Finite Group Databasebeta

← #414232

Subgroups — #414232

Order 512.

1142 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic, complemented
2C2×#1753C211yesmaximal, characteristic
3#15220C212yesmaximal
4#11328C211yesmaximal, characteristic, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11096C211yesmaximal, characteristic
9#10881C212yesmaximal, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10722C212yesmaximal
12C23×M16C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#382C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1942—24no—
25#1753—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1443—22no—
37#1441—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C22×M16—22no—
44C22×M16—42no—
45C22×M16—42no—
46C22×M16Q814yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—22no—
64C2×#107—24no—
65C2×#107—22no—
66C2×#107—42no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8Q814yes—
70C2×#97—22no—
71C2×#97—42no—
72C2×C4×C8—42no—
73C2×C4×C8D414yes—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—42no—
78#382—24no—
79#382—42no—
80#380—24no—
81#380—24no—
82#380—42no—
83#380—42no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96#340—24no—
97#340—22no—
98#340—24no—
99#340—22no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340—42no—
104#336—22no—
105#336—22no—
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×Q812yes—
125C23×C4C2×Q812yes—
126C23×C4C2411yescharacteristic
127C2×M16—28no—
128C2×M16—48no—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—48no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8C2×Q812yes—
147C22×C8C2×Q812yes—
148C22×C8C2×Q812yes—
149C22×C8C2×Q812yes—
150C22×C8#4112yes—
151C22×C8#4112yes—
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—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4#4114yes—
181C2×C4⋊C4#4114yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C2×D412yes—
199C2×C42#4112yes—
200#107—28no—
201#107—44no—
202C22⋊C8—28no—
203#97—44no—
204C4×C8—44no—
205#95—22no—
206#95—22no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—82no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24C4⋊Q812yes—
226C24C22×D411yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
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—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4C4⋊Q812yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q811yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2×#4112yes—
305C22×C4C2×#4111yescharacteristic
306C22×C4H(5,𝔽2)12yes—
307C22×C4H(5,𝔽2)12yes—
308M16—416no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8C22⋊Q814yes—
328C2×C8C22⋊Q814yes—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—88no—
346C4⋊C4—88no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—88no—
350C22⋊C4—88no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—41no—
355C42—41no—
356C42—42no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23C2×#12111yescharacteristic
381C23C2×C4⋊Q811yescharacteristic
382C23#53211yescharacteristic
383C23#54612yes—
384C23#54612yes—
385C23#57512yes—
386C23#57512yes—
387C23#57611yescharacteristic
388C23#57611yescharacteristic
389C23#57811yescharacteristic
390C23#57811yescharacteristic
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
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—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4#52511yescharacteristic
453C2×C4#52912yes—
454C2×C4#53211yescharacteristic
455C2×C4#53311yescharacteristic
456C2×C4#53311yescharacteristic
457C2×C4#53312yes—
458C2×C4#53411yescharacteristic
459C2×C4#53511yescharacteristic
460C2×C4#53812yes—
461C2×C4#54312yes—
462C2×C4#55312yes—
463C2×C4#55312yes—
464C2×C4#55312yes—
465C2×C4#55312yes—
466C2×C4#55412yes—
467C2×C4#55412yes—
468C8—24no—
469C8—24no—
470C8—42no—
471C8—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22#253511yescharacteristic
484C22#254311yescharacteristic
485C22#254912yes—
486C22#255912yes—
487C22#285711yescharacteristic
488C22#305411yescharacteristic
489C22#305411yescharacteristic
490C22#305811yescharacteristic
491C22#305911yescharacteristic
492C22#316411yescharacteristic
493C22#316411yescharacteristic
494C22#316612yes—
495C22#317511yescharacteristic
496C22#317511yescharacteristic
497C22#318312yes—
498C22#330912yes—
499C22#331312yes—
500C22#331512yes—
501C22#331912yes—
502C22#332212yes—
503C22#332212yes—
504C22#332712yes—
505C22#332712yes—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—84no—
510C4—81no—
511C4—84no—
512C4—81no—
513C4—162no—
514C4—162no—
515C4#239611yescharacteristic
516C4#256611yescharacteristic
517C4#257911yescharacteristic
518C4#258712yes—
519C4#260812yes—
520C4#262611yescharacteristic
521C2—82no—
522C2#1516511yescharacteristic
523C2#2814611yescharacteristic
524C2#2814811yescharacteristic
525C2#2856211yescharacteristic
526C2#2856411yescharacteristic
527C2#2878612yes—
528C2#2878712yes—
529C2#2906312yes—
530C2#2906512yes—
531C2#3321111yescharacteristic
532C2#3321511yescharacteristic
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
1. C2×#1942C2semidirect
4. #11328C2semidirect
5. #11256C2semidirect
6. #11208C2semidirect
9. #10881C2semidirect
10. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#17532
#152202
#113282
#112562
#112082
#110962
#108812
#108422
#107222