Finite Group Databasebeta

← #479229

Subgroups — #479229

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15181C212yesmaximal
2#11328C211yesmaximal, characteristic, complemented
3#11292C212yesmaximal, complemented
4#11256C211yesmaximal, characteristic, complemented
5#11200C212yesmaximal, complemented
6#11199C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10C22×C4×D4C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C2×#389C2212yes—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#340C2212yes—
21C2×#340C2211yescharacteristic
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30#1442—22no—
31#1439—22no—
32#1439—22no—
33#1438—22no—
34C4⋊#95C2212yes—
35C23×D4D411yescharacteristic
36C24×C4C2312yes—
37C23×C8C2312yes—
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—42no—
43C2×C4×D4—42no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D412yes—
55C22×C42C2311yescharacteristic
56C2×#107—42no—
57C2×#107—41no—
58C2×#107—41no—
59C2×#107—42no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—42no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8—42no—
65C2×C22⋊C8—42no—
66#389—42no—
67#389—42no—
68#380—24no—
69#380—24no—
70#380—42no—
71#380—42no—
72#380—42no—
73#380—42no—
74#380—42no—
75#380—42no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83#340—22no—
84#340—22no—
85#340—24no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#340—42no—
91#336—41no—
92#336—41no—
93C25C2×D412yes—
94C22×D4—24no—
95C22×D4—22no—
96C22×D4—42no—
97C22×D4—41no—
98C22×D4—41no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8#4112yes—
137C22×C8#4112yes—
138C4×D4—44no—
139C4×D4—48no—
140C4×D4—44no—
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—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4#4114yes—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—22no—
189#107—44no—
190#107—44no—
191C22⋊C8—48no—
192C22⋊C8—48no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—82no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24#13714yes—
215C24C22×D411yescharacteristic
216C2×D4—48no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—48no—
220C2×D4—44no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
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—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4#13512yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C4H(5,𝔽2)12yes—
299C22×C4H(5,𝔽2)12yes—
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C42−1+411yescharacteristic
303C22×C42−1+411yescharacteristic
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—44no—
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—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—88no—
339C22⋊C4—48no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C22⋊C4—88no—
343C22⋊C4—88no—
344C42—41no—
345C42—42no—
346C42—41no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23#53412yes—
383C23#53411yescharacteristic
384C23#54612yes—
385C23#54711yescharacteristic
386C23#54711yescharacteristic
387C23C2×#13511yescharacteristic
388C23C2×#13712yes—
389C23#57511yescharacteristic
390C23#57611yescharacteristic
391C23#57612yes—
392C23#57811yescharacteristic
393D4—416no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—82no—
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—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#53812yes—
448C2×C4#53812yes—
449C2×C4#53911yescharacteristic
450C2×C4#53911yescharacteristic
451C2×C4#53912yes—
452C2×C4#54012yes—
453C2×C4#54212yes—
454C2×C4#55312yes—
455C2×C4#55511yescharacteristic
456C2×C4#55512yes—
457C2×C4#55911yescharacteristic
458C2×C4#56011yescharacteristic
459C2×C4#56011yescharacteristic
460C2×C4#56012yes—
461C2×C4H(3,𝔽4)11yescharacteristic
462C2×C4#56212yes—
463C2×C4#56212yes—
464C2×C4#56311yescharacteristic
465C8—42no—
466C8—42no—
467C8—82no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22#268612yes—
484C22#269911yescharacteristic
485C22#270812yes—
486C22#272411yescharacteristic
487C22#281211yescharacteristic
488C22#319012yes—
489C22#319111yescharacteristic
490C22#319511yescharacteristic
491C22#319612yes—
492C22#319912yes—
493C22#319911yescharacteristic
494C22#320312yes—
495C22#320311yescharacteristic
496C22#330512yes—
497C22#331212yes—
498C22#331412yes—
499C22#332212yes—
500C22#334111yescharacteristic
501C22#334211yescharacteristic
502C22#334612yes—
503C22#334612yes—
504C22#335211yescharacteristic
505C22#335511yescharacteristic
506C4—42no—
507C4—84no—
508C4—82no—
509C4—81no—
510C4—162no—
511C4—162no—
512C4—162no—
513C4#278511yescharacteristic
514C4#279511yescharacteristic
515C4#280312yes—
516C4#282411yescharacteristic
517C4#282611yescharacteristic
518C4#283712yes—
519C2—84no—
520C2#1719311yescharacteristic
521C2#3132712yes—
522C2#3134812yes—
523C2#3171911yescharacteristic
524C2#3173211yescharacteristic
525C2#3200912yes—
526C2#3201812yes—
527C2#3246211yescharacteristic
528C2#3246711yescharacteristic
529C2#3271111yescharacteristic
530C2#3271511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11328C2semidirect
3. #11292C2semidirect
4. #11256C2semidirect
5. #11200C2semidirect
6. #11199C2semidirect
7. #11096C2semidirect
8. #11096C2semidirect

Maximal subgroups

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

subgroupindex
#151812
#113282
#112922
#112562
#112002
#111992
#110962
#109642