Finite Group Databasebeta

← #412924

Subgroups — #412924

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23671C212yesmaximal
2C2×#1951C211yesmaximal, characteristic
3C2×#1945C211yesmaximal, characteristic
4#21157C211yesmaximal, characteristic
5#20802C211yesmaximal, characteristic
6#11342C212yesmaximal
7#11276C212yesmaximal
8#11216C212yesmaximal
9#10869C211yesmaximal, characteristic
10#10861C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#424C2212yes—
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2317C2212yes—
23#2316C2212yes—
24#2088—22no—
25#2088—22no—
26#2087—22no—
27#2086—22no—
28#1951—24no—
29#1945—24no—
30#1944—22no—
31#1941—21no—
32#1941—21no—
33#1847—22no—
34#1846—21no—
35#1844—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2211yescharacteristic
39#1736C2212yes—
40#1736C2212yes—
41#1735C2211yescharacteristic
42C2×#340C2212yes—
43C2×#339C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C4⋊#95C2212yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2312yes—
54C2×#117C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C2×#108—22no—
59C2×#108—22no—
60C2×#108D414yes—
61#428—24no—
62#428—24no—
63#428—42no—
64#428—42no—
65#427—22no—
66#427—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107D412yes—
70C2×#106—21no—
71C2×#106—21no—
72C2×#106—22no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106D411yescharacteristic
80C2×#106D411yescharacteristic
81C2×#106Q814yes—
82#424—24no—
83#424—42no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86#405—22no—
87#405—22no—
88#405—22no—
89#405Q814yes—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#385—42no—
104#384—22no—
105#382—22no—
106#380—42no—
107C2×#95—42no—
108#343—24no—
109#343—24no—
110#341—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340D412yes—
118#340D412yes—
119#339—24no—
120#339—22no—
121#339Q814yes—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2411yescharacteristic
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—22no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—21no—
139C2×M16C2×D414yes—
140C2×M16C2×Q812yes—
141C2×M16C2×Q812yes—
142C2×M16#4112yes—
143C2×M16#4112yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D412yes—
157C22×C8C2×Q812yes—
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162#117—21no—
163#117—21no—
164#117—22no—
165#117—21no—
166#117—21no—
167#117—21no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—21no—
172#117C2×Q812yes—
173#117C2×Q812yes—
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—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×Q814yes—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42C2×Q812yes—
207#108—28no—
208#107—24no—
209#107—44no—
210#107—44no—
211#106—22no—
212#106—22no—
213#106—24no—
214#106—24no—
215#106—44no—
216#106—44no—
217#106—42no—
218#106—42no—
219C4⋊C8—44no—
220C4⋊C8—44no—
221#97—24no—
222#97—42no—
223C4×C8—24no—
224C4×C8—42no—
225C4×C8—44no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—84no—
231C24H(5,𝔽2)11yescharacteristic
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×Q811yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C22×Q811yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C42−1+411yescharacteristic
265C22×C42−1+411yescharacteristic
266M16—22no—
267M16—22no—
268M16—22no—
269M16—24no—
270M16—22no—
271M16—24no—
272M16—22no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16—42no—
277M16#12514yes—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8#12112yes—
295C2×C8#12112yes—
296C2×C8C22⋊Q812yes—
297C2×C8C22⋊Q812yes—
298C2×C8C22⋊Q812yes—
299C2×C8C22⋊Q812yes—
300C2×C8#12314yes—
301C2×C8#12514yes—
302C2×C8#12514yes—
303C2×C8C4⋊Q814yes—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
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—
324C4⋊C4C4⋊Q812yes—
325C4⋊C4C4⋊Q812yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—41no—
337C42—41no—
338C42—42no—
339C42—42no—
340C42—41no—
341C42—41no—
342C23—22no—
343C23—22no—
344C23#54011yescharacteristic
345C23#54211yescharacteristic
346C23#54411yescharacteristic
347C23D4211yescharacteristic
348C23#55011yescharacteristic
349C23#55411yescharacteristic
350C23#55411yescharacteristic
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—162no—
378C2×C4—162no—
379C2×C4C2×#12111yescharacteristic
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4C2×#12311yescharacteristic
383C2×C4C2×#12511yescharacteristic
384C2×C4C2×#12511yescharacteristic
385C2×C4C2×#12511yescharacteristic
386C2×C4C2×C4⋊Q811yescharacteristic
387C2×C4C2×C4⋊Q811yescharacteristic
388C2×C4#53211yescharacteristic
389C2×C4#53211yescharacteristic
390C2×C4#53311yescharacteristic
391C2×C4#53311yescharacteristic
392C2×C4#54111yescharacteristic
393C2×C4#54211yescharacteristic
394C2×C4#54411yescharacteristic
395C2×C4D4211yescharacteristic
396C2×C4#54611yescharacteristic
397C2×C4#54711yescharacteristic
398C2×C4#54711yescharacteristic
399C2×C4#54811yescharacteristic
400C2×C4D4×Q811yescharacteristic
401C2×C4D4×Q811yescharacteristic
402C2×C4D4×Q811yescharacteristic
403C2×C4#55211yescharacteristic
404C2×C4#55311yescharacteristic
405C2×C4#55711yescharacteristic
406C2×C4Q8211yescharacteristic
407C8—24no—
408C8—42no—
409C8—44no—
410C8—42no—
411C8#39814yes—
412C8#39814yes—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22#242311yescharacteristic
422C22#242811yescharacteristic
423C22#244011yescharacteristic
424C22#245211yescharacteristic
425C22#250711yescharacteristic
426C22#255211yescharacteristic
427C22#264711yescharacteristic
428C22#329012yes—
429C22#329112yes—
430C4—44no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—84no—
437C4—162no—
438C4C2×#39811yescharacteristic
439C4#242211yescharacteristic
440C4#242911yescharacteristic
441C4#244211yescharacteristic
442C4#245111yescharacteristic
443C4#250611yescharacteristic
444C4#255111yescharacteristic
445C4#285911yescharacteristic
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#1513711yescharacteristic
451C2#2544012yes—
452C2#2608011yescharacteristic
453C2#2608111yescharacteristic
454C2#2650012yes—
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
#236712
C2×#19512
C2×#19452
#211572
#208022
#113422
#112762
#112162
#108692
#108612
#107062