Finite Group Databasebeta

← #410434

Subgroups — #410434

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1927C212yesmaximal
2#14261C211yesmaximal, characteristic
3#13779C211yesmaximal, characteristic
4#11323C211yesmaximal, characteristic
5#11268C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11190C212yesmaximal
8#11052C211yesmaximal, characteristic
9#11050C211yesmaximal, characteristic
10#10807C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#1927—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1443—22no—
34#1442—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×Q8C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C42—22no—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#106—24no—
55C2×#106—24no—
56C2×#106—42no—
57C2×#103—24no—
58C2×#103—24no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—42no—
63#391—22no—
64#391—22no—
65#391—22no—
66#391—42no—
67#391D414yes—
68#389—42no—
69#384—24no—
70#382—24no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73#376—22no—
74#376—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87C2×#95D412yes—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—42no—
92#340—42no—
93#336—24no—
94#336—24no—
95#336—22no—
96#336—22no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8C2×D412yes—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
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#4114yes—
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—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4#4114yes—
160C2×C4⋊C4#4114yes—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—44no—
172C2×C42—82no—
173C2×C42—82no—
174C2×C42C2×D412yes—
175C2×C42#4112yes—
176C2×C42#4112yes—
177C2×C42#4112yes—
178#106—48no—
179#103—216no—
180#103—48no—
181#95—28no—
182#95—28no—
183#95—24no—
184#95—24no—
185#95—42no—
186#95—42no—
187#95—42no—
188#95—44no—
189#95—44no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194C24C22×D411yescharacteristic
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
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—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4#12114yes—
270C22×C4#12314yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4H(5,𝔽2)12yes—
279C22×C42−1+411yescharacteristic
280C22×C42−1+411yescharacteristic
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—88no—
310C4⋊C4—84no—
311C42—24no—
312C42—24no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—88no—
320C23C2×#12111yescharacteristic
321C23C2×#12311yescharacteristic
322C23#52511yescharacteristic
323C23#52511yescharacteristic
324C23#53211yescharacteristic
325C23#53511yescharacteristic
326C23#53611yescharacteristic
327C23#57712yes—
328C23#57712yes—
329C23#57712yes—
330C23#57812yes—
331Q8—44no—
332Q8—44no—
333Q8—82no—
334Q8—82no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4#52512yes—
396C2×C4#52912yes—
397C2×C4#52912yes—
398C2×C4#52912yes—
399C2×C4#54212yes—
400C2×C4#54212yes—
401C2×C4#54612yes—
402C2×C4#54711yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4#55212yes—
405C2×C4#55211yescharacteristic
406C2×C4#55211yescharacteristic
407C2×C4#55312yes—
408C2×C4#55312yes—
409C2×C4#55512yes—
410C2×C4#55612yes—
411C8—44no—
412C8—82no—
413C22#239211yescharacteristic
414C22#248611yescharacteristic
415C22#250011yescharacteristic
416C22#253511yescharacteristic
417C22#260411yescharacteristic
418C22#263211yescharacteristic
419C22#285311yescharacteristic
420C22#304312yes—
421C22#304612yes—
422C22#309712yes—
423C22#309912yes—
424C22#312012yes—
425C22#312112yes—
426C22#312312yes—
427C22#312512yes—
428C22#317312yes—
429C22#317812yes—
430C22#322111yescharacteristic
431C22#322111yescharacteristic
432C22#322211yescharacteristic
433C22#322311yescharacteristic
434C22#324212yes—
435C22#324412yes—
436C4—44no—
437C4—44no—
438C4—84no—
439C4—82no—
440C4—84no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—161no—
445C4—161no—
446C4#246812yes—
447C4#247412yes—
448C4#252512yes—
449C4#252912yes—
450C2#1507811yescharacteristic
451C2#2369412yes—
452C2#2765712yes—
453C2#2789811yescharacteristic
454C2#2790211yescharacteristic
455C2#2816611yescharacteristic
456C2#2817311yescharacteristic
457C2#2952512yes—
458C2#3007311yescharacteristic
459C2#3007411yescharacteristic
460C2#3310512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19272
#142612
#137792
#113232
#112682
#112202
#111902
#110522
#110502
#108072
#107962