Finite Group Databasebeta

← #365659

Subgroups — #365659

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2C2×#1994C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic
5#14687C211yesmaximal, characteristic
6#11276C214yesmaximal
7#10899C214yesmaximal
8#10857C211yesmaximal, characteristic
9C4×#339C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2360—22no—
15#2360—22no—
16#2355—21no—
17#2355—21no—
18Q8×C4⋊C4—21no—
19Q8×C4⋊C4—21no—
20#2330—22no—
21#2317C2212yes—
22C4×C4⋊Q8—21no—
23C4×C4⋊Q8—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382—22no—
27#2274C2212yes—
28C4×#117C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#1994C418yescomplemented
31#1992—22no—
32#1990—21no—
33#1990—21no—
34#1948—21no—
35#1948—21no—
36#1947—21no—
37#1947—21no—
38C2×#343C2214yes—
39#1736C2212yes—
40#1736C2214yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2214yes—
44C4⋊#95—22no—
45C4⋊#95C2214yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533Q814yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125D412yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×#117C2311yescharacteristic
62C2×#117C2312yes—
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#428—28no—
75C2×#107—22no—
76C2×#106—22no—
77#423—22no—
78#423C2×C414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C4×M16—22no—
82C4×M16—22no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#391—22no—
86#391—22no—
87#389—22no—
88#389—22no—
89#389—24no—
90#389—24no—
91#389—22no—
92#389—22no—
93#387—24no—
94#387—24no—
95#385—24no—
96#384—22no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105#382—22no—
106#382—48no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—24no—
120#376—21no—
121C2×#95—28no—
122C43—21no—
123C43—22no—
124C43C2×C414yes—
125#343—28no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—24no—
130#339—24no—
131#339—24no—
132#339C2×C4116yes—
133C22×Q8—21no—
134C22×Q8—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D414yes—
140C23×C4C2411yescharacteristic
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—24no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16C22×C414yes—
150C2×M16#4114yes—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8C4⋊C414yes—
156C4⋊Q8C4⋊C414yes—
157#125—24no—
158#125C4⋊C418yes—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—84no—
168C4×Q8—84no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117C22×C414yes—
175#117C2×Q814yes—
176#117#4114yes—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C414yes—
233C2×C42C22×C412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#107—44no—
242#106—44no—
243C4⋊C8—24no—
244C4⋊C8—24no—
245#97—42no—
246C4×C8—42no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—416no—
256C24C22×D411yescharacteristic
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—28no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q811yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4112yes—
302C22×C4C2×#4112yes—
303C22×C4C2×#4111yescharacteristic
304C22×C42−1+412yes—
305C22×C42−1+412yes—
306M16—28no—
307M16—24no—
308M16—24no—
309M16—44no—
310M16—42no—
311M16—42no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—84no—
343C4⋊C4—84no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4C2×C4⋊C414yes—
347C4⋊C4C2×C4⋊C414yes—
348C4⋊C4#12412yes—
349C4⋊C4#12412yes—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—42no—
354C42—216no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—24no—
360C42—48no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—84no—
369C42—84no—
370C42C2×C4⋊C412yes—
371C42C2×C4⋊C412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42C22⋊Q812yes—
375C42C22⋊Q812yes—
376C42#12312yes—
377C42#12312yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23C2×C4×D411yescharacteristic
383C23C2×#12011yescharacteristic
384C23C2×C22⋊Q811yescharacteristic
385C23#54712yes—
386C23#54812yes—
387Q8—48no—
388Q8—48no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—88no—
426C2×C4—88no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#38512yes—
431C2×C4#38512yes—
432C2×C4#38512yes—
433C2×C4#38512yes—
434C2×C4#45312yes—
435C2×C4#45412yes—
436C2×C4#45512yes—
437C2×C4#45612yes—
438C2×C4C22×C4⋊C411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C4×#4112yes—
441C2×C4#51912yes—
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C2×#12411yescharacteristic
445C2×C4#52912yes—
446C2×C4#53211yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#54112yes—
450C2×C4#54812yes—
451C2×C4#54812yes—
452C2×C4#55212yes—
453C2×C4#55612yes—
454C2×C4#55712yes—
455C2×C4#55712yes—
456C8—44no—
457C8—44no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22C2×#38511yescharacteristic
467C22#236212yes—
468C22#243211yescharacteristic
469C22#243611yescharacteristic
470C22#245312yes—
471C22C2×#45311yescharacteristic
472C22C2×#45511yescharacteristic
473C22C2×#45411yescharacteristic
474C22C2×#45611yescharacteristic
475C4—22no—
476C4—22no—
477C4—48no—
478C4—42no—
479C4—42no—
480C4—44no—
481C4—44no—
482C4—81no—
483C4—84no—
484C4—81no—
485C4—162no—
486C4#185914yes—
487C4#186014yes—
488C4C4×#12411yescharacteristic
489C4#232711yescharacteristic
490C4#235112yes—
491C4#242112yes—
492C4#256511yescharacteristic
493C4#256811yescharacteristic
494C2—22no—
495C2—22no—
496C2#1397811yescharacteristic
497C2C2×#185911yescharacteristic
498C2C2×#186011yescharacteristic
499C2#2568711yescharacteristic
500C2#2568811yescharacteristic
501C2#2592311yescharacteristic
502C2#2592411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1994C4semidirect

Maximal subgroups

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

subgroupindex
#342142
C2×#19942
#213222
#211712
#146872
#112762
#108992
#108572
C4×#3392