Finite Group Databasebeta

← #381435

Subgroups — #381435

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic, complemented
2C2×#2013C211yesmaximal, characteristic
3C2×#1880C212yesmaximal
4#14260C211yesmaximal, characteristic
5#11268C211yesmaximal, characteristic, complemented
6#11067C212yesmaximal, complemented
7#11051C211yesmaximal, characteristic
8#11023C212yesmaximal
9#11010C211yesmaximal, characteristic, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×SD16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19#2327—28no—
20C2×#391C2212yes—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387—22no—
26C2×#387—22no—
27C2×#382C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#2013C418yescomplemented
30#1880—28no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1440—22no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×Q8C2312yes—
43C23×D4C2311yescharacteristic
44C22×SD16—24no—
45C22×SD16—24no—
46C22×SD16—42no—
47C22×SD16—42no—
48C22×SD16—42no—
49C22×SD16—42no—
50C22×SD16C2×C418yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8D418yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103C2×C418yes—
87C2×#102—24no—
88C2×#102—42no—
89C2×#102—42no—
90#391—24no—
91#391—24no—
92#390—24no—
93#390—22no—
94#390—42no—
95#390C2×C414yes—
96#389—42no—
97#387—216no—
98#387—216no—
99#382—24no—
100#376—24no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D412yes—
108C2×#95D412yes—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336—42no—
115#336C2×C414yes—
116C25—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8C22×C414yes—
129C22×Q8C2×D414yes—
130C22×Q8#4114yes—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C2×SD16—216no—
152C2×SD16—48no—
153C2×SD16—48no—
154C2×SD16—48no—
155C2×SD16—48no—
156C2×SD16—48no—
157C2×SD16—48no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
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×C8C22×C414yes—
172C4⋊Q8—216no—
173C4⋊Q8—216no—
174C4×Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C4×Q8—48no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—28no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42C22×C412yes—
227C2×C42C2×D414yes—
228C2×C42#4112yes—
229#103—48no—
230#103—48no—
231#103—48no—
232#102—48no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—42no—
239#95—82no—
240#95—82no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8C4×D418yes—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4C4×D414yes—
349C22×C4#12014yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360SD16—432no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—232no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—416no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—84no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—416no—
397C42—42no—
398C42—42no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23C2×C4×D411yescharacteristic
423C23C2×#12011yescharacteristic
424C23#52511yescharacteristic
425C23#53511yescharacteristic
426C23#53511yescharacteristic
427C23#53511yescharacteristic
428C23#53711yescharacteristic
429C23#57712yes—
430C23#57712yes—
431C23#57712yes—
432C23#57812yes—
433Q8—28no—
434Q8—28no—
435Q8—44no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441D4—44no—
442D4—44no—
443D4—82no—
444D4—82no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4C2×C4×D412yes—
499C2×C4#51811yescharacteristic
500C2×C4#51812yes—
501C2×C4#52011yescharacteristic
502C2×C4#52512yes—
503C2×C4#53412yes—
504C2×C4#53412yes—
505C2×C4#53512yes—
506C2×C4#53712yes—
507C2×C4#53812yes—
508C2×C4#53812yes—
509C2×C4#53812yes—
510C2×C4#53912yes—
511C2×C4#54012yes—
512C2×C4#54011yescharacteristic
513C2×C4#54211yescharacteristic
514C8—44no—
515C8—82no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—84no—
520C22—82no—
521C22—161no—
522C22—161no—
523C22—162no—
524C22—162no—
525C22—161no—
526C22—161no—
527C22C4×#12011yescharacteristic
528C22#232411yescharacteristic
529C22#232811yescharacteristic
530C22#261311yescharacteristic
531C22#261511yescharacteristic
532C22#263111yescharacteristic
533C22#263611yescharacteristic
534C22#294712yes—
535C22#294912yes—
536C22#301012yes—
537C22#301312yes—
538C22#312112yes—
539C22#312612yes—
540C22#322112yes—
541C22#322112yes—
542C22#322211yescharacteristic
543C22#322211yescharacteristic
544C22#322211yescharacteristic
545C22#322311yescharacteristic
546C22#322512yes—
547C22#322512yes—
548C22#325812yes—
549C22#326012yes—
550C4—44no—
551C4—44no—
552C4—48no—
553C4—82no—
554C4—84no—
555C4—81no—
556C4—81no—
557C4—161no—
558C4—161no—
559C4#232112yes—
560C4#232512yes—
561C4#263012yes—
562C4#263512yes—
563C2—162no—
564C2#1429911yescharacteristic
565C2#2146212yes—
566C2#2216212yes—
567C2#2227511yescharacteristic
568C2#2227811yescharacteristic
569C2#2963112yes—
570C2#2967111yescharacteristic
571C2#2967611yescharacteristic
572C2#3005011yescharacteristic
573C2#3005111yescharacteristic
574C2#3013912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2327C2semidirect
5. #11268C2semidirect
6. #11067C2semidirect
9. #11010C2semidirect
10. #10796C2semidirect
11. #10789C2semidirect
29. #2013C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
C2×#20132
C2×#18802
#142602
#112682
#110672
#110512
#110232
#110102
#107962
#107892