Finite Group Databasebeta

← #413018

Subgroups — #413018

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23749C212yesmaximal
2#21157C211yesmaximal, characteristic
3#21156C211yesmaximal, characteristic
4#11303C212yesmaximal
5#11296C212yesmaximal
6#11276C212yesmaximal
7#10870C211yesmaximal, characteristic
8#10869C211yesmaximal, characteristic
9#10863C211yesmaximal, characteristic
10#10860C211yesmaximal, characteristic
11#10707C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2317C2212yes—
20#2316C2212yes—
21#2101—22no—
22#2101—22no—
23#2100—22no—
24#2099—22no—
25#1944—22no—
26#1944—21no—
27#1944—21no—
28#1942—22no—
29#1941—21no—
30#1941—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—22no—
65C2×#107D412yes—
66C2×#106—21no—
67C2×#106—21no—
68C2×#106—22no—
69C2×#106D411yescharacteristic
70C2×#106D411yescharacteristic
71C2×#106Q814yes—
72#405—22no—
73#405—22no—
74#405—22no—
75#405—22no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
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#340D412yes—
108#340D412yes—
109#339—22no—
110#339—22no—
111#339—24no—
112#336—22no—
113#336—22no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2411yescharacteristic
118C2×M16—21no—
119C2×M16—21no—
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×M16#4112yes—
135C2×M16#4112yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8C2×D412yes—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×Q812yes—
149C22×C8#4112yes—
150C22×C8#4112yes—
151C22×C8#4112yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154#117—22no—
155#117—21no—
156#117—21no—
157#117—21no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—22no—
162#117—22no—
163#117—21no—
164#117—22no—
165#117—21no—
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—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×Q814yes—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42#4112yes—
199#108—44no—
200#107—24no—
201#107—42no—
202#107—42no—
203#106—24no—
204#106—24no—
205#106—22no—
206#106—22no—
207#97—42no—
208#97—42no—
209#97—44no—
210C4×C8—42no—
211C4×C8—42no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95—42no—
216#95—84no—
217C24H(5,𝔽2)11yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4C22×Q811yescharacteristic
241C22×C4C2×#4111yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C42−1+411yescharacteristic
250C22×C42−1+411yescharacteristic
251C22×C42−1+411yescharacteristic
252M16—22no—
253M16—22no—
254M16—22no—
255M16—22no—
256M16—41no—
257M16—41no—
258M16—41no—
259M16—41no—
260M16—42no—
261M16—42no—
262M16—42no—
263M16—42no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8C22⋊Q812yes—
281C2×C8C22⋊Q812yes—
282C2×C8#12312yes—
283C2×C8#12312yes—
284C2×C8#12412yes—
285C2×C8#12412yes—
286C2×C8#12514yes—
287C2×C8#12514yes—
288C2×C8#12614yes—
289C2×C8#12614yes—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—162no—
309C4⋊C4—162no—
310C4⋊C4—162no—
311C4⋊C4—162no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—41no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—42no—
324C42—41no—
325C42—42no—
326C42—41no—
327C42—41no—
328C23—22no—
329C23—22no—
330C23#54211yescharacteristic
331C23#54211yescharacteristic
332C23#54211yescharacteristic
333C23D4211yescharacteristic
334C23#55311yescharacteristic
335C23#55411yescharacteristic
336C23#55511yescharacteristic
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—162no—
364C2×C4—162no—
365C2×C4C2×C22⋊Q811yescharacteristic
366C2×C4C2×#12311yescharacteristic
367C2×C4C2×#12411yescharacteristic
368C2×C4C2×#12511yescharacteristic
369C2×C4C2×#12511yescharacteristic
370C2×C4C2×#12611yescharacteristic
371C2×C4C2×#12611yescharacteristic
372C2×C4#52911yescharacteristic
373C2×C4#52911yescharacteristic
374C2×C4#52911yescharacteristic
375C2×C4#53211yescharacteristic
376C2×C4#53211yescharacteristic
377C2×C4#53211yescharacteristic
378C2×C4#54111yescharacteristic
379C2×C4#54111yescharacteristic
380C2×C4#54111yescharacteristic
381C2×C4#54711yescharacteristic
382C2×C4#54711yescharacteristic
383C2×C4#54711yescharacteristic
384C2×C4#54811yescharacteristic
385C2×C4#54811yescharacteristic
386C2×C4D4×Q811yescharacteristic
387C2×C4#55211yescharacteristic
388C2×C4#55211yescharacteristic
389C2×C4#55211yescharacteristic
390C2×C4#55611yescharacteristic
391C2×C4#55711yescharacteristic
392C2×C4#55711yescharacteristic
393C8—42no—
394C8—42no—
395C8—42no—
396C8—82no—
397C8#40014yes—
398C8#40014yes—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22#242211yescharacteristic
408C22#245211yescharacteristic
409C22#249111yescharacteristic
410C22#249511yescharacteristic
411C22#250811yescharacteristic
412C22#250911yescharacteristic
413C22#264511yescharacteristic
414C22#328812yes—
415C22#328912yes—
416C4—82no—
417C4—84no—
418C4—81no—
419C4—81no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4—162no—
424C4C2×#40011yescharacteristic
425C4#242111yescharacteristic
426C4#245311yescharacteristic
427C4#248911yescharacteristic
428C4#249611yescharacteristic
429C4#250611yescharacteristic
430C4#250611yescharacteristic
431C4#286011yescharacteristic
432C2—21no—
433C2—21no—
434C2—21no—
435C2—21no—
436C2#1513911yescharacteristic
437C2#2505512yes—
438C2#2650312yes—
439C2#2781011yescharacteristic
440C2#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
#237492
#211572
#211562
#113032
#112962
#112762
#108702
#108692
#108632
#108602
#107072