Finite Group Databasebeta

← #474126

Subgroups — #474126

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1939C211yesmaximal, characteristic
2C2×#1933C211yesmaximal, characteristic
3#15193C211yesmaximal, characteristic
4#15156C211yesmaximal, characteristic
5#11228C212yesmaximal
6#11227C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11122C212yesmaximal
9#11077C212yesmaximal
10#11077C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#400—21no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1939—24no—
26#1933—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1445—22no—
35#1445—21no—
36#1444—21no—
37#1443—22no—
38#1441—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×#106—24no—
59C2×#106—24no—
60C2×#106—42no—
61C2×C4⋊C8—24no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×C4⋊C8—42no—
65C2×C4⋊C8—42no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103Q814yes—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#400—44no—
77#391—24no—
78#391—22no—
79#391—22no—
80#391—42no—
81#389—24no—
82#389—22no—
83#389—22no—
84#389—42no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95D412yes—
102#340—22no—
103#340—22no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—24no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8C2×Q812yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8C2×Q812yes—
146C22×C8#4112yes—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4SD1614yes—
182C2×C4⋊C4C2×Q812yes—
183C2×C4⋊C4#4114yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194#106—48no—
195C4⋊C8—44no—
196C4⋊C8—48no—
197#103—28no—
198#103—44no—
199#103—48no—
200C4×C8—44no—
201#95—24no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—42no—
207#95—42no—
208#95—42no—
209#95—44no—
210#95—42no—
211#95—44no—
212#95—42no—
213#95—81no—
214#95—81no—
215#95—82no—
216C24C22×D411yescharacteristic
217C2×Q8—22no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
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—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C2×SD1614yes—
293C22×C4C2×SD1612yes—
294C22×C4#13512yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)12yes—
301C22×C4H(5,𝔽2)12yes—
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C42−1+412yes—
304C22×C42−1+411yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8C22⋊Q814yes—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—88no—
343C4⋊C4—84no—
344C42—22no—
345C42—22no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C23C2×#12111yescharacteristic
352C23#53712yes—
353C23#54612yes—
354C23D4×Q811yescharacteristic
355C23#55011yescharacteristic
356C23C22×SD1611yescharacteristic
357C23C2×#13511yescharacteristic
358C23#57512yes—
359C23#57711yescharacteristic
360C23#57711yescharacteristic
361C23#57812yes—
362Q8—42no—
363Q8—42no—
364Q8—44no—
365Q8—81no—
366Q8—82no—
367Q8—81no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
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—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#46514yes—
428C2×C4C2×C22⋊Q811yescharacteristic
429C2×C4#53211yescharacteristic
430C2×C4#53411yescharacteristic
431C2×C4#53511yescharacteristic
432C2×C4#53611yescharacteristic
433C2×C4#53711yescharacteristic
434C2×C4#53912yes—
435C2×C4#53912yes—
436C2×C4#54112yes—
437C2×C4#54212yes—
438C2×C4#54412yes—
439C2×C4#55112yes—
440C2×C4#55412yes—
441C2×C4#55411yescharacteristic
442C2×C4#55611yescharacteristic
443C2×C4#55912yes—
444C2×C4#56012yes—
445C2×C4#56212yes—
446C8—24no—
447C8—42no—
448C8—82no—
449C22#253811yescharacteristic
450C22#254411yescharacteristic
451C22#261811yescharacteristic
452C22#270412yes—
453C22#270512yes—
454C22C2×#46511yescharacteristic
455C22#305411yescharacteristic
456C22#305911yescharacteristic
457C22#306011yescharacteristic
458C22#324612yes—
459C22#325012yes—
460C22#325812yes—
461C22#325812yes—
462C22#329812yes—
463C22#330212yes—
464C22#332312yes—
465C22#332612yes—
466C22Q8×SD1611yescharacteristic
467C22#338111yescharacteristic
468C22#339012yes—
469C22#339211yescharacteristic
470C22#339411yescharacteristic
471C22#340212yes—
472C4—24no—
473C4—42no—
474C4—44no—
475C4—82no—
476C4—82no—
477C4—84no—
478C4—82no—
479C4—161no—
480C4—162no—
481C4—161no—
482C4#256411yescharacteristic
483C4#262811yescharacteristic
484C4#266411yescharacteristic
485C4#279812yes—
486C4#280712yes—
487C4#281311yescharacteristic
488C2#1689911yescharacteristic
489C2#2826411yescharacteristic
490C2#2827111yescharacteristic
491C2#2860011yescharacteristic
492C2#2860411yescharacteristic
493C2#2977411yescharacteristic
494C2#2977711yescharacteristic
495C2#3186812yes—
496C2#3187312yes—
497C2#3190012yes—
498C2#3190512yes—
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×#19392
C2×#19332
#151932
#151562
#112282
#112272
#112202
#111222
#110772
#108612