Finite Group Databasebeta

← #414855

Subgroups — #414855

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4#21145C211yesmaximal, characteristic
5#20647C211yesmaximal, characteristic
6C2×C4×#101C211yesmaximal, characteristic
7#11270C212yesmaximal
8#11239C212yesmaximal
9#11164C212yesmaximal
10#11062C212yesmaximal
11#10853C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—22no—
15C2×#477—22no—
16C2×#477—24no—
17C2×#477—24no—
18C22×#107C2212yes—
19C2×#424C2212yes—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#419C2212yes—
23C22×#103C2212yes—
24C22×#101C2211yescharacteristic
25C22×#101C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28#2285C2211yescharacteristic
29#1997—24no—
30#1993—22no—
31#1990—22no—
32#1940—22no—
33#1939—22no—
34#1805—22no—
35#1803—22no—
36C4×#101—24no—
37C2×#343C2211yescharacteristic
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#343C2211yescharacteristic
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46#1729C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D414yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8D414yes—
71C2×#117C2312yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#477—216no—
79#477—216no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83#424—24no—
84#424—42no—
85#423—22no—
86#423—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D412yes—
93C2×C4⋊C8D412yes—
94#419—24no—
95#419—42no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#101—22no—
101C2×#101—22no—
102C2×#101—22no—
103C2×#101—22no—
104C2×#101—22no—
105C2×#101Q814yescomplemented
106C4×M16—22no—
107C4×M16—22no—
108C4×M16—22no—
109C4×M16—22no—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#384D412yes—
115#382D412yes—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120#343—24no—
121#343—24no—
122#343—22no—
123#343—22no—
124#341—24no—
125#341—42no—
126#340—24no—
127#340—24no—
128#340—24no—
129#336—24no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—41no—
143C22×Q8—41no—
144C22×Q8—42no—
145C22×Q8—84no—
146C22×Q8C2×D412yes—
147C22×Q8C2×Q812yes—
148C22×Q8#4112yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4C2411yescharacteristic
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16C2×Q812yes—
182C2×M16C2×Q812yes—
183C2×M16#4112yes—
184C2×M16#4112yes—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C2×D414yes—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—42no—
202C4⋊Q8—42no—
203C4⋊Q8—42no—
204C4⋊Q8—42no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8—42no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—22no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—42no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4112yes—
253C2×C42#4112yes—
254#107—216no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—48no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260#103—216no—
261#103—48no—
262#101—28no—
263#101—28no—
264#97—42no—
265#97—42no—
266C4×C8—42no—
267C4×C8—42no—
268#95—22no—
269#95—22no—
270C24C2×#4111yescharacteristic
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—88no—
304C2×Q8C22⋊Q814yes—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—82no—
322C22×C4—84no—
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12312yes—
328C22×C4#12312yes—
329C22×C4#12312yes—
330C22×C4#12312yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q811yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C42−1+411yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—42no—
365C2×C8—24no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C42—42no—
406C42—42no—
407C42—44no—
408C42—44no—
409C42#12012yes—
410C42#12012yes—
411C42#12112yes—
412C42#12112yes—
413C42#12112yes—
414C42#12112yes—
415C42#12412yes—
416C42#12412yes—
417C23—22no—
418C23—22no—
419C23C2×#12111yescharacteristic
420C23C2×#12311yescharacteristic
421C23C2×#12311yescharacteristic
422C23#52911yescharacteristic
423C23#53311yescharacteristic
424C23#54111yescharacteristic
425C23#54211yescharacteristic
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—42no—
433Q8—42no—
434Q8—88no—
435Q8—88no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4#39412yes—
462C2×C4#39412yes—
463C2×C4#39412yes—
464C2×C4#39412yes—
465C2×C4#47314yes—
466C2×C4C2×#12011yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4#52511yescharacteristic
471C2×C4C2×#12411yescharacteristic
472C2×C4#52911yescharacteristic
473C2×C4#53311yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53711yescharacteristic
476C2×C4#54111yescharacteristic
477C2×C4#54211yescharacteristic
478C2×C4#54611yescharacteristic
479C2×C4#54611yescharacteristic
480C2×C4#54711yescharacteristic
481C2×C4#54711yescharacteristic
482C2×C4#54811yescharacteristic
483C2×C4#54811yescharacteristic
484C2×C4#54811yescharacteristic
485C2×C4#54811yescharacteristic
486C2×C4D4×Q811yescharacteristic
487C2×C4D4×Q811yescharacteristic
488C2×C4D4×Q811yescharacteristic
489C2×C4#55011yescharacteristic
490C2×C4#55211yescharacteristic
491C2×C4#55211yescharacteristic
492C2×C4#55411yescharacteristic
493C2×C4#55611yescharacteristic
494C8—44no—
495C8—42no—
496C8—42no—
497C8—42no—
498C8—42no—
499C8—82no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22C2×#39411yescharacteristic
509C22#242111yescharacteristic
510C22#242311yescharacteristic
511C22#242711yescharacteristic
512C22#242911yescharacteristic
513C22#262511yescharacteristic
514C22#285511yescharacteristic
515C22C2×#47312yes—
516C22#306812yes—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—44no—
521C4—84no—
522C4—82no—
523C4—84no—
524C4#204714yes—
525C4#240711yescharacteristic
526C4#240911yescharacteristic
527C4#241511yescharacteristic
528C4#241711yescharacteristic
529C4#242611yescharacteristic
530C4#242911yescharacteristic
531C4#248911yescharacteristic
532C4#249211yescharacteristic
533C2—21no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2#1517911yescharacteristic
538C2C2×#204711yescharacteristic
539C2#2352111yescharacteristic
540C2#2536012yes—
541C2#2549112yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×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
105. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#19972
#213242
#211452
#206472
C2×C4×#1012
#112702
#112392
#111642
#110622
#108532