Finite Group Databasebeta

← #414059

Subgroups — #414059

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23734C211yesmaximal, characteristic
2#23452C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#20383C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic, complemented
6#11354C211yesmaximal, characteristic, complemented
7#11249C212yesmaximal, complemented
8#11105C212yesmaximal
9#10896C212yesmaximal, complemented
10#10864C211yesmaximal, characteristic, complemented
11#10774C212yesmaximal
12C22×#131C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#108C2212yes—
15C2×#428C2212yes—
16C2×#417C2212yes—
17C2×#417C2212yes—
18C2×#408C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2297C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2096—21no—
24#2095—21no—
25#2094—21no—
26#2093—21no—
27#2024—21no—
28#2024—21no—
29#2023—21no—
30#2023—21no—
31#1990—21no—
32#1990—21no—
33#1990—21no—
34#1990—21no—
35#1753—21no—
36#1753—21no—
37C8×C22⋊C4—21no—
38C8×C22⋊C4—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#341C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47#1729C2211yescharacteristic
48#1728C2211yescharacteristic
49#1728C2211yescharacteristic
50#1728C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C23×Q8—21no—
53C2×#131—22no—
54C2×#131—42no—
55C2×#131—42no—
56C2×#131Q814yescomplemented
57C22×M16—21no—
58C22×M16C2312yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C23×C8—21no—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—24no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—42no—
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C42C2311yescharacteristic
78C2×#108—24no—
79C2×#108—24no—
80C2×#108—42no—
81#428—24no—
82#428—42no—
83C2×#107—22no—
84C2×#106—22no—
85#417—24no—
86#417—24no—
87#417—42no—
88#417—42no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#102—22no—
92C2×#102—22no—
93#408—22no—
94#408Q814yescomplemented
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97#405—42no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—42no—
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8D411yescharacteristic
105#391D411yescharacteristic
106#391D411yescharacteristic
107#390D411yescharacteristic
108#390D411yescharacteristic
109#386—21no—
110#386—21no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117#343—24no—
118#343—42no—
119#343—42no—
120#343—42no—
121#341—24no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#336—21no—
129#336—21no—
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41C2×Q812yes—
143C2×#41C2×Q812yes—
144C22×Q8—44no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—82no—
148C22×Q8—82no—
149C22×Q8C2×D411yescharacteristic
150C22×D4—21no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
158#131—48no—
159#131—48no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16C2×Q812yes—
181C2×M16C2×Q812yes—
182C2×M16#4114yes—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—24no—
191C22×C8—22no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×Q812yes—
199C22×C8C2×Q812yes—
200C22×C8#4112yes—
201C22×C8#4112yes—
202C4⋊Q8—24no—
203C4⋊Q8—24no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—42no—
207C4⋊Q8—42no—
208#117—22no—
209#117—22no—
210C2×C4⋊C4—22no—
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—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245#108—48no—
246#107—44no—
247#106—44no—
248#103—44no—
249#103—44no—
250#102—44no—
251#102—44no—
252C22⋊C8—24no—
253C22⋊C8—24no—
254#97—42no—
255C4×C8—41no—
256C4×C8—41no—
257C4×C8—41no—
258C4×C8—41no—
259C4×C8—42no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264C24—21no—
265C24C2×#4111yescharacteristic
266#41—24no—
267#41—24no—
268#41—24no—
269#41—24no—
270#41—42no—
271#41—42no—
272#41—42no—
273#41—42no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—816no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—82no—
284C2×Q8C4⋊Q812yes—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4C4⋊Q812yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C22×Q811yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4112yes—
329C22×C4C2×#4112yes—
330C22×C4H(5,𝔽2)12yes—
331C22×C4H(5,𝔽2)12yes—
332M16—24no—
333M16—24no—
334M16—24no—
335M16—24no—
336M16—48no—
337M16—42no—
338M16—42no—
339M16—42no—
340M16—42no—
341M16—42no—
342M16—42no—
343M16—42no—
344M16—42no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—48no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8C22⋊Q812yes—
371C2×C8C22⋊Q812yes—
372C2×C8C22⋊Q812yes—
373C2×C8C22⋊Q812yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—22no—
400C42—22no—
401C42—41no—
402C42—41no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23C2×#12111yescharacteristic
415C23#53311yescharacteristic
416C23#53311yescharacteristic
417C23#55112yes—
418C23#55912yes—
419Q8—24no—
420Q8—42no—
421Q8—88no—
422Q8—88no—
423D4—24no—
424D4—44no—
425D4—44no—
426D4—42no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—84no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—162no—
459C2×C4—162no—
460C2×C4#47112yes—
461C2×C4#47112yes—
462C2×C4#47212yes—
463C2×C4#47312yes—
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4#52511yescharacteristic
467C2×C4#52912yes—
468C2×C4C2×C4⋊Q811yescharacteristic
469C2×C4#53211yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4#53312yes—
472C2×C4#53411yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53812yes—
475C2×C4#54312yes—
476C2×C4#54612yes—
477C2×C4#54612yes—
478C2×C4#55112yes—
479C2×C4#55412yes—
480C2×C4#55412yes—
481C2×C4#55912yes—
482C8—22no—
483C8—22no—
484C8—22no—
485C8—22no—
486C8—42no—
487C8—42no—
488C8—44no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22#253411yescharacteristic
506C22#254411yescharacteristic
507C22#258912yes—
508C22#260712yes—
509C22#285711yescharacteristic
510C22C2×#47111yescharacteristic
511C22C2×#47111yescharacteristic
512C22C2×#47211yescharacteristic
513C22C2×#47311yescharacteristic
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—42no—
519C4—81no—
520C4—81no—
521C4—84no—
522C4—162no—
523C4#239611yescharacteristic
524C4#254912yes—
525C4#255712yes—
526C4#256411yescharacteristic
527C4#258011yescharacteristic
528C4#262611yescharacteristic
529C2—22no—
530C2—22no—
531C2—82no—
532C2#1516111yescharacteristic
533C2#2813411yescharacteristic
534C2#2813511yescharacteristic
535C2#2863111yescharacteristic
536C2#2863211yescharacteristic
537C2#3322011yescharacteristic
538C2#3322211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
3. C2×#1990C2semidirect
5. #11382C2semidirect
6. #11354C2semidirect
7. #11249C2semidirect
9. #10896C2semidirect
10. #10864C2semidirect
56. C2×#131Q8semidirect
94. #408Q8semidirect

Maximal subgroups

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

subgroupindex
#237342
#234522
C2×#19902
#203832
#113822
#113542
#112492
#111052
#108962
#108642
#107742