Finite Group Databasebeta

← #414871

Subgroups — #414871

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#21323C211yesmaximal, characteristic
3#21302C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6C2×C4×#101C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11249C212yesmaximal
9#11164C212yesmaximal
10#11128C212yesmaximal
11#10858C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078—24no—
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19C22×#108C2212yes—
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419C2212yes—
24C22×#103C2212yes—
25C22×#101C2211yescharacteristic
26C22×#101C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×#382C2211yescharacteristic
30#1993—21no—
31#1993—21no—
32#1991—22no—
33#1985—22no—
34#1984—22no—
35#1950—24no—
36#1805—22no—
37#1805—22no—
38C4×#101—24no—
39C2×#343C2211yescharacteristic
40C2×#343C2212yes—
41C2×#343C2211yescharacteristic
42C2×#343C2212yes—
43C2×#341C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48#1729C2212yes—
49C23×Q8C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125D412yes—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8D414yes—
75C2×#117C2312yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#479—216no—
82#477—28no—
83#475—28no—
84C2×#108—24no—
85C2×#108—24no—
86C2×#108—42no—
87#428—24no—
88#428—24no—
89C2×#107—22no—
90C2×#106—22no—
91#423—22no—
92#423D414yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8D412yes—
98C2×C4⋊C8D412yes—
99#419—24no—
100#419—24no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×#101—22no—
107C2×#101—22no—
108C2×#101—22no—
109C2×#101—22no—
110C2×#101—22no—
111C2×#101Q814yescomplemented
112C4×M16—22no—
113C4×M16—22no—
114C4×M16—22no—
115C4×M16—22no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120#382—21no—
121#382—21no—
122#382—21no—
123#382—21no—
124C2×#95—21no—
125C2×#95—21no—
126#343—24no—
127#343—22no—
128#343—22no—
129#343—42no—
130#341—24no—
131#340—24no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336—24no—
137#336—24no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8C2×D412yes—
152C22×Q8C2×Q812yes—
153C22×Q8#4112yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4#4111yescharacteristic
158C23×C4#4111yescharacteristic
159C23×C4C2411yescharacteristic
160C2×M16—22no—
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—22no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16C2×D414yes—
186C2×M16C2×Q812yes—
187C2×M16C2×Q812yes—
188C2×M16#4112yes—
189C2×M16#4112yes—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2×D414yes—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206#125—24no—
207#125—24no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×Q8—42no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C22⋊C4—22no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4112yes—
263C2×C42#4112yes—
264#108—48no—
265#107—28no—
266#106—28no—
267C4⋊C8—24no—
268C4⋊C8—24no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271#103—28no—
272#103—28no—
273#103—48no—
274#101—28no—
275#101—28no—
276#97—42no—
277#97—42no—
278C4×C8—42no—
279C4×C8—42no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—41no—
284C24C2×#4111yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—44no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—82no—
316C2×Q8C22⋊Q814yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12312yes—
342C22×C4#12312yes—
343C22×C4#12412yes—
344C22×C4#12412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q811yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357C22×C42−1+411yescharacteristic
358C22×C42−1+411yescharacteristic
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—22no—
364M16—24no—
365M16—24no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—42no—
370M16—41no—
371M16—41no—
372M16—42no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C42—42no—
422C42—42no—
423C42—44no—
424C42—44no—
425C42#12012yes—
426C42#12012yes—
427C42#12112yes—
428C42#12112yes—
429C42#12112yes—
430C42#12112yes—
431C42#12312yes—
432C42#12312yes—
433C23—22no—
434C23—22no—
435C23C2×#12111yescharacteristic
436C23C2×#12311yescharacteristic
437C23C2×#12411yescharacteristic
438C23#52911yescharacteristic
439C23#53311yescharacteristic
440C23#54111yescharacteristic
441C23#54211yescharacteristic
442Q8—24no—
443Q8—24no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—84no—
451Q8—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—162no—
479C2×C4—162no—
480C2×C4#39412yes—
481C2×C4#39412yes—
482C2×C4#39412yes—
483C2×C4#39412yes—
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×C22⋊Q811yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52911yescharacteristic
491C2×C4#53311yescharacteristic
492C2×C4#53611yescharacteristic
493C2×C4#53711yescharacteristic
494C2×C4#54111yescharacteristic
495C2×C4#54211yescharacteristic
496C2×C4#54611yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54811yescharacteristic
501C2×C4#54811yescharacteristic
502C2×C4#54811yescharacteristic
503C2×C4#54811yescharacteristic
504C2×C4D4×Q811yescharacteristic
505C2×C4D4×Q811yescharacteristic
506C2×C4D4×Q811yescharacteristic
507C2×C4#55011yescharacteristic
508C2×C4#55211yescharacteristic
509C2×C4#55211yescharacteristic
510C2×C4#55411yescharacteristic
511C2×C4#55611yescharacteristic
512C8—44no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—42no—
517C8—44no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22C2×#39411yescharacteristic
527C22#242611yescharacteristic
528C22#242911yescharacteristic
529C22#248911yescharacteristic
530C22#249211yescharacteristic
531C22#262511yescharacteristic
532C22#285511yescharacteristic
533C22#306712yes—
534C22#306912yes—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—162no—
544C4#240711yescharacteristic
545C4#240911yescharacteristic
546C4#241511yescharacteristic
547C4#241711yescharacteristic
548C4#242111yescharacteristic
549C4#242311yescharacteristic
550C4#242711yescharacteristic
551C4#242911yescharacteristic
552C2—21no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2#1517911yescharacteristic
557C2#2353011yescharacteristic
558C2#2353211yescharacteristic
559C2#2531712yes—
560C2#2549412yes—
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
111. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#213232
#213022
C2×#19502
C2×#18052
C2×C4×#1012
#112902
#112492
#111642
#111282
#108582