Finite Group Databasebeta

← #97662

Subgroups — #97662

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×Q8C211yesmaximal, characteristic
2#9870C211yesmaximal, characteristic, complemented
3#8564C211yesmaximal, characteristic, complemented
4#8563C211yesmaximal, characteristic
5C4×Q8⋊C8C211yesmaximal, characteristic, complemented
6C4×D4⋊C8C211yesmaximal, characteristic
7#7570C211yesmaximal, characteristic, complemented
8C2×D4×Q8C411yescharacteristic
9C22×C4×Q8—21no—
10#2365—21no—
11#2365C411yescharacteristic
12#2360—22no—
13#2360—21no—
14Q8×C4⋊C4C411yescharacteristic
15D4×C4⋊C4—21no—
16D4×C4⋊C4C411yescharacteristic
17Q8×C22⋊C4—21no—
18C4×C4⋊Q8—21no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×C22⋊Q8—22no—
21C4×C22⋊Q8—21no—
22C42×Q8C2211yescharacteristic
23C42×D4—21no—
24C42×D4C2211yescharacteristic
25#1842C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C42×C8C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31C2×D4⋊C8—21no—
32C2×D4⋊C8—21no—
33#1279—21no—
34#1279—21no—
35#1279—22no—
36#1279—21no—
37#1279—21no—
38#1271—22no—
39C23×Q8—21no—
40D4×Q8—22no—
41D4×Q8—22no—
42D4×Q8—22no—
43D4×Q8C814yescomplemented
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—42no—
54C2×C4×Q8—42no—
55C2×C4×Q8—42no—
56C2×C4×Q8—42no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C42—22no—
68C22×C42—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#391—22no—
80#391—22no—
81#391—21no—
82#391—21no—
83#389—22no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—22no—
88#389—21no—
89#389D412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#384—21no—
97#384C2×C411yescharacteristic
98#380—22no—
99#380—21no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C4⋊C4D411yescharacteristic
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C43—21no—
121C43C2311yescharacteristic
122#336—21no—
123#336—21no—
124Q8⋊C8—24no—
125Q8⋊C8—42no—
126D4⋊C8—24no—
127D4⋊C8—42no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8C22⋊C411yescharacteristic
137C22×D4C22⋊C411yescharacteristic
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—41no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8C2×C812yes—
152C4⋊Q8M1612yes—
153C22⋊Q8—24no—
154C22⋊Q8—28no—
155C22⋊Q8—24no—
156C22⋊Q8—28no—
157C4×Q8—21no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×Q8—22no—
162C4×Q8—41no—
163C4×Q8—44no—
164C4×Q8—44no—
165C4×Q8—42no—
166C4×Q8—44no—
167C4×Q8—44no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×Q8C2×C812yes—
176C4×Q8M1612yes—
177C4×Q8D812yes—
178C4×Q8SD1612yes—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4C2×C812yes—
190C4×D4M1612yes—
191C4×D4SD1612yes—
192C4×D4Q1612yes—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—42no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—41no—
260C4×C8—41no—
261C4×C8—41no—
262C4×C8—41no—
263C4×C8—41no—
264C4×C8—41no—
265C4×C8—41no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—22no—
278C24—21no—
279C2×Q8—28no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8C22⋊C812yes—
296C2×Q8#10212yes—
297C2×Q8#10412yes—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C22⋊C812yes—
302C2×D4#10312yes—
303C2×D4#10412yes—
304C22×C4—21no—
305C22×C4—28no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—44no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—81no—
334C22×C4—82no—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—81no—
348C2×C8—81no—
349C2×C8—81no—
350C2×C8—81no—
351C2×C8—81no—
352C2×C8—81no—
353C2×C8—81no—
354C4⋊C4—28no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—82no—
382C4⋊C4C22⋊C812yes—
383C4⋊C4C22⋊C812yes—
384C4⋊C4C22⋊C814yes—
385C4⋊C4#10212yes—
386C4⋊C4#10312yes—
387C22⋊C4—28no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C42—22no—
395C42—22no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—22no—
401C42—22no—
402C42—21no—
403C42—21no—
404C42—41no—
405C42—42no—
406C42—44no—
407C42—44no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—42no—
414C42—41no—
415C42—41no—
416C42—42no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—81no—
421C42C22×C811yescharacteristic
422C42C2×M1611yescharacteristic
423C42C2×M1611yescharacteristic
424C42C2×M1611yescharacteristic
425C42C2×D811yescharacteristic
426C42C2×SD1611yescharacteristic
427C42C2×SD1611yescharacteristic
428C42C2×Q1611yescharacteristic
429C42AGL(1,ℤ/8)11yescharacteristic
430C42AGL(1,ℤ/8)11yescharacteristic
431C42#13711yescharacteristic
432C42#13711yescharacteristic
433C23—22no—
434C23—22no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23#37911yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8—416no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8D4⋊C814yescomplemented
450D4—22no—
451D4—22no—
452D4—22no—
453D4Q8⋊C814yescomplemented
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—416no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—88no—
502C2×C4C2×C22⋊C811yescharacteristic
503C2×C4C2×C22⋊C811yescharacteristic
504C2×C4C2×C22⋊C811yescharacteristic
505C2×C4#40711yescharacteristic
506C2×C4#40711yescharacteristic
507C2×C4#40711yescharacteristic
508C2×C4C2×#10211yescharacteristic
509C2×C4C2×#10311yescharacteristic
510C2×C4#41811yescharacteristic
511C2×C4#41911yescharacteristic
512C2×C4C2×#10411yescharacteristic
513C2×C4C2×#10411yescharacteristic
514C2×C4#42111yescharacteristic
515C2×C4#42111yescharacteristic
516C2×C4#44711yescharacteristic
517C2×C4#44711yescharacteristic
518C2×C4#44811yescharacteristic
519C2×C4#44811yescharacteristic
520C2×C4#45011yescharacteristic
521C2×C4#45011yescharacteristic
522C2×C4#45111yescharacteristic
523C2×C4#45111yescharacteristic
524C2×C4#45411yescharacteristic
525C2×C4#45611yescharacteristic
526C8—44no—
527C8—82no—
528C8—82no—
529C8—161no—
530C22—24no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22#177911yescharacteristic
538C22#178611yescharacteristic
539C22#178711yescharacteristic
540C22#178911yescharacteristic
541C22#178911yescharacteristic
542C22#179311yescharacteristic
543C22#179411yescharacteristic
544C4—24no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—82no—
558C4—81no—
559C4—81no—
560C4C2×D4⋊C811yescharacteristic
561C4C2×Q8⋊C811yescharacteristic
562C4#147811yescharacteristic
563C4#147911yescharacteristic
564C4#148611yescharacteristic
565C4#148711yescharacteristic
566C4#149011yescharacteristic
567C4#149111yescharacteristic
568C4#162711yescharacteristic
569C4#163711yescharacteristic
570C4#164911yescharacteristic
571C4#165111yescharacteristic
572C2—44no—
573C2#831011yescharacteristic
574C2#831111yescharacteristic
575C2#831911yescharacteristic
576C2#843611yescharacteristic
577C2#843911yescharacteristic
578C2#844211yescharacteristic
579C2#844711yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC23#379
prime radicalC22×D4C22⋊C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9870C2semidirect
3. #8564C2semidirect
5. C4×Q8⋊C8C2semidirect
7. #7570C2semidirect
43. D4×Q8C8semidirect
449. Q8D4⋊C8semidirect
453. D4Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×D4×Q82
#98702
#85642
#85632
C4×Q8⋊C82
C4×D4⋊C82
#75702