Finite Group Databasebeta

← #374602

Subgroups — #374602

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic
2C2×#2906C211yesmaximal, characteristic
3C2×#1998C211yesmaximal, characteristic, complemented
4C2×#1989C211yesmaximal, characteristic, complemented
5C4×#417C211yesmaximal, characteristic
6#11216C212yesmaximal, complemented
7#11216C212yesmaximal, complemented
8#10853C211yesmaximal, characteristic, complemented
9#10847C211yesmaximal, characteristic, complemented
10#10774C212yesmaximal
11#10774C212yesmaximal
12C22×#131C2212yes—
13C22×#125C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#3080—24no—
16#3080C418yescomplemented
17C2×#479—21no—
18C2×#479—21no—
19C2×#478—21no—
20C2×#478—21no—
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23#2906—24no—
24#2906—24no—
25C2×#424—22no—
26C2×#424C2212yes—
27C2×#423—21no—
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C22×C4⋊C8—21no—
31C22×C4⋊C8C2211yescharacteristic
32C2×#417C2211yescharacteristic
33C2×#417C2211yescharacteristic
34C2×#417C2212yes—
35C2×C4×M16C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1998—24no—
38#1989—21no—
39#1989—21no—
40#1989—22no—
41#1763—21no—
42#1762—21no—
43C4×#103—21no—
44C4×#102—21no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2212yes—
48C2×#343C2212yes—
49C2×#341C2212yes—
50C2×#341C2212yes—
51#1736C2212yes—
52C2×#340C2212yes—
53C2×#340C2212yes—
54C22×#41C2311yescharacteristic
55C2×#131—24no—
56C2×#131—24no—
57C2×#131—24no—
58C2×#131—24no—
59C2×#131—42no—
60C2×#131—42no—
61C22×M16—22no—
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2312yes—
65C22×M16C2312yes—
66C23×C8—22no—
67C23×C8C2312yes—
68C2×#125—22no—
69C2×#125—21no—
70C2×#125—21no—
71C2×#125—22no—
72C2×#125—21no—
73C2×#125—21no—
74C2×#125—22no—
75C2×#125C2×C414yes—
76C4×#41—22no—
77C4×#41—22no—
78C4×#41C2×C414yes—
79C4×#41D414yes—
80C2×C4×Q8D411yescharacteristic
81C2×C4×Q8D411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C2×#117C2311yescharacteristic
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C42—21no—
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#479—28no—
97#479—28no—
98#478—28no—
99#478—28no—
100#428—22no—
101#428—22no—
102#428C2×C414yes—
103#428C2×C414yes—
104C2×#107—21no—
105C2×#107—21no—
106C2×#106—21no—
107C2×#106—21no—
108#424—216no—
109#424—24no—
110#424—24no—
111#423—28no—
112#423—22no—
113#423—22no—
114#423C2×C414yes—
115#423D414yes—
116C2×C4⋊C8—28no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8D411yescharacteristic
126C2×C4⋊C8D411yescharacteristic
127C2×C4⋊C8D412yes—
128#417—22no—
129#417—22no—
130#417—24no—
131#417C2×C414yes—
132#417C2×C414yes—
133#417Q818yes—
134C2×#103—21no—
135C2×#103—21no—
136C2×#103—22no—
137C2×#102—21no—
138C2×#102—21no—
139C2×#102—22no—
140C4×M16—22no—
141C4×M16—22no—
142C2×#97—21no—
143C2×C4×C8—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C2×#95—21no—
149#343—24no—
150#343—24no—
151#343—42no—
152#343—42no—
153#341—42no—
154#341—42no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#340—22no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41C22×C412yes—
166C2×#41C2×D412yes—
167C2×#41C2×Q814yes—
168C2×#41#4114yes—
169C22×Q8C2×D411yescharacteristic
170C22×D4—21no—
171C22×D4C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4#4111yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4C2411yescharacteristic
181#131—48no—
182#131—48no—
183#131—48no—
184#131—48no—
185C2×M16—28no—
186C2×M16—22no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C2×D414yes—
207C2×M16C2×Q814yes—
208C22×C8—28no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223#125—24no—
224#125—24no—
225#125—24no—
226#125—24no—
227#125—42no—
228#125—42no—
229#125—42no—
230#125—42no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8C22⋊C414yes—
234C4×Q8C22⋊C414yes—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4C22⋊C414yes—
242C4×D4C22⋊C414yes—
243#117—21no—
244#117—21no—
245#117—24no—
246#117—24no—
247#117C22×C412yes—
248#117C2×D412yes—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C412yes—
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×Q814yes—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—41no—
288C2×C42—41no—
289C2×C42—82no—
290C2×C42C22×C412yes—
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42#4112yes—
298#107—24no—
299#107—24no—
300#106—24no—
301#106—24no—
302C4⋊C8—28no—
303C4⋊C8—28no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306C4⋊C8—22no—
307C4⋊C8—24no—
308C4⋊C8—22no—
309C4⋊C8—24no—
310C4⋊C8—24no—
311#103—24no—
312#103—24no—
313#103—28no—
314#102—24no—
315#102—24no—
316#102—28no—
317#97—42no—
318C4×C8—42no—
319#95—41no—
320#95—41no—
321C24—21no—
322C24C2×#4111yescharacteristic
323#41—24no—
324#41—22no—
325#41—22no—
326#41—24no—
327#41—24no—
328#41—24no—
329#41—24no—
330#41C4×Q818yes—
331#41#12414yes—
332#41C4⋊Q814yes—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8C2×C22⋊C412yes—
336C2×Q8C22⋊Q814yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—21no—
342C2×D4—22no—
343C2×D4C2×C22⋊C412yes—
344C2×D4C22⋊Q814yes—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12312yes—
381C22×C4#12312yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×Q812yes—
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392M16—24no—
393M16—24no—
394M16—24no—
395M16—24no—
396M16—24no—
397M16—24no—
398M16—416no—
399M16—42no—
400M16—41no—
401M16—41no—
402M16—42no—
403M16—41no—
404M16—41no—
405M16—42no—
406M16—42no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—416no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4C2×C22⋊C412yes—
449C4⋊C4C2×C22⋊C412yes—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C42—22no—
455C42—22no—
456C42—24no—
457C42—24no—
458C42—82no—
459C42—82no—
460C42C2×C22⋊C412yes—
461C42C2×C22⋊C412yes—
462C42C4×D412yes—
463C42C4×D412yes—
464C42#12012yes—
465C42#12012yes—
466C42#12112yes—
467C42#12112yes—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—24no—
472C23—22no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23C2×C4×D411yescharacteristic
477C23C2×#12111yescharacteristic
478C23C2×#12311yescharacteristic
479C23#55212yes—
480C23#55412yes—
481Q8—22no—
482Q8—24no—
483Q8#39114yes—
484D4—28no—
485D4—24no—
486D4—24no—
487D4—22no—
488D4—24no—
489D4#39114yes—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4#38612yes—
527C2×C4#38612yes—
528C2×C4#38612yes—
529C2×C4#38612yes—
530C2×C4C22×C22⋊C411yescharacteristic
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×C4×Q812yes—
533C2×C4#52012yes—
534C2×C4C2×#12011yescharacteristic
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×C22⋊Q812yes—
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×C4⋊Q811yescharacteristic
539C2×C4#53511yescharacteristic
540C2×C4#53611yescharacteristic
541C2×C4#53712yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4D4×Q812yes—
545C2×C4D4×Q812yes—
546C2×C4#55212yes—
547C2×C4#55412yes—
548C8—42no—
549C8—42no—
550C8—44no—
551C8—42no—
552C8—42no—
553C8—44no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—44no—
566C22—44no—
567C22C2×#38611yescharacteristic
568C22#236012yes—
569C22#241712yes—
570C22#250211yescharacteristic
571C22#250311yescharacteristic
572C22#306712yes—
573C22#306811yescharacteristic
574C22#306911yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—24no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4—82no—
584C4—84no—
585C4C2×#39111yescharacteristic
586C4#231111yescharacteristic
587C4#236012yes—
588C4#241712yes—
589C4#243411yescharacteristic
590C4#243511yescharacteristic
591C2—22no—
592C2—21no—
593C2—21no—
594C2—44no—
595C2#1413911yescharacteristic
596C2#2100311yescharacteristic
597C2#2100411yescharacteristic
598C2#2485912yes—
599C2#2486012yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1998C2semidirect
4. C2×#1989C2semidirect
6. #11216C2semidirect
7. #11216C2semidirect
8. #10853C2semidirect
9. #10847C2semidirect
16. #3080C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
C2×#29062
C2×#19982
C2×#19892
C4×#4172
#112162
#108532
#108472
#107742