Finite Group Databasebeta

← #470339

Subgroups — #470339

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4#15192C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11324C212yesmaximal
8#11219C212yesmaximal
9#11067C212yesmaximal
10#11050C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×#125—21no—
13C22×#107C2212yes—
14C22×#107C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18#2552—24no—
19#2552—24no—
20C2×#398—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2095—24no—
30#1928—24no—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1445—21no—
41#1443—22no—
42#1442—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C2×#125—28no—
50C2×#125—48no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—22no—
69C2×#107—42no—
70C2×#107—42no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—22no—
77C2×#103—42no—
78C2×#103—42no—
79#398—216no—
80#391—24no—
81#391—22no—
82#391—22no—
83#391—42no—
84#389—28no—
85#389—24no—
86#389—24no—
87#384—24no—
88#384—24no—
89#384—22no—
90#384D414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—24no—
107#340—22no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—22no—
113#336—22no—
114#336—42no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8#4112yes—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8#4112yes—
152C22×C8#4114yes—
153#125—432no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—81no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4Q1614yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D414yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211#107—48no—
212#107—44no—
213#103—28no—
214#103—48no—
215#103—44no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—44no—
224#95—42no—
225#95—81no—
226#95—82no—
227C24C22×D411yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
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—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
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—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4#12012yes—
299C22×C4#12012yes—
300C22×C4C2×Q1614yes—
301C22×C4C2×Q1612yes—
302C22×C4AGL(1,ℤ/8)14yes—
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4H(5,𝔽2)12yes—
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C42−1+412yes—
312C22×C42−1+412yes—
313C22×C42−1+411yescharacteristic
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—416no—
341C4⋊C4—416no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—84no—
355C4⋊C4—88no—
356C42—22no—
357C42—22no—
358C42—24no—
359C42—48no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C23C2×#12011yescharacteristic
365C23#53612yes—
366C23#54611yescharacteristic
367C23#54711yescharacteristic
368C23#55012yes—
369C23C22×Q1611yescharacteristic
370C23C2×AGL(1,ℤ/8)12yes—
371C23C2×#13711yescharacteristic
372C23#57611yescharacteristic
373C23#57711yescharacteristic
374C23#57812yes—
375Q8—42no—
376Q8—42no—
377Q8—44no—
378Q8—81no—
379Q8—81no—
380Q8—82no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4#45114yes—
439C2×C4#52511yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4#53511yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#53612yes—
445C2×C4#53711yescharacteristic
446C2×C4#53912yes—
447C2×C4#53912yes—
448C2×C4#53912yes—
449C2×C4#54112yes—
450C2×C4#54612yes—
451C2×C4#54811yescharacteristic
452C2×C4#55011yescharacteristic
453C2×C4#55212yes—
454C2×C4#55212yes—
455C2×C4#56212yes—
456C2×C4#56312yes—
457C8—42no—
458C8—44no—
459C8—81no—
460C22#240711yescharacteristic
461C22#243111yescharacteristic
462C22#261411yescharacteristic
463C22#269412yes—
464C22#272112yes—
465C22C2×#45111yescharacteristic
466C22#300311yescharacteristic
467C22#301111yescharacteristic
468C22#301211yescharacteristic
469C22#323012yes—
470C22#323912yes—
471C22#324212yes—
472C22#324312yes—
473C22#329911yescharacteristic
474C22#330812yes—
475C22#333311yescharacteristic
476C22#333811yescharacteristic
477C22#334312yes—
478C22#335611yescharacteristic
479C22#339512yes—
480C22#339712yes—
481C22#339812yes—
482C22#339912yes—
483C4—24no—
484C4—44no—
485C4—44no—
486C4—42no—
487C4—82no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4—161no—
492C4—162no—
493C4#261111yescharacteristic
494C4#263311yescharacteristic
495C4#266411yescharacteristic
496C4#269012yes—
497C4#269412yes—
498C4#282011yescharacteristic
499C2#1668211yescharacteristic
500C2#2412211yescharacteristic
501C2#2413211yescharacteristic
502C2#2563411yescharacteristic
503C2#2563711yescharacteristic
504C2#2964211yescharacteristic
505C2#2964911yescharacteristic
506C2#3157612yes—
507C2#3159212yes—
508C2#3238012yes—
509C2#3239612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#25522
C2×#20952
C2×#19282
#151922
#113572
#113462
#113242
#112192
#110672
#110502
#108072