Finite Group Databasebeta

← #224740

Subgroups — #224740

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2341C211yesmaximal, characteristic
2#22510C212yesmaximal, complemented
3#9653C214yesmaximal, complemented
4#9649C212yesmaximal, complemented
5#8559C212yesmaximal, complemented
6#8557C211yesmaximal, characteristic, complemented
7#8152C212yesmaximal, complemented
8C4×#323C211yesmaximal, characteristic, complemented
9C23×#41C411yescharacteristic
10C22×C4×Q8C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#446—22no—
13#2341—22no—
14#2341—22no—
15#2341—24no—
16C2×#390C412yes—
17C2×#390C411yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#390C2212yes—
20C2×#380C412yes—
21C2×#380C411yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24#1985—24no—
25#1984—22no—
26#1797C2212yes—
27#1797C2214yes—
28#1749C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#323C2214yes—
31C2×#323C2212yes—
32C2×#323C2212yes—
33C2×#323C2214yes—
34#1280C2212yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C22×#41—22no—
38C22×#41—21no—
39C22×#41—21no—
40C22×#41—24no—
41C22×#41—24no—
42C22×#41—44no—
43C22×#41—44no—
44C23×Q8C2×C411yescharacteristic
45C23×D4C2×C412yes—
46C23×D4C2×C411yescharacteristic
47C24×C4C2×C412yes—
48C24×C4C2×C411yescharacteristic
49C2×C4×Q8—22no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—44no—
53C2×C4×Q8—44no—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D414yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2311yescharacteristic
68C22×C42C2×C412yes—
69C22×C42C2×C411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72#446—48no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—24no—
75C2×C22⋊C8C2312yes—
76C2×C22⋊C8C2314yes—
77C2×C22⋊C8C2312yes—
78C2×#97—24no—
79C2×C4×C8—22no—
80#390—24no—
81#390—22no—
82#390—22no—
83#390—24no—
84#390—22no—
85#390—24no—
86#390—22no—
87#390—24no—
88#380—24no—
89#380—22no—
90#380—22no—
91#380—24no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—24no—
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C412yes—
98C2×#95C2311yescharacteristic
99#336—24no—
100#336—22no—
101#336—24no—
102#336—22no—
103#323—28no—
104#323—24no—
105#323—42no—
106#323—44no—
107C25C22×C412yes—
108C25C22×C411yescharacteristic
109C2×#41—216no—
110C2×#41—28no—
111C2×#41—28no—
112C2×#41—432no—
113C2×#41—432no—
114C2×#41—48no—
115C2×#41—44no—
116C2×#41—48no—
117C2×#41—44no—
118C2×#41—44no—
119C2×#41—44no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C411yescharacteristic
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—44no—
137C22×D4—42no—
138C22×D4M1614yes—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4M1614yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2411yescharacteristic
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—24no—
163C4×Q8—28no—
164C4×Q8—44no—
165C4×Q8—48no—
166C4×Q8—48no—
167C4×Q8—44no—
168C4×Q8—44no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C22⋊C414yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—42no—
206C2×C42—44no—
207C2×C42—44no—
208C2×C42—42no—
209C2×C42C2×D412yes—
210C2×C42C2×D414yes—
211C2×C42C2×D412yes—
212C2×C42C2×D414yes—
213C4⋊C8—44no—
214C4⋊C8—48no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—24no—
218C22⋊C8—22no—
219C22⋊C8—24no—
220C22⋊C8#4114yescomplemented
221#97—48no—
222C4×C8—44no—
223#95—22no—
224#95—24no—
225#95—22no—
226C24—22no—
227C24—21no—
228C24—22no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—21no—
233C24—22no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24—42no—
239C24—41no—
240C24—42no—
241C24—41no—
242C24—42no—
243C24—41no—
244C24—42no—
245C24—41no—
246C24C2×M1614yes—
247C24C2×M1612yes—
248C24C23×C411yescharacteristic
249#41—464no—
250#41—432no—
251#41—432no—
252C2×Q8—216no—
253C2×Q8—216no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8#9914yes—
262C2×Q8#10014yes—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—416no—
268C2×D4—416no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—416no—
310C22×C4—416no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C414yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×M1612yes—
336C22×C4C2×M1612yes—
337C22×C4C2×M1612yes—
338C22×C4#13112yes—
339C22×C4C22×D412yes—
340C22×C4C2×#4112yes—
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—44no—
348C2×C8—42no—
349C2×C8—44no—
350C2×C8—42no—
351C2×C8—44no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—48no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—84no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—24no—
370C42—28no—
371C42—42no—
372C42—44no—
373C42—42no—
374C42—44no—
375C42—44no—
376C42—42no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—24no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—22no—
386C23—24no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—42no—
391C23—44no—
392C23—42no—
393C23—44no—
394C23—42no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—81no—
412C23—82no—
413C23—81no—
414C23—82no—
415C23—81no—
416C23—82no—
417C23—81no—
418C23—82no—
419C23—81no—
420C23—82no—
421C23—82no—
422C23—81no—
423C23—81no—
424C23—82no—
425C23—82no—
426C23—81no—
427C23C22×C22⋊C411yescharacteristic
428C23C4×#4111yescharacteristic
429C23#52011yescharacteristic
430C23C22×M1611yescharacteristic
431C23C22×M1611yescharacteristic
432C23C2×#13111yescharacteristic
433C23#56811yescharacteristic
434Q8—216no—
435Q8—28no—
436Q8—28no—
437D4—48no—
438D4—416no—
439D4—416no—
440D4—48no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—48no—
465C2×C4—416no—
466C2×C4—416no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4#40712yes—
491C2×C4#40712yes—
492C2×C4C2×#9914yes—
493C2×C4C2×#9912yes—
494C2×C4C2×#10014yes—
495C2×C4C2×#10012yes—
496C2×C4#54812yes—
497C2×C4#55012yes—
498C8—44no—
499C8—82no—
500C8—84no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—42no—
514C22—44no—
515C22—44no—
516C22—42no—
517C22—44no—
518C22—42no—
519C22—82no—
520C22—84no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22—82no—
525C22—84no—
526C22—82no—
527C22—81no—
528C22—82no—
529C22—81no—
530C22—82no—
531C22—81no—
532C22—82no—
533C22—81no—
534C22—82no—
535C22#234111yescharacteristic
536C22C2×#40711yescharacteristic
537C22#288011yescharacteristic
538C22C22×#9911yescharacteristic
539C22#288411yescharacteristic
540C22C22×#10011yescharacteristic
541C22#288811yescharacteristic
542C22#296511yescharacteristic
543C22#297011yescharacteristic
544C22#298911yescharacteristic
545C22#299011yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—24no—
550C4—24no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—81no—
557C4#146214yes—
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—82no—
563C2—84no—
564C2C2×#146211yescharacteristic
565C2#1737211yescharacteristic
566C2#1741911yescharacteristic
567C2#1742511yescharacteristic
568C2#2253211yescharacteristic
569C2#2253411yescharacteristic
570C2#2253611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. #9653C2semidirect
4. #9649C2semidirect
5. #8559C2semidirect
6. #8557C2semidirect
7. #8152C2semidirect
8. C4×#323C2semidirect
220. C22⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#23412
#225102
#96532
#96492
#85592
#85572
#81522
C4×#3232