Finite Group Databasebeta

← #413016

Subgroups — #413016

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23749C212yesmaximal
2#21158C211yesmaximal, characteristic
3#21156C211yesmaximal, characteristic
4#11303C212yesmaximal
5#11296C212yesmaximal
6#11276C212yesmaximal
7#10870C211yesmaximal, characteristic
8#10870C211yesmaximal, characteristic
9#10862C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2317C2212yes—
20#2316C2212yes—
21#2101—22no—
22#2101—22no—
23#2100—22no—
24#2099—22no—
25#1944—22no—
26#1943—22no—
27#1942—21no—
28#1942—21no—
29#1942—21no—
30#1942—21no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#341C2211yescharacteristic
34#1736C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37#1736C2212yes—
38#1735C2211yescharacteristic
39C2×#340C2212yes—
40C2×#339C2211yescharacteristic
41C2×#339C2212yes—
42C2×#339C2211yescharacteristic
43#1729C2212yes—
44C4⋊#95C2212yes—
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56C2×#108—22no—
57C2×#108—22no—
58C2×#108—22no—
59#427—22no—
60#427—22no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107D411yescharacteristic
69C2×#107D411yescharacteristic
70C2×#107Q814yes—
71C2×#106D412yes—
72#405—22no—
73#405—22no—
74#405—22no—
75#405—22no—
76C2×#97—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#385—42no—
90#384—22no—
91#382—22no—
92#380—42no—
93C2×#95—42no—
94#343—24no—
95#343—24no—
96#341—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340D411yescharacteristic
108#340D411yescharacteristic
109#340D411yescharacteristic
110#340D411yescharacteristic
111#339—22no—
112#339—22no—
113#339—24no—
114#336—22no—
115#336—22no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4C2411yescharacteristic
120C2×M16—21no—
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16#4112yes—
137C2×M16#4112yes—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×Q812yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154C22×C8#4112yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157#117—22no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—21no—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—22no—
166#117—21no—
167#117—22no—
168#117—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×Q814yes—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42#4112yes—
203#108—44no—
204#107—24no—
205#107—24no—
206#107—22no—
207#107—22no—
208#107—42no—
209#107—42no—
210#106—24no—
211#97—42no—
212#97—42no—
213C4×C8—42no—
214C4×C8—42no—
215C4×C8—44no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—84no—
221C24H(5,𝔽2)11yescharacteristic
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×Q811yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C42−1+411yescharacteristic
257C22×C42−1+411yescharacteristic
258C22×C42−1+411yescharacteristic
259M16—22no—
260M16—22no—
261M16—22no—
262M16—22no—
263M16—41no—
264M16—41no—
265M16—41no—
266M16—41no—
267M16—42no—
268M16—42no—
269M16—42no—
270M16—42no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8C22⋊Q812yes—
291C2×C8C22⋊Q812yes—
292C2×C8#12312yes—
293C2×C8#12312yes—
294C2×C8#12412yes—
295C2×C8#12412yes—
296C2×C8#12514yes—
297C2×C8#12514yes—
298C2×C8#12614yes—
299C2×C8#12614yes—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—162no—
321C4⋊C4—162no—
322C4⋊C4—162no—
323C4⋊C4—162no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—42no—
336C42—41no—
337C42—42no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#54211yescharacteristic
345C23#54211yescharacteristic
346C23#54211yescharacteristic
347C23D4211yescharacteristic
348C23#55311yescharacteristic
349C23#55411yescharacteristic
350C23#55511yescharacteristic
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—162no—
385C2×C4—162no—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×#12311yescharacteristic
388C2×C4C2×#12411yescharacteristic
389C2×C4C2×#12511yescharacteristic
390C2×C4C2×#12511yescharacteristic
391C2×C4C2×#12611yescharacteristic
392C2×C4C2×#12611yescharacteristic
393C2×C4#52911yescharacteristic
394C2×C4#52911yescharacteristic
395C2×C4#52911yescharacteristic
396C2×C4#53211yescharacteristic
397C2×C4#53211yescharacteristic
398C2×C4#53211yescharacteristic
399C2×C4#54111yescharacteristic
400C2×C4#54111yescharacteristic
401C2×C4#54111yescharacteristic
402C2×C4#54711yescharacteristic
403C2×C4#54711yescharacteristic
404C2×C4#54711yescharacteristic
405C2×C4#54811yescharacteristic
406C2×C4#54811yescharacteristic
407C2×C4D4×Q811yescharacteristic
408C2×C4#55211yescharacteristic
409C2×C4#55211yescharacteristic
410C2×C4#55211yescharacteristic
411C2×C4#55611yescharacteristic
412C2×C4#55711yescharacteristic
413C2×C4#55711yescharacteristic
414C8—42no—
415C8—42no—
416C8—41no—
417C8—41no—
418C8—82no—
419C8#40014yes—
420C8#40014yes—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22#242211yescharacteristic
434C22#245211yescharacteristic
435C22#249111yescharacteristic
436C22#249511yescharacteristic
437C22#250811yescharacteristic
438C22#250911yescharacteristic
439C22#264511yescharacteristic
440C22#329011yescharacteristic
441C22#329011yescharacteristic
442C22#329211yescharacteristic
443C22#329311yescharacteristic
444C4—44no—
445C4—84no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—82no—
450C4—81no—
451C4—82no—
452C4—162no—
453C4C2×#40011yescharacteristic
454C4#242111yescharacteristic
455C4#245311yescharacteristic
456C4#248911yescharacteristic
457C4#249611yescharacteristic
458C4#250611yescharacteristic
459C4#250611yescharacteristic
460C4#286011yescharacteristic
461C2—21no—
462C2—21no—
463C2—21no—
464C2—21no—
465C2#1513911yescharacteristic
466C2#2505211yescharacteristic
467C2#2505311yescharacteristic
468C2#2650111yescharacteristic
469C2#2650211yescharacteristic
470C2#2780811yescharacteristic
471C2#2780911yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×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
#237492
#211582
#211562
#113032
#112962
#112762
#108702
#108622
#108592
#107062