Finite Group Databasebeta

← #474600

Subgroups — #474600

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14264C212yesmaximal
2#11328C212yesmaximal, complemented
3#11257C212yesmaximal, complemented
4#11220C212yesmaximal, complemented
5#11219C212yesmaximal, complemented
6#11096C211yesmaximal, characteristic
7#11096C211yesmaximal, characteristic
8#11095C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1444—22no—
30#1442—22no—
31#1439—22no—
32#1439—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C24×C4C2311yescharacteristic
36C23×C8C2312yes—
37C23×C8C2311yescharacteristic
38C23×C8C2311yescharacteristic
39C2×#117—42no—
40C2×#117—42no—
41C2×#117—41no—
42C2×#117—41no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D412yes—
54C22×C42C2312yes—
55C2×#107—24no—
56C2×#107—41no—
57C2×#107—42no—
58C2×#107—41no—
59C2×#106—24no—
60C2×#106—42no—
61C2×#106—42no—
62C2×C22⋊C8—41no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8Q814yes—
67C2×C22⋊C8Q814yes—
68#389—42no—
69#389—42no—
70#389—42no—
71#389—42no—
72#380—24no—
73#380—24no—
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—
83C2×#95—22no—
84#340—24no—
85#340—24no—
86#340—42no—
87#340—42no—
88#340—41no—
89#340—42no—
90#340—41no—
91#336—42no—
92#336—42no—
93C25C2×D411yescharacteristic
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×Q812yes—
110C23×C4C2×Q812yes—
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8C2×Q814yes—
127C22×C8C2×Q812yes—
128C22×C8C2×Q812yes—
129#117—44no—
130#117—48no—
131#117—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
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—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4#4114yes—
165C2×C4⋊C4#4114yes—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4—82no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181#107—44no—
182#107—44no—
183#106—48no—
184C22⋊C8—44no—
185C22⋊C8—44no—
186#95—28no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194C24—22no—
195C24—22no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—22no—
201C24—22no—
202C24—21no—
203C24—21no—
204C24C4⋊Q812yes—
205C24C22×D411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
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×C4C4⋊Q812yes—
271C22×C4C22×Q811yescharacteristic
272C22×C4C22×Q811yescharacteristic
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)12yes—
277C22×C4H(5,𝔽2)12yes—
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C42−1+411yescharacteristic
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C4⋊C4—88no—
318C22⋊C4—48no—
319C22⋊C4—88no—
320C22⋊C4—88no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C23—24no—
326C23—24no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23#52512yes—
349C23C2×C4⋊Q811yescharacteristic
350C23#53412yes—
351C23#53411yescharacteristic
352C23#53711yescharacteristic
353C23#57611yescharacteristic
354C23#57611yescharacteristic
355C23#57712yes—
356C23#57712yes—
357C23#57811yescharacteristic
358C23#57811yescharacteristic
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#53812yes—
413C2×C4#53912yes—
414C2×C4#54012yes—
415C2×C4#54212yes—
416C2×C4#54311yescharacteristic
417C2×C4#54311yescharacteristic
418C2×C4#54411yescharacteristic
419C2×C4#54411yescharacteristic
420C2×C4#55111yescharacteristic
421C2×C4#55312yes—
422C2×C4#55412yes—
423C2×C4#55412yes—
424C2×C4#55412yes—
425C2×C4#55412yes—
426C2×C4#55911yescharacteristic
427C2×C4#56011yescharacteristic
428C2×C4#56012yes—
429C2×C4H(3,𝔽4)11yescharacteristic
430C8—42no—
431C8—44no—
432C8—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—44no—
437C22—44no—
438C22—82no—
439C22—82no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22#258312yes—
445C22#260011yescharacteristic
446C22#263012yes—
447C22#265911yescharacteristic
448C22#281211yescharacteristic
449C22#311812yes—
450C22#311912yes—
451C22#312012yes—
452C22#312112yes—
453C22#316312yes—
454C22#316411yescharacteristic
455C22#316411yescharacteristic
456C22#319812yes—
457C22#319812yes—
458C22#319911yescharacteristic
459C22#320112yes—
460C22#320112yes—
461C22#320211yescharacteristic
462C22#320411yescharacteristic
463C22#320411yescharacteristic
464C22#326012yes—
465C22#326211yescharacteristic
466C22#326211yescharacteristic
467C4—42no—
468C4—84no—
469C4—82no—
470C4—82no—
471C4—84no—
472C4—162no—
473C4—162no—
474C4#274811yescharacteristic
475C4#276311yescharacteristic
476C4#278012yes—
477C4#279012yes—
478C4#280411yescharacteristic
479C4#280711yescharacteristic
480C2—82no—
481C2#1698711yescharacteristic
482C2#2930812yes—
483C2#2930912yes—
484C2#2942111yescharacteristic
485C2#2942611yescharacteristic
486C2#3003212yes—
487C2#3003412yes—
488C2#3030011yescharacteristic
489C2#3030311yescharacteristic
490C2#3271211yescharacteristic
491C2#3271411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11328C2semidirect
3. #11257C2semidirect
4. #11220C2semidirect
5. #11219C2semidirect

Maximal subgroups

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

subgroupindex
#142642
#113282
#112572
#112202
#112192
#110962
#110952
#107132