Finite Group Databasebeta

← #414506

Subgroups — #414506

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3#14261C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic
5#11315C212yesmaximal
6#11208C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11052C211yesmaximal, characteristic
9#10932C212yesmaximal
10#10907C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×#376C2211yescharacteristic
23#2083—24no—
24#2072—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1443—22no—
35#1442—22no—
36#1436—22no—
37#1436—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C23×Q8C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C42—22no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—24no—
57C2×#107—24no—
58C2×#107—42no—
59C2×#103—24no—
60C2×#103—24no—
61C2×#103—24no—
62C2×#103—24no—
63C2×#103—42no—
64C2×#103—42no—
65#391—22no—
66#391—22no—
67#391—22no—
68#391—22no—
69#391—24no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384D414yes—
74#382—24no—
75C4×C4⋊C4—42no—
76C4×C4⋊C4—42no—
77#376—24no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#336—24no—
98#336—22no—
99#336—22no—
100#336—24no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8#4112yes—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2411yescharacteristic
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—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8#4114yes—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4#4114yes—
167C2×C4⋊C4#4112yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—44no—
178C2×C42—82no—
179C2×C42—82no—
180C2×C42C2×D412yes—
181C2×C42#4112yes—
182C2×C42#4114yes—
183#107—48no—
184#103—48no—
185#103—48no—
186#95—24no—
187#95—24no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—44no—
192#95—42no—
193#95—42no—
194#95—42no—
195#95—44no—
196#95—82no—
197#95—82no—
198#95—82no—
199#95—82no—
200C24C22×D411yescharacteristic
201C2×Q8—24no—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—22no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
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—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
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—41no—
265C22×C4—41no—
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#12112yes—
275C22×C4#12112yes—
276C22×C4#12314yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C42−1+412yes—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42—88no—
327C23C2×#12111yescharacteristic
328C23C2×#12311yescharacteristic
329C23#52511yescharacteristic
330C23#52511yescharacteristic
331C23#53211yescharacteristic
332C23#53511yescharacteristic
333C23#53611yescharacteristic
334C23#57612yes—
335C23#57712yes—
336C23#57812yes—
337C23#57812yes—
338Q8—44no—
339Q8—44no—
340Q8—82no—
341Q8—82no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4#52912yes—
403C2×C4#52912yes—
404C2×C4#52912yes—
405C2×C4#53212yes—
406C2×C4#53912yes—
407C2×C4#53912yes—
408C2×C4#54611yescharacteristic
409C2×C4#54611yescharacteristic
410C2×C4#54812yes—
411C2×C4#55112yes—
412C2×C4#55212yes—
413C2×C4#55311yescharacteristic
414C2×C4#55311yescharacteristic
415C2×C4#55312yes—
416C2×C4#55612yes—
417C2×C4#55612yes—
418C8—44no—
419C8—82no—
420C22#239511yescharacteristic
421C22#253511yescharacteristic
422C22#254511yescharacteristic
423C22#259911yescharacteristic
424C22#260311yescharacteristic
425C22#262311yescharacteristic
426C22#285411yescharacteristic
427C22#304111yescharacteristic
428C22#304211yescharacteristic
429C22#304411yescharacteristic
430C22#304411yescharacteristic
431C22#309712yes—
432C22#309812yes—
433C22#311912yes—
434C22#312012yes—
435C22#312412yes—
436C22#312712yes—
437C22#317512yes—
438C22#317812yes—
439C22#322312yes—
440C22#322412yes—
441C22#324212yes—
442C22#324312yes—
443C4—42no—
444C4—42no—
445C4—44no—
446C4—82no—
447C4—84no—
448C4—82no—
449C4—84no—
450C4—82no—
451C4—161no—
452C4—161no—
453C4#255812yes—
454C4#256012yes—
455C4#257712yes—
456C4#259412yes—
457C2#1517411yescharacteristic
458C2#2377312yes—
459C2#2816711yescharacteristic
460C2#2817011yescharacteristic
461C2#2866311yescharacteristic
462C2#2866611yescharacteristic
463C2#2940712yes—
464C2#2951112yes—
465C2#2991812yes—
466C2#3311911yescharacteristic
467C2#3312011yescharacteristic
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×#20832
C2×#20722
#142612
#137792
#113152
#112082
#111242
#110522
#109322
#109072
#107962