Finite Group Databasebeta

← #374210

Subgroups — #374210

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3#20911C212yesmaximal
4C2×#1740C211yesmaximal, characteristic
5#11225C212yesmaximal, complemented
6#11074C212yesmaximal, complemented
7#10947C212yesmaximal, complemented
8#10889C212yesmaximal, complemented
9#10716C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13#2924—24no—
14#2924—24no—
15#2921—22no—
16#2920—22no—
17#2905C2212yes—
18C2×#419C2214yes—
19C2×#407C2212yes—
20#2873C2212yes—
21#2855—22no—
22#2625—22no—
23#2503C412yescomplemented
24#2502C412yescomplemented
25#2499C412yescomplemented
26#2499C412yescomplemented
27#2484—22no—
28#2483—22no—
29#2317C2211yescharacteristic
30#2317C2211yescharacteristic
31#2310C2211yescharacteristic
32#2310C2211yescharacteristic
33#2280C2211yescharacteristic
34#1873—24no—
35#1872—24no—
36#1740—22no—
37#1740C2212yes—
38#1740C2212yes—
39C2×#342—22no—
40#1736C2214yes—
41C2×H(3,ℤ/4)C2214yes—
42C2×H(3,ℤ/4)C2214yes—
43C23×Q8C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16C2314yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—28no—
61C2×#117—22no—
62C2×#117—44no—
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C2×#117D412yes—
66C2×#117D412yes—
67C2×#117C2314yes—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42—22no—
81C22×C42D412yes—
82#433—216no—
83#432—24no—
84#432—24no—
85#431—24no—
86#431—24no—
87#424—28no—
88#423—24no—
89#419—28no—
90#419—44no—
91C2×#103—28no—
92#407—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96C2×C22⋊C8D414yes—
97C2×C22⋊C8D414yes—
98#405—28no—
99C4×M16—24no—
100#400—24no—
101#398—24no—
102#397—24no—
103#397—24no—
104#391—22no—
105#391—22no—
106#391C2×C412yescomplemented
107#391C2×C412yescomplemented
108#391C2×C412yescomplemented
109#391C2×C412yescomplemented
110#389—22no—
111#389—22no—
112#387—24no—
113#387—24no—
114#385—24no—
115#385—22no—
116#385—22no—
117#384C2×C412yescomplemented
118#384C2×C412yescomplemented
119#382C2×C412yescomplemented
120#382C2×C412yescomplemented
121#381—24no—
122#381—21no—
123#381—22no—
124#381—22no—
125#380—21no—
126#380—21no—
127#379—21no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4C2×C412yes—
130#342—24no—
131#342—24no—
132#340—28no—
133H(3,ℤ/4)—28no—
134H(3,ℤ/4)—44no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C22×C412yes—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—42no—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C2×M16—24no—
162C2×M16—24no—
163C2×M16—24no—
164C2×M16—24no—
165C2×M16—24no—
166C2×M16—24no—
167C22×C8C2×D418yes—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8—42no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—42no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—24no—
183#117—24no—
184#117—24no—
185#117—24no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—24no—
191#117—24no—
192#117—22no—
193#117—22no—
194#117—28no—
195#117—24no—
196#117—24no—
197#117—48no—
198#117—48no—
199#117—44no—
200#117#4118yes—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C2×D412yes—
243C2×C22⋊C4C2×D412yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—48no—
250C2×C42—42no—
251C2×C42C22⋊C412yes—
252C2×C42C22⋊C412yes—
253C2×C42C22×C412yes—
254C2×C42C22×C412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C4⋊C8—28no—
258C4⋊C8—28no—
259#103—416no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C22⋊C8—24no—
264#97—28no—
265C4×C8—28no—
266#95—22no—
267#95—22no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—42no—
278C24—81no—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C24#12112yes—
282C24#12112yes—
283C24C22×D411yescharacteristic
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—44no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—44no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C412yes—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12012yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4#12312yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4114yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C42−1+412yes—
359M16—44no—
360M16—44no—
361M16—44no—
362M16—44no—
363C2×C8—28no—
364C2×C8—28no—
365C2×C8—28no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—28no—
374C4⋊C4—48no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—28no—
390C22⋊C4—44no—
391C22⋊C4—48no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—44no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—82no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—24no—
409C42—24no—
410C42—28no—
411C42—48no—
412C42—44no—
413C23—21no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—21no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—84no—
432C23—82no—
433C23—82no—
434C23#38612yes—
435C23#38611yescharacteristic
436C23#38611yescharacteristic
437C23C22×C22⋊C411yescharacteristic
438C23C2×#12112yes—
439C23#52512yes—
440C23#53511yescharacteristic
441C23#53611yescharacteristic
442Q8—48no—
443Q8—82no—
444Q8—82no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—816no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#38611yescharacteristic
494C2×C4#38611yescharacteristic
495C2×C4#38612yes—
496C2×C4C2×C4×D412yes—
497C2×C4C4×#4112yes—
498C2×C4#52012yes—
499C2×C4C2×#12011yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4#53212yes—
502C2×C4#53712yes—
503C2×C4#54614yes—
504C2×C4#54714yes—
505C2×C4#55114yes—
506C2×C4#55214yes—
507C8—48no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—22no—
516C22—21no—
517C22—22no—
518C22—21no—
519C22—22no—
520C22—42no—
521C22—41no—
522C22—41no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—82no—
528C22—81no—
529C22—81no—
530C22—82no—
531C22#190912yes—
532C22C2×#38611yescharacteristic
533C22#229711yescharacteristic
534C22#231111yescharacteristic
535C22#241112yes—
536C22#241312yes—
537C4—21no—
538C4—22no—
539C4—21no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—82no—
546C4—88no—
547C4—82no—
548C4—82no—
549C4#235914yes—
550C4#243012yes—
551C4#249812yes—
552C2—22no—
553C2—21no—
554C2—21no—
555C2—22no—
556C2—22no—
557C2—41no—
558C2—42no—
559C2#1413511yescharacteristic
560C2C2×#190911yescharacteristic
561C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11225C2semidirect
6. #11074C2semidirect
7. #10947C2semidirect
8. #10889C2semidirect
23. #2503C4semidirect
24. #2502C4semidirect
25. #2499C4semidirect
26. #2499C4semidirect
106. #391C2×C4semidirect
107. #391C2×C4semidirect
108. #391C2×C4semidirect
109. #391C2×C4semidirect
117. #384C2×C4semidirect
118. #384C2×C4semidirect
119. #382C2×C4semidirect
120. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#353112
#209112
C2×#17402
#112252
#110742
#109472
#108892
#107162