Finite Group Databasebeta

← #436997

Subgroups — #436997

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15150C211yesmaximal, characteristic
2#14130C211yesmaximal, characteristic
3#11315C212yesmaximal, complemented
4#11292C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#11199C212yesmaximal
7#10921C211yesmaximal, characteristic
8#10880C212yesmaximal, complemented
9#10730C212yesmaximal
10C23×C4⋊C4C2211yescharacteristic
11C22×C22⋊C8C2212yes—
12C22×C4×C8C2211yescharacteristic
13C2×#397—21no—
14C2×#397—21no—
15C2×#389C2212yes—
16C2×#385—21no—
17C2×#384C2211yescharacteristic
18C2×#380C2212yes—
19C2×#380C2211yescharacteristic
20C2×C4×C22⋊C4C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31#1445—21no—
32#1445—21no—
33#1444—21no—
34#1442—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—24no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—41no—
49C22×C4⋊C4—41no—
50C22×C4⋊C4—42no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—42no—
64C2×C4×C8—41no—
65C2×C4×C8—42no—
66#397—44no—
67#397—44no—
68#389—24no—
69#389—42no—
70#385—44no—
71#384—41no—
72#384—41no—
73#380—24no—
74#380—24no—
75#380—24no—
76C4×C22⋊C4—41no—
77C4×C22⋊C4—41no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#340—42no—
96#340—42no—
97#336—24no—
98#336—41no—
99C25C2×D411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—44no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—82no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4#4114yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8#4114yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
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—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4#4114yes—
174C2×C4⋊C4#4114yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4D814yes—
186C2×C22⋊C4SD1614yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C22⋊C8—48no—
196C4×C8—44no—
197#95—24no—
198#95—41no—
199#95—41no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—44no—
205#95—42no—
206#95—44no—
207#95—82no—
208C24—21no—
209C24—21no—
210C24—22no—
211C24—22no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24#12412yes—
217C24C2×D812yes—
218C24C2×SD1612yes—
219C24C22×D411yescharacteristic
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—88no—
286C22×C4—88no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4#12412yes—
290C22×C4C2×D812yes—
291C22×C4C2×D812yes—
292C22×C4C2×SD1612yes—
293C22×C4C2×SD1612yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4C2×#4112yes—
297C22×C4C2×#4112yes—
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)12yes—
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+412yes—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—88no—
334C4⋊C4—84no—
335C4⋊C4—84no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—48no—
339C22⋊C4—88no—
340C42—42no—
341C42—42no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C23—24no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23#48614yes—
369C23#52511yescharacteristic
370C23C2×#12411yescharacteristic
371C23#53411yescharacteristic
372C23#54612yes—
373C23#54712yes—
374C23C22×D811yescharacteristic
375C23C22×SD1611yescharacteristic
376C23#57511yescharacteristic
377C23#57511yescharacteristic
378C23#57712yes—
379C23#57812yes—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—816no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#52912yes—
439C2×C4#52912yes—
440C2×C4#53812yes—
441C2×C4#53912yes—
442C2×C4#53912yes—
443C2×C4#53912yes—
444C2×C4#55111yescharacteristic
445C2×C4#55112yes—
446C2×C4#55212yes—
447C2×C4#55312yes—
448C2×C4#55312yes—
449C2×C4#55312yes—
450C2×C4#55612yes—
451C2×C4#55911yescharacteristic
452C2×C4H(3,𝔽4)11yescharacteristic
453C2×C4#56311yescharacteristic
454C8—44no—
455C8—82no—
456C22—28no—
457C22—24no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22#246612yes—
468C22#248812yes—
469C22#259111yescharacteristic
470C22#268411yescharacteristic
471C22#272411yescharacteristic
472C22#310011yescharacteristic
473C22#310311yescharacteristic
474C22#312012yes—
475C22C2×#48611yescharacteristic
476C22#313712yes—
477C22#314211yescharacteristic
478C22#318611yescharacteristic
479C22#318711yescharacteristic
480C22#320011yescharacteristic
481C22#320211yescharacteristic
482C22#329412yes—
483C22#329812yes—
484C22#331112yes—
485C22#331312yes—
486C22#333412yes—
487C22#333512yes—
488C22#335112yes—
489C22#335212yes—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—84no—
496C4—84no—
497C4—162no—
498C4—162no—
499C4#257712yes—
500C4#259512yes—
501C4#278212yes—
502C4#278712yes—
503C2—44no—
504C2#1557911yescharacteristic
505C2#2703012yes—
506C2#2703412yes—
507C2#2769512yes—
508C2#2772412yes—
509C2#2932811yescharacteristic
510C2#2934411yescharacteristic
511C2#3126011yescharacteristic
512C2#3126211yescharacteristic
513C2#3245811yescharacteristic
514C2#3245911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11315C2semidirect
4. #11292C2semidirect
5. #11267C2semidirect
8. #10880C2semidirect

Maximal subgroups

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

subgroupindex
#151502
#141302
#113152
#112922
#112672
#111992
#109212
#108802
#107302