Finite Group Databasebeta

← #380100

Subgroups — #380100

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1875C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic, complemented
4#14257C211yesmaximal, characteristic
5#13795C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11257C212yesmaximal, complemented
8#11189C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10713C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#1875C418yescomplemented
25#1874—24no—
26#1853—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1439—22no—
35#1438—22no—
36C4⋊#95—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C22⋊#95—22no—
42C23×D4—21no—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121C2×C414yes—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—42no—
58C2×#117—42no—
59C2×#117—42no—
60C2×#117—42no—
61C2×#117C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D414yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#106—24no—
83C2×#106—42no—
84C2×#106—42no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102C2×C418yescomplemented
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2×C414yes—
97#390—42no—
98#390—42no—
99#380—22no—
100#380—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340C2×C418yescomplemented
113#336—42no—
114#336—42no—
115C25—21no—
116C25C2×D411yescharacteristic
117C22×D4—24no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—82no—
130C22×D4—82no—
131C22×D4C22×C412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
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×C8C22×C414yes—
170#121—28no—
171#121—28no—
172#121—44no—
173#121—44no—
174#117—48no—
175#117—48no—
176#117—48no—
177#117—48no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240#106—48no—
241#102—48no—
242#102—48no—
243C22⋊C8—28no—
244C22⋊C8—44no—
245#95—42no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95—82no—
250#95—82no—
251#95—82no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C4×D412yes—
272C24#12112yes—
273C24#12312yes—
274C24C22×D411yescharacteristic
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—48no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—88no—
293C2×D4—88no—
294C2×D4—84no—
295C2×D4—84no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—48no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4#11714yes—
361C22×C4C4×D414yes—
362C22×C4C4×D412yes—
363C22×C4#12112yes—
364C22×C4#12312yes—
365C22×C4#12414yes—
366C22×C4#12614yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C22⋊C4—84no—
415C42—48no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—82no—
448C23—82no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23#38814yes—
466C23C2×C4×D411yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×#12311yescharacteristic
469C23#53411yescharacteristic
470C23#53411yescharacteristic
471C23#53511yescharacteristic
472C23#53611yescharacteristic
473C23#57612yes—
474C23#57712yes—
475C23#57712yes—
476C23#57712yes—
477D4—44no—
478D4—44no—
479D4—816no—
480D4—82no—
481D4—82no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—816no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4#38814yes—
537C2×C4C2×#11711yescharacteristic
538C2×C4C2×C4×D411yescharacteristic
539C2×C4#51812yes—
540C2×C4#52012yes—
541C2×C4C2×#12411yescharacteristic
542C2×C4C2×#12611yescharacteristic
543C2×C4#53412yes—
544C2×C4#53712yes—
545C2×C4#53812yes—
546C2×C4#53812yes—
547C2×C4#53812yes—
548C2×C4#53812yes—
549C2×C4#53912yes—
550C2×C4#53912yes—
551C2×C4#53912yes—
552C2×C4#53912yes—
553C8—44no—
554C8—82no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—84no—
567C22—82no—
568C22—84no—
569C22—82no—
570C22—82no—
571C22—161no—
572C22—161no—
573C22—161no—
574C22—161no—
575C22—162no—
576C22—162no—
577C22C2×#38811yescharacteristic
578C22#232111yescharacteristic
579C22#232211yescharacteristic
580C22#261711yescharacteristic
581C22#261811yescharacteristic
582C22#262211yescharacteristic
583C22#262311yescharacteristic
584C22#294612yes—
585C22#295212yes—
586C22#304111yescharacteristic
587C22#304211yescharacteristic
588C22#304311yescharacteristic
589C22#304311yescharacteristic
590C22#309612yes—
591C22#309912yes—
592C22#319812yes—
593C22#319812yes—
594C22#320012yes—
595C22#320412yes—
596C22#322112yes—
597C22#322412yes—
598C22#324112yes—
599C22#324412yes—
600C4—24no—
601C4—84no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—161no—
607C4—161no—
608C4#231812yes—
609C4#232512yes—
610C4#263912yes—
611C4#264312yes—
612C2—44no—
613C2—162no—
614C2#1425911yescharacteristic
615C2#2106311yescharacteristic
616C2#2106811yescharacteristic
617C2#2205312yes—
618C2#2209012yes—
619C2#2972211yescharacteristic
620C2#2972711yescharacteristic
621C2#2976311yescharacteristic
622C2#2976811yescharacteristic
623C2#2989912yes—
624C2#2991612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1853C2semidirect
5. #13795C2semidirect
6. #11315C2semidirect
7. #11257C2semidirect
8. #11189C2semidirect
9. #11121C2semidirect
10. #10731C2semidirect
11. #10713C2semidirect
24. #1875C4semidirect
90. C2×#102C2×C4semidirect
112. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18752
C2×#18742
C2×#18532
#142572
#137952
#113152
#112572
#111892
#111212
#107312
#107132