Finite Group Databasebeta

← #415649

Subgroups — #415649

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1864C212yesmaximal
2#14277C211yesmaximal, characteristic, complemented
3#13785C211yesmaximal, characteristic
4#11323C211yesmaximal, characteristic, complemented
5#11268C211yesmaximal, characteristic, complemented
6#11231C212yesmaximal, complemented
7#11076C212yesmaximal
8#11051C211yesmaximal, characteristic
9#11049C211yesmaximal, characteristic
10#10884C212yesmaximal, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1864—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1444—22no—
34#1442—22no—
35#1437—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C22⋊#95—22no—
40C23×D4C2311yescharacteristic
41C24×C4—21no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4×D4—22no—
45C2×C4×D4—42no—
46C2×C4×D4—42no—
47C2×C4×D4D414yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#106—24no—
63C2×#106—42no—
64C2×#106—42no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—42no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—42no—
73C2×#102—42no—
74#390—22no—
75#390—22no—
76#390Q814yes—
77#390Q814yes—
78#389—24no—
79#389—24no—
80#384—42no—
81#382—24no—
82C4×C4⋊C4—42no—
83#376—42no—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D412yes—
92C2×#95D412yes—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—22no—
100#336—22no—
101C25—21no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4C2×D412yes—
110C22×D4C2×Q812yes—
111C22×D4C2×Q812yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—82no—
121C23×C4—82no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
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—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8#4114yes—
141C4×D4—48no—
142C4×D4—48no—
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—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×Q812yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C4⋊C4#4114yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—42no—
189C2×C42—82no—
190C2×C42C2×D412yes—
191C2×C42C2×Q812yes—
192C2×C42C2×Q812yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195C2×C42#4112yes—
196#106—48no—
197C4⋊C8—48no—
198#102—216no—
199#102—48no—
200#95—28no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—44no—
205#95—82no—
206#95—82no—
207#95—82no—
208#95—82no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24C22×D411yescharacteristic
221C2×D4—24no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4C4⋊Q814yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—48no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
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—42no—
286C22×C4—42no—
287C22×C4—88no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4#12112yes—
292C22×C4#12112yes—
293C22×C4C22⋊Q814yes—
294C22×C4C22⋊Q814yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C22×C4H(5,𝔽2)12yes—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
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—88no—
335C22⋊C4—48no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—84no—
345C23—24no—
346C23—42no—
347C23—42no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—44no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C2×#12111yescharacteristic
369C23C2×C22⋊Q811yescharacteristic
370C23C2×C22⋊Q811yescharacteristic
371C23#52511yescharacteristic
372C23#53211yescharacteristic
373C23#53411yescharacteristic
374C23#53511yescharacteristic
375C23#57612yes—
376C23#57712yes—
377C23#57712yes—
378C23#57712yes—
379D4—28no—
380D4—44no—
381D4—44no—
382D4—44no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—816no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
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#52512yes—
438C2×C4#52912yes—
439C2×C4C2×C4⋊Q811yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4#53311yescharacteristic
442C2×C4#53311yescharacteristic
443C2×C4#53412yes—
444C2×C4#53812yes—
445C2×C4#55112yes—
446C2×C4#55312yes—
447C2×C4#55312yes—
448C2×C4#55412yes—
449C2×C4#55412yes—
450C2×C4#55412yes—
451C2×C4#55412yes—
452C2×C4#55912yes—
453C8—44no—
454C8—82no—
455C22—44no—
456C22—48no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—84no—
462C22—82no—
463C22—82no—
464C22—84no—
465C22—84no—
466C22#239211yescharacteristic
467C22#256411yescharacteristic
468C22#256611yescharacteristic
469C22#257911yescharacteristic
470C22#258011yescharacteristic
471C22#261711yescharacteristic
472C22#285711yescharacteristic
473C22#304212yes—
474C22#304512yes—
475C22#308112yes—
476C22#308112yes—
477C22#308412yes—
478C22#308412yes—
479C22#311912yes—
480C22#312512yes—
481C22#317312yes—
482C22#317912yes—
483C22#319812yes—
484C22#320012yes—
485C22#322011yescharacteristic
486C22#322111yescharacteristic
487C22#322211yescharacteristic
488C22#322211yescharacteristic
489C4—42no—
490C4—42no—
491C4—44no—
492C4—44no—
493C4—82no—
494C4—82no—
495C4—82no—
496C4—84no—
497C4—162no—
498C4#258312yes—
499C4#258712yes—
500C4#258912yes—
501C4#260212yes—
502C2—84no—
503C2#1520411yescharacteristic
504C2#2369312yes—
505C2#2917412yes—
506C2#2919911yescharacteristic
507C2#2920411yescharacteristic
508C2#2924912yes—
509C2#2926511yescharacteristic
510C2#2926911yescharacteristic
511C2#2972311yescharacteristic
512C2#2972511yescharacteristic
513C2#3320512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14277C2semidirect
4. #11323C2semidirect
5. #11268C2semidirect
6. #11231C2semidirect
10. #10884C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18642
#142772
#137852
#113232
#112682
#112312
#110762
#110512
#110492
#108842
#108332