Finite Group Databasebeta

← #406777

Subgroups — #406777

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20845C211yesmaximal, characteristic
2#20835C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1821C211yesmaximal, characteristic
5C2×#1811C211yesmaximal, characteristic
6C2×#1736C211yesmaximal, characteristic
7#11303C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#11242C212yesmaximal, complemented
10#11216C212yesmaximal, complemented
11#10728C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428C2212yes—
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22#2280C2211yescharacteristic
23#1854—22no—
24#1854—22no—
25#1851—22no—
26#1851—22no—
27#1824—22no—
28#1823—22no—
29#1821—24no—
30#1811—24no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2212yes—
36#1736—22no—
37#1736—22no—
38#1736C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1729C2211yescharacteristic
44#1729C2211yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—24no—
50C22×M16—42no—
51C22×M16D412yes—
52C22×M16D412yes—
53C22×M16C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—24no—
64C2×#117—42no—
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4Q814yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C42—21no—
80C2×#108—24no—
81C2×#108—24no—
82C2×#108—42no—
83#428—24no—
84#428—42no—
85C2×#107—22no—
86C2×#106—22no—
87#424—24no—
88#424—42no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91#407—22no—
92#407—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96#407D414yes—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103#381—21no—
104#380D412yes—
105#380D412yes—
106#379—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109#343—24no—
110#343—24no—
111#343—24no—
112#343—42no—
113#341—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—24no—
119#336—22no—
120#336—22no—
121C25#4111yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—42no—
132C23×C4—84no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×Q812yes—
139C23×C4#4111yescharacteristic
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—22no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—48no—
169C2×M16—48no—
170C2×M16C2×D414yes—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8#4114yes—
185#117—24no—
186#117—24no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—22no—
191#117—44no—
192#117—44no—
193#117—42no—
194#117—42no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×Q814yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—44no—
231C2×C42—41no—
232C2×C42—41no—
233#108—48no—
234#107—44no—
235#106—44no—
236C4⋊C8—44no—
237C4⋊C8—44no—
238C22⋊C8—24no—
239C22⋊C8—24no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244#95—22no—
245#95—22no—
246C24—22no—
247C24—22no—
248C24—42no—
249C24—42no—
250C24#12111yescharacteristic
251C24#12111yescharacteristic
252C24C22⋊Q812yes—
253C24#12612yes—
254C24C2×#4111yescharacteristic
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4—88no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—84no—
293C22×C4—88no—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4C22⋊Q812yes—
304C22×C4C22⋊Q812yes—
305C22×C4#12612yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×Q811yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4112yes—
314C22×C4H(5,𝔽2)12yes—
315C22×C42−1+412yes—
316M16—24no—
317M16—24no—
318M16—48no—
319M16—48no—
320M16—42no—
321M16—42no—
322M16—42no—
323M16—42no—
324M16—42no—
325M16—42no—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—42no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8#12314yes—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C42—44no—
372C42—44no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—21no—
381C23—21no—
382C23—24no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—48no—
394C23—44no—
395C23—44no—
396C23#39612yes—
397C23#39612yes—
398C23C2×#12111yescharacteristic
399C23C2×C22⋊Q811yescharacteristic
400C23C2×#12611yescharacteristic
401C23#55112yes—
402C23#55212yes—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4—88no—
431C2×C4—84no—
432C2×C4—88no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4—82no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4#39612yes—
439C2×C4#39612yes—
440C2×C4C2×#12111yescharacteristic
441C2×C4C2×#12111yescharacteristic
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×C22⋊Q811yescharacteristic
444C2×C4C2×#12311yescharacteristic
445C2×C4#52511yescharacteristic
446C2×C4#52912yes—
447C2×C4#53312yes—
448C2×C4#53411yescharacteristic
449C2×C4#53611yescharacteristic
450C2×C4#53912yes—
451C2×C4#54412yes—
452C2×C4#54612yes—
453C2×C4#54612yes—
454C2×C4#54712yes—
455C2×C4#54712yes—
456C2×C4#55112yes—
457C2×C4#55212yes—
458C8—24no—
459C8—42no—
460C8—44no—
461C8—81no—
462C8—81no—
463C8—82no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—48no—
478C22—48no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22C2×#39611yescharacteristic
484C22#243812yes—
485C22#248611yescharacteristic
486C22#249312yes—
487C22#253411yescharacteristic
488C22#306812yes—
489C22#306912yes—
490C4—22no—
491C4—22no—
492C4—82no—
493C4—84no—
494C4—82no—
495C4—84no—
496C4#241211yescharacteristic
497C4#241311yescharacteristic
498C4#242512yes—
499C4#243812yes—
500C4#262711yescharacteristic
501C4#285511yescharacteristic
502C2—22no—
503C2—22no—
504C2—42no—
505C2—42no—
506C2#1493511yescharacteristic
507C2#2360211yescharacteristic
508C2#2360311yescharacteristic
509C2#2767612yes—
510C2#2813912yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11303C2semidirect
8. #11249C2semidirect
9. #11242C2semidirect
10. #11216C2semidirect

Maximal subgroups

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

subgroupindex
#208452
#208352
#207002
C2×#18212
C2×#18112
C2×#17362
#113032
#112492
#112422
#112162
#107282