Finite Group Databasebeta

← #413411

Subgroups — #413411

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23748C212yesmaximal
2#21156C211yesmaximal, characteristic
3#20802C211yesmaximal, characteristic
4#11290C212yesmaximal
5#11276C212yesmaximal
6#11217C212yesmaximal
7#10870C211yesmaximal, characteristic
8#10869C211yesmaximal, characteristic
9#10863C211yesmaximal, characteristic
10#10860C211yesmaximal, characteristic
11#10707C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#427C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2317C2212yes—
21#2317C2212yes—
22#2101—22no—
23#2101—22no—
24#2100—22no—
25#2099—22no—
26#1944—21no—
27#1944—21no—
28#1942—22no—
29#1847—22no—
30#1846—21no—
31#1844—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—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107D412yes—
69C2×#106—21no—
70C2×#106—21no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106D411yescharacteristic
75C2×#106D411yescharacteristic
76#405—22no—
77#405—22no—
78#405—22no—
79#405Q814yes—
80C2×#97—21no—
81C2×#97—21no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
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#340D412yes—
110#340D412yes—
111#339—22no—
112#339—42no—
113#339Q814yes—
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—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16C2×Q812yes—
135C2×M16C2×Q812yes—
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×D412yes—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154C22×C8#4112yes—
155C22×C8#4112yes—
156#117—21no—
157#117—21no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—21no—
166#117C2×Q812yes—
167#117C2×Q812yes—
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—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×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C2×Q812yes—
201#108—44no—
202#107—24no—
203#107—44no—
204#107—42no—
205#107—42no—
206#106—22no—
207#106—22no—
208#106—44no—
209#106—42no—
210#106—42no—
211#97—24no—
212#97—42no—
213#97—44no—
214C4×C8—24no—
215C4×C8—42no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—84no—
221C24H(5,𝔽2)11yescharacteristic
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×Q811yescharacteristic
245C22×C4C22×Q811yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4H(5,𝔽2)11yescharacteristic
254C22×C42−1+411yescharacteristic
255C22×C42−1+411yescharacteristic
256M16—22no—
257M16—22no—
258M16—22no—
259M16—22no—
260M16—22no—
261M16—41no—
262M16—41no—
263M16—42no—
264M16—42no—
265M16—42no—
266M16—42no—
267M16#12514yes—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8C22⋊Q812yes—
285C2×C8C22⋊Q812yes—
286C2×C8#12312yes—
287C2×C8#12312yes—
288C2×C8#12412yes—
289C2×C8#12412yes—
290C2×C8#12514yes—
291C2×C8#12514yes—
292C2×C8#12614yes—
293C2×C8#12614yes—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—162no—
312C4⋊C4—162no—
313C4⋊C4—162no—
314C4⋊C4—162no—
315C4⋊C4C4⋊Q814yes—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—41no—
327C42—41no—
328C42—42no—
329C42—41no—
330C42—42no—
331C42—41no—
332C23—22no—
333C23—22no—
334C23#54011yescharacteristic
335C23#54211yescharacteristic
336C23#54411yescharacteristic
337C23D4211yescharacteristic
338C23#55411yescharacteristic
339C23#55511yescharacteristic
340C23#55911yescharacteristic
341C2×C4—24no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—162no—
368C2×C4—162no—
369C2×C4C2×C22⋊Q811yescharacteristic
370C2×C4C2×#12311yescharacteristic
371C2×C4C2×#12411yescharacteristic
372C2×C4C2×#12511yescharacteristic
373C2×C4C2×#12511yescharacteristic
374C2×C4C2×#12511yescharacteristic
375C2×C4C2×#12611yescharacteristic
376C2×C4C2×#12611yescharacteristic
377C2×C4C2×C4⋊Q811yescharacteristic
378C2×C4#53211yescharacteristic
379C2×C4#53211yescharacteristic
380C2×C4#53311yescharacteristic
381C2×C4#53311yescharacteristic
382C2×C4#54111yescharacteristic
383C2×C4#54211yescharacteristic
384C2×C4#54411yescharacteristic
385C2×C4#54611yescharacteristic
386C2×C4#54711yescharacteristic
387C2×C4#54711yescharacteristic
388C2×C4#54811yescharacteristic
389C2×C4D4×Q811yescharacteristic
390C2×C4#55011yescharacteristic
391C2×C4#55211yescharacteristic
392C2×C4#55311yescharacteristic
393C2×C4#55411yescharacteristic
394C2×C4#55611yescharacteristic
395C2×C4#55611yescharacteristic
396C2×C4#55711yescharacteristic
397C8—24no—
398C8—42no—
399C8—42no—
400C8—82no—
401C8#40014yes—
402C8#40014yes—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22#244011yescharacteristic
412C22#244511yescharacteristic
413C22#249111yescharacteristic
414C22#249711yescharacteristic
415C22#251111yescharacteristic
416C22#257111yescharacteristic
417C22#264711yescharacteristic
418C22#328812yes—
419C22#328912yes—
420C4—44no—
421C4—82no—
422C4—81no—
423C4—82no—
424C4—82no—
425C4—81no—
426C4—162no—
427C4—162no—
428C4C2×#40011yescharacteristic
429C4#244211yescharacteristic
430C4#244611yescharacteristic
431C4#249211yescharacteristic
432C4#249611yescharacteristic
433C4#251211yescharacteristic
434C4#255211yescharacteristic
435C4#285911yescharacteristic
436C2—21no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2#1514611yescharacteristic
441C2#2608412yes—
442C2#2627512yes—
443C2#2781011yescharacteristic
444C2#2781111yescharacteristic
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
#208022
#112902
#112762
#112172
#108702
#108692
#108632
#108602
#107072