512yes—
282C24#13712yes—
283C24#13712yes—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—84no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×Q8—84no—
301C2×Q8C2×C22⋊C414yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
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—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4#12011yescharacteristic
359C22×C4#12011yescharacteristic
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#12312yes—
364C22×C4#12312yes—
365C22×C4C2×Q1614yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372Q16—88no—
373C2×C8—24no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—42no—
377C2×C8—84no—
378C2×C8—84no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—28no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4C2×C22⋊C414yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C42—44no—
408C42—82no—
409C42—84no—
410C42—84no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23#38612yes—
426C23#38612yes—
427C23C4×Q1614yes—
428C23#44114yes—
429C23#45112yes—
430C23#45112yes—
431C23#45212yes—
432C23#45212yes—
433C23#46214yes—
434C23#46414yes—
435C23C2×C4×D411yescharacteristic
436C23C2×#12011yescharacteristic
437C23C2×#12111yescharacteristic
438C23C22×Q1611yescharacteristic
439C23C2×#13511yescharacteristic
440C23C2×#13711yescharacteristic
441C23C2×#13711yescharacteristic
442Q8—44no—
443Q8—48no—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—816no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—164no—
505C2×C4#38612yes—
506C2×C4#38612yes—
507C2×C4C22×C22⋊C411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×#12311yescharacteristic
511C8—44no—
512C8—82no—
513C22—21no—
514C22—21no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—24no—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22#187814yes—
534C22#187814yes—
535C22C2×#38611yescharacteristic
536C22C2×C4×Q1611yescharacteristic
537C22C2×#44111yescharacteristic
538C22C2×#45111yescharacteristic
539C22C2×#45211yescharacteristic
540C22#301812yes—
541C22#302612yes—
542C22C2×#46211yescharacteristic
543C22C2×#46411yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—22no—
549C4—22no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—84no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4—82no—
558C2—21no—
559C2—24no—
560C2—24no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2#1838312yes—
565C2#1838512yes—
566C2C2×#187811yescharacteristic
567C2#2094211yescharacteristic
568C2#2096011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9046C2semidirect
7. #9040C2semidirect
13. #3189C4semidirect
83. #484C2×C4semidirect
87. #451C2×C4semidirect
243. C2×C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#342122
#217592
C22×#3282
#141592
#90462
#90402
C2×#12942
#79072