Finite Group Databasebeta

← #413410

Subgroups — #413410

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23748C212yesmaximal
2#21156C211yesmaximal, characteristic
3#20803C211yesmaximal, characteristic
4#11290C212yesmaximal
5#11276C212yesmaximal
6#11217C212yesmaximal
7#10870C211yesmaximal, characteristic
8#10870C211yesmaximal, characteristic
9#10862C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#427C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2317C2212yes—
21#2317C2212yes—
22#2101—22no—
23#2101—22no—
24#2100—22no—
25#2099—22no—
26#1944—22no—
27#1942—21no—
28#1942—21no—
29#1847—22no—
30#1846—21no—
31#1845—21no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2211yescharacteristic
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#428—24no—
60#428—42no—
61#427—22no—
62#427—22no—
63C2×#107—21no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107D411yescharacteristic
73C2×#107D411yescharacteristic
74C2×#106—22no—
75C2×#106D412yes—
76#405—22no—
77#405—22no—
78#405—22no—
79#405Q814yes—
80C2×#97—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#385—42no—
92#385—42no—
93#384—22no—
94#384—22no—
95#382—22no—
96#382—22no—
97C2×#95—42no—
98#343—24no—
99#343—42no—
100#341—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340D411yescharacteristic
110#340D411yescharacteristic
111#340D411yescharacteristic
112#340D411yescharacteristic
113#339—22no—
114#339—42no—
115#339Q814yes—
116#336—22no—
117#336—22no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C2411yescharacteristic
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×M16C2×Q812yes—
137C2×M16C2×Q812yes—
138C2×M16#4112yes—
139C2×M16#4112yes—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×Q812yes—
154C22×C8#4112yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158C22×C8#4112yes—
159#117—21no—
160#117—21no—
161#117—21no—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—21no—
169#117C2×Q812yes—
170#117C2×Q812yes—
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—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C2×Q812yes—
205#108—44no—
206#107—22no—
207#107—22no—
208#107—44no—
209#107—42no—
210#107—42no—
211#107—42no—
212#107—42no—
213#106—24no—
214#106—44no—
215#97—24no—
216#97—42no—
217C4×C8—24no—
218C4×C8—42no—
219C4×C8—44no—
220#95—42no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—84no—
225C24H(5,𝔽2)11yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4H(5,𝔽2)11yescharacteristic
261C22×C42−1+411yescharacteristic
262C22×C42−1+411yescharacteristic
263M16—22no—
264M16—22no—
265M16—22no—
266M16—22no—
267M16—22no—
268M16—41no—
269M16—41no—
270M16—42no—
271M16—42no—
272M16—42no—
273M16—42no—
274M16#12514yes—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8C22⋊Q812yes—
295C2×C8C22⋊Q812yes—
296C2×C8#12312yes—
297C2×C8#12312yes—
298C2×C8#12412yes—
299C2×C8#12412yes—
300C2×C8#12514yes—
301C2×C8#12514yes—
302C2×C8#12614yes—
303C2×C8#12614yes—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—162no—
323C4⋊C4—162no—
324C4⋊C4—162no—
325C4⋊C4—162no—
326C4⋊C4C4⋊Q812yes—
327C4⋊C4C4⋊Q812yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C42—42no—
341C42—41no—
342C42—42no—
343C42—41no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23#54011yescharacteristic
349C23#54211yescharacteristic
350C23#54411yescharacteristic
351C23D4211yescharacteristic
352C23#55411yescharacteristic
353C23#55511yescharacteristic
354C23#55911yescharacteristic
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—162no—
389C2×C4—162no—
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×#12311yescharacteristic
392C2×C4C2×#12411yescharacteristic
393C2×C4C2×#12511yescharacteristic
394C2×C4C2×#12511yescharacteristic
395C2×C4C2×#12511yescharacteristic
396C2×C4C2×#12611yescharacteristic
397C2×C4C2×#12611yescharacteristic
398C2×C4C2×C4⋊Q811yescharacteristic
399C2×C4#53211yescharacteristic
400C2×C4#53211yescharacteristic
401C2×C4#53311yescharacteristic
402C2×C4#53311yescharacteristic
403C2×C4#54111yescharacteristic
404C2×C4#54211yescharacteristic
405C2×C4#54411yescharacteristic
406C2×C4#54611yescharacteristic
407C2×C4#54711yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4#54811yescharacteristic
410C2×C4D4×Q811yescharacteristic
411C2×C4#55011yescharacteristic
412C2×C4#55211yescharacteristic
413C2×C4#55311yescharacteristic
414C2×C4#55411yescharacteristic
415C2×C4#55611yescharacteristic
416C2×C4#55611yescharacteristic
417C2×C4#55711yescharacteristic
418C8—24no—
419C8—42no—
420C8—41no—
421C8—41no—
422C8—82no—
423C8#40014yes—
424C8#40014yes—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22#244011yescharacteristic
438C22#244511yescharacteristic
439C22#249111yescharacteristic
440C22#249711yescharacteristic
441C22#251111yescharacteristic
442C22#257111yescharacteristic
443C22#264711yescharacteristic
444C22#329011yescharacteristic
445C22#329011yescharacteristic
446C22#329211yescharacteristic
447C22#329311yescharacteristic
448C4—44no—
449C4—42no—
450C4—42no—
451C4—82no—
452C4—82no—
453C4—81no—
454C4—81no—
455C4—162no—
456C4—162no—
457C4C2×#40011yescharacteristic
458C4#244211yescharacteristic
459C4#244611yescharacteristic
460C4#249211yescharacteristic
461C4#249611yescharacteristic
462C4#251211yescharacteristic
463C4#255211yescharacteristic
464C4#285911yescharacteristic
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2#1514611yescharacteristic
470C2#2608211yescharacteristic
471C2#2608311yescharacteristic
472C2#2627111yescharacteristic
473C2#2627211yescharacteristic
474C2#2780811yescharacteristic
475C2#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
#237482
#211562
#208032
#112902
#112762
#112172
#108702
#108622
#108592
#107062