Finite Group Databasebeta

← #411939

Subgroups — #411939

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2028C212yesmaximal
3#20647C212yesmaximal
4#13786C211yesmaximal, characteristic
5#11392C212yesmaximal
6#11296C212yesmaximal
7#11055C212yesmaximal
8#10847C211yesmaximal, characteristic
9#10840C212yesmaximal
10C2×#544C2212yes—
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2212yes—
13C2×#419C2214yes—
14#2857—22no—
15#2659—22no—
16#2552—22no—
17#2551—22no—
18#2550—24no—
19#2316C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22#2285C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#2028—28no—
26#1805—24no—
27#1803—22no—
28#1803—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31#1736C2214yes—
32C2×#339C2214yes—
33C2×#339C2214yes—
34#1436—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37#1434—22no—
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40#544—24no—
41#544—24no—
42#544—44no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—24no—
45C2×#125—24no—
46C2×C22⋊Q8—24no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—24no—
51C2×C4×Q8Q814yescomplemented
52C2×#117C2314yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42D412yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62#423—24no—
63#423—24no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66#419—28no—
67#419—44no—
68C2×#103—24no—
69C2×#103—24no—
70#398—28no—
71#395—28no—
72#391—24no—
73#391—22no—
74#391—22no—
75#391D414yes—
76#391Q814yescomplemented
77#389—24no—
78#389—24no—
79#389—24no—
80#389—24no—
81#384—22no—
82#384D412yes—
83#384D412yes—
84#382D414yes—
85#380—24no—
86#380—22no—
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92#376—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D412yes—
100#343—24no—
101#343—24no—
102#340—24no—
103#340—24no—
104#339—28no—
105#339—44no—
106C22×Q8—24no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8C2×D412yes—
115C22×Q8C2×Q812yes—
116C22×Q8C2×Q812yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C22×C8—24no—
133C4⋊Q8—48no—
134C4⋊Q8—48no—
135C4⋊Q8—84no—
136#125—48no—
137#125—48no—
138C22⋊Q8—48no—
139C22⋊Q8—48no—
140C4×Q8—28no—
141C4×Q8—28no—
142C4×Q8—44no—
143C4×Q8—44no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×Q812yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4#4112yes—
173C2×C4⋊C4#4112yes—
174C2×C22⋊C4—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—82no—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×Q812yes—
192C2×C42C2×Q812yes—
193C2×C42#4114yes—
194C2×C42#4112yes—
195C4⋊C8—44no—
196C4⋊C8—44no—
197#103—48no—
198#103—48no—
199#95—28no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—28no—
204#95—28no—
205#95—42no—
206#95—42no—
207#95—82no—
208#95—82no—
209C24C22×D411yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—48no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×Q8C4⋊Q814yescomplemented
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4#12011yescharacteristic
270C22×C4#12012yes—
271C22×C4#12011yescharacteristic
272C22×C4#12114yes—
273C22×C4C22⋊Q814yes—
274C22×C4C22⋊Q812yes—
275C22×C4C22⋊Q812yes—
276C22×C4#12412yes—
277C22×C4#12412yes—
278C22×C4C22×D412yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×Q811yescharacteristic
281C22×C4C22×Q811yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+412yes—
288M16—44no—
289M16—44no—
290M16—44no—
291M16—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—44no—
305C4⋊C4—48no—
306C4⋊C4—416no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C42—24no—
318C42—24no—
319C42—44no—
320C42—44no—
321C42—44no—
322C42—42no—
323C42—42no—
324C42—48no—
325C42—82no—
326C42C22⋊Q814yes—
327C42C22⋊Q814yes—
328C42#12414yes—
329C42#12414yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23C2×#12011yescharacteristic
335C23C2×#12111yescharacteristic
336C23C2×C22⋊Q811yescharacteristic
337C23C2×C22⋊Q811yescharacteristic
338C23C2×#12411yescharacteristic
339C23#53511yescharacteristic
340C23#53611yescharacteristic
341Q8—28no—
342Q8—44no—
343Q8—44no—
344Q8—44no—
345Q8—168no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#39314yes—
399C2×C4#39312yes—
400C2×C4#39312yes—
401C2×C4C2×C22⋊Q812yes—
402C2×C4C2×#12412yes—
403C2×C4C2×C4⋊Q811yescharacteristic
404C2×C4#53211yescharacteristic
405C2×C4#53311yescharacteristic
406C2×C4#53311yescharacteristic
407C2×C4#54212yes—
408C2×C4#54412yes—
409C2×C4D4212yes—
410C2×C4#54612yes—
411C2×C4#54712yes—
412C2×C4#54812yes—
413C2×C4D4×Q812yes—
414C2×C4#55412yes—
415C2×C4#55512yes—
416C2×C4#55612yes—
417C8—84no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22C2×#39311yescharacteristic
431C22#240811yescharacteristic
432C22#240911yescharacteristic
433C22#254711yescharacteristic
434C22#256311yescharacteristic
435C22#262811yescharacteristic
436C22#285711yescharacteristic
437C22#301911yescharacteristic
438C22#302011yescharacteristic
439C22#302211yescharacteristic
440C22#302211yescharacteristic
441C4—22no—
442C4—22no—
443C4—24no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4—82no—
448C4—82no—
449C4—84no—
450C4—84no—
451C4—82no—
452C4#243912yes—
453C4#244012yes—
454C4#249112yes—
455C4#249212yes—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#1512511yescharacteristic
461C2#2346211yescharacteristic
462C2#2346411yescharacteristic
463C2#2422811yescharacteristic
464C2#2422911yescharacteristic
465C2#2431811yescharacteristic
466C2#2431911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
51. C2×C4×Q8Q8semidirect
76. #391Q8semidirect
228. C2×Q8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#20282
#206472
#137862
#113922
#112962
#110552
#108472
#108402