Finite Group Databasebeta

← #358040

Subgroups — #358040

Order 512.

2074 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C22×#424C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#13795C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11232C214yesmaximal, complemented
9#11231C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#428—24no—
18C2×#428—24no—
19C2×#428—24no—
20C22×#107C2212yes—
21C22×#106C2212yes—
22C2×#424—24no—
23C2×#424—24no—
24C2×#424C418yescomplemented
25C22×C4⋊C8C2214yes—
26#1736—24no—
27#1735—24no—
28C2×#340—24no—
29C2×#340C2214yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2214yes—
33C4⋊#95—22no—
34C4⋊#95—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2214yes—
37C4⋊#95C2212yes—
38C22⋊#95—22no—
39C22⋊#95—22no—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16C2×C414yes—
48C22×M16D414yes—
49C22×M16D414yes—
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8C2×C414yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C414yes—
74C2×#117D414yes—
75C2×#117Q814yes—
76C2×#117Q814yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4D414yes—
90C22×C4⋊C4D412yes—
91C22×C4⋊C4Q814yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D412yes—
100C22×C42—21no—
101C22×C42—21no—
102C22×C42C2312yes—
103#428—232no—
104C2×#107—24no—
105C2×#107—24no—
106C2×#107—24no—
107C2×#106—24no—
108C2×#106—24no—
109C2×#106—24no—
110#424—232no—
111C2×C4⋊C8—28no—
112C2×C4⋊C8—28no—
113C2×C4⋊C8C2×C4116yescomplemented
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—24no—
118#340—28no—
119#340—28no—
120#340—24no—
121#340—24no—
122C25C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4C22×C412yes—
139C23×C4C2×D414yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4C2×Q812yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C2×M16—28no—
156C2×M16—28no—
157C2×M16—28no—
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16C22⋊C4116yes—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—28no—
167C22×C8—28no—
168C22×C8—28no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8—44no—
177C22×C8—44no—
178C22×C8—44no—
179C22×C8C22×C414yes—
180C22×C8C22×C414yes—
181C22×C8C2×D414yes—
182C22×C8C2×D414yes—
183#117—28no—
184#117—28no—
185#117—28no—
186#117—28no—
187#117—28no—
188#117—28no—
189#117—28no—
190#117—28no—
191#117—44no—
192#117—44no—
193#117—44no—
194#117—44no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
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—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C4⋊C418yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×Q814yes—
241C2×C4⋊C4C2×Q814yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4C4⋊C418yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42C22×C414yes—
262C2×C42C2×Q814yes—
263#107—216no—
264#106—216no—
265C4⋊C8—232no—
266#95—84no—
267#95—84no—
268#95—84no—
269#95—84no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24C2×C4⋊C412yes—
275C24#12112yes—
276C24#12112yes—
277C24#12112yes—
278C24#12112yes—
279C24C22⋊Q812yes—
280C24C22⋊Q812yes—
281C24C22×D411yescharacteristic
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—88no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4C2×C22⋊C414yes—
326C22×C4C2×C4⋊C418yes—
327C22×C4C2×C4⋊C412yes—
328C22×C4#11714yes—
329C22×C4C4×D414yes—
330C22×C4C4×D414yes—
331C22×C4C4×Q814yes—
332C22×C4C4×Q814yes—
333C22×C4#12014yes—
334C22×C4#12114yes—
335C22×C4#12114yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C22⋊Q814yes—
341C22×C4C22⋊Q814yes—
342C22×C4C22⋊Q814yes—
343C22×C4C22⋊Q814yes—
344C22×C4C22⋊Q814yes—
345C22×C4C22⋊Q814yes—
346C22×C4C22⋊Q812yes—
347C22×C4C22⋊Q812yes—
348C22×C4#12314yes—
349C22×C4#12314yes—
350C22×C4#12314yes—
351C22×C4#12414yes—
352C22×C4#12514yes—
353C22×C4#12514yes—
354C22×C4#12514yes—
355C22×C4#12514yes—
356C22×C4#12614yes—
357C22×C4#12614yes—
358C22×C4C4⋊Q814yes—
359C22×C4C4⋊Q814yes—
360C22×C4C4⋊Q814yes—
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375M16—232no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—416no—
390C2×C8C2×C22⋊C418yes—
391C4⋊C4—216no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C22⋊C4—216no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—48no—
422C42—48no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23#38014yes—
440C23#38914yes—
441C23#38914yes—
442C23#39414yes—
443C23#39614yes—
444C23#39614yes—
445C23#39814yes—
446C23C22×C4⋊C411yescharacteristic
447C23C2×#12111yescharacteristic
448C23C2×#12111yescharacteristic
449C23C2×#12111yescharacteristic
450C23C2×#12111yescharacteristic
451C23C2×C22⋊Q811yescharacteristic
452C23C2×C22⋊Q811yescharacteristic
453C23#57612yes—
454C23#57712yes—
455C23#57712yes—
456C23#57812yes—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—48no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—816no—
477C2×C4—816no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#38014yes—
494C2×C4#38618yes—
495C2×C4#38718yes—
496C2×C4#38718yes—
497C2×C4#38914yes—
498C2×C4#38914yes—
499C2×C4#39118yes—
500C2×C4#39414yes—
501C2×C4#39518yes—
502C2×C4#39614yes—
503C2×C4#39614yes—
504C2×C4#39818yes—
505C2×C4#39818yes—
506C2×C4#39814yes—
507C2×C4#40018yes—
508C2×C4C22×C22⋊C411yescharacteristic
509C2×C4C2×#11711yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×C4×Q811yescharacteristic
513C2×C4C2×C4×Q811yescharacteristic
514C2×C4C2×#12011yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4C2×C22⋊Q811yescharacteristic
518C2×C4C2×C22⋊Q811yescharacteristic
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×C22⋊Q811yescharacteristic
521C2×C4C2×C22⋊Q811yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4C2×#12311yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4C2×#12411yescharacteristic
527C2×C4C2×#12511yescharacteristic
528C2×C4C2×#12511yescharacteristic
529C2×C4C2×#12511yescharacteristic
530C2×C4C2×#12511yescharacteristic
531C2×C4C2×#12611yescharacteristic
532C2×C4C2×#12611yescharacteristic
533C2×C4C2×C4⋊Q811yescharacteristic
534C2×C4C2×C4⋊Q811yescharacteristic
535C2×C4C2×C4⋊Q811yescharacteristic
536C8—28no—
537C8—44no—
538C22—24no—
539C22—24no—
540C22—28no—
541C22—28no—
542C22—24no—
543C22—24no—
544C22—28no—
545C22—28no—
546C22—24no—
547C22—24no—
548C22#144118yes—
549C22C2×#38011yescharacteristic
550C22C2×#38911yescharacteristic
551C22C2×#38911yescharacteristic
552C22C2×#39411yescharacteristic
553C22C2×#39611yescharacteristic
554C22C2×#39611yescharacteristic
555C22C2×#39811yescharacteristic
556C22#291212yes—
557C22#291312yes—
558C22#304111yescharacteristic
559C22#304211yescharacteristic
560C22#304311yescharacteristic
561C22#304411yescharacteristic
562C22#304512yes—
563C22#304612yes—
564C22#306212yes—
565C22#306212yes—
566C22#306312yes—
567C22#306312yes—
568C22#308112yes—
569C22#308212yes—
570C22#308312yes—
571C22#308412yes—
572C4—48no—
573C4—84no—
574C4—84no—
575C4—82no—
576C4—84no—
577C4—82no—
578C4#144118yes—
579C4C2×#38611yescharacteristic
580C4C2×#38711yescharacteristic
581C4C2×#38711yescharacteristic
582C4C2×#39111yescharacteristic
583C4C2×#39511yescharacteristic
584C4C2×#39811yescharacteristic
585C4C2×#39811yescharacteristic
586C4C2×#40011yescharacteristic
587C2—28no—
588C2C2×#144111yescharacteristic
589C2#2066412yes—
590C2#2117311yescharacteristic
591C2#2117411yescharacteristic
592C2#2117512yes—
593C2#2354411yescharacteristic
594C2#2354511yescharacteristic
595C2#2361411yescharacteristic
596C2#2361511yescharacteristic
597C2#2361612yes—
598C2#2367112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
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
8. #11232C2semidirect
9. #11231C2semidirect
24. C2×#424C4semidirect
113. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C22×#4242
C2×#17362
C2×#17352
C22×#3402
#137952
#112322
#112312