Finite Group Databasebeta

← #360943

Subgroups — #360943

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21039C212yesmaximal
2#20803C211yesmaximal, characteristic
3#20455C211yesmaximal, characteristic
4C2×#1772C211yesmaximal, characteristic
5#11276C212yesmaximal
6#11249C212yesmaximal
7#10899C212yesmaximal
8#10870C211yesmaximal, characteristic
9#10870C211yesmaximal, characteristic
10#10862C211yesmaximal, characteristic
11#10859C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C2×#427C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2317C2212yes—
23#2274C2212yes—
24#1921—22no—
25#1920—22no—
26#1919—22no—
27#1918—22no—
28#1847—22no—
29#1846—21no—
30#1845—21no—
31#1776—22no—
32C4×#107—21no—
33C4×#107—21no—
34#1772C418yescomplemented
35C2×#343C2212yes—
36C2×#343C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41C2×#339C2212yes—
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44#1729C2212yes—
45C4⋊#95C2212yes—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57C2×#108—22no—
58C2×#108—22no—
59C2×#108—22no—
60C2×#108—22no—
61C2×#108—22no—
62C2×#108C2×C414yes—
63#428—24no—
64#428—42no—
65#427—22no—
66#427—22no—
67#427C2×C414yes—
68#427Q814yes—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—22no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107D411yescharacteristic
78C2×#107D411yescharacteristic
79C2×#107D411yescharacteristic
80C2×#107D411yescharacteristic
81C2×#107Q814yes—
82C2×#106—22no—
83C2×#106D412yes—
84C2×#106D411yescharacteristic
85C2×#106D411yescharacteristic
86#405—22no—
87#405—22no—
88#405C2×C414yes—
89#405C2×C414yes—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8C2×C414yes—
97#389—22no—
98#389—22no—
99#385—42no—
100#384—22no—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C22⋊C4—42no—
107C2×#95—42no—
108#343—24no—
109#343—42no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#339—42no—
119#339C2×C414yes—
120#339C2×C414yes—
121#336—22no—
122#336—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2411yescharacteristic
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16C22×C412yes—
142C2×M16C22×C412yes—
143C2×M16#4112yes—
144C2×M16#4112yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8C22×C412yes—
153C22×C8C22×C412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×Q812yes—
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C22×C8#4112yes—
163C22×C8#4112yes—
164#117—21no—
165#117—21no—
166#117—21no—
167#117—21no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117C22×C412yes—
173#117C22×C412yes—
174#117C2×Q812yes—
175#117C2×Q812yes—
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—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×Q814yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C22×C412yes—
210#108—28no—
211#108—44no—
212#107—24no—
213#107—24no—
214#107—22no—
215#107—22no—
216#107—44no—
217#107—42no—
218#107—42no—
219#107C4⋊C414yes—
220#107C4⋊C414yes—
221#106—24no—
222#106—44no—
223#106C4⋊C414yes—
224#106C4⋊C414yes—
225#97—24no—
226#97—24no—
227C4×C8—28no—
228C4×C8—24no—
229C4×C8—24no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—84no—
235C24H(5,𝔽2)11yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×Q811yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C42−1+411yescharacteristic
272C22×C42−1+411yescharacteristic
273M16—22no—
274M16—22no—
275M16—22no—
276M16—22no—
277M16—22no—
278M16—41no—
279M16—41no—
280M16—42no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16#11714yes—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8C2×C4⋊C412yes—
303C2×C8C2×C4⋊C412yes—
304C2×C8#11714yes—
305C2×C8C4×D412yes—
306C2×C8C4×D412yes—
307C2×C8C4×Q814yes—
308C2×C8C22⋊Q812yes—
309C2×C8C22⋊Q812yes—
310C2×C8#12312yes—
311C2×C8#12312yes—
312C2×C8#12514yes—
313C2×C8#12614yes—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—162no—
331C4⋊C4—162no—
332C4⋊C4C2×C4⋊C412yes—
333C4⋊C4C2×C4⋊C412yes—
334C4⋊C4C2×C4⋊C412yes—
335C4⋊C4C2×C4⋊C412yes—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—162no—
353C42—162no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23#52011yescharacteristic
359C23#54011yescharacteristic
360C23#54211yescharacteristic
361C23D4211yescharacteristic
362C23#55411yescharacteristic
363C23#55411yescharacteristic
364C23#55911yescharacteristic
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—162no—
399C2×C4—162no—
400C2×C4C22×C4⋊C411yescharacteristic
401C2×C4C2×#11711yescharacteristic
402C2×C4C2×#11711yescharacteristic
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×C4×Q811yescharacteristic
405C2×C4#52011yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#12511yescharacteristic
409C2×C4C2×#12611yescharacteristic
410C2×C4#53211yescharacteristic
411C2×C4#53211yescharacteristic
412C2×C4#53311yescharacteristic
413C2×C4#53311yescharacteristic
414C2×C4#54111yescharacteristic
415C2×C4#54211yescharacteristic
416C2×C4#54611yescharacteristic
417C2×C4#54711yescharacteristic
418C2×C4#54811yescharacteristic
419C2×C4D4×Q811yescharacteristic
420C2×C4D4×Q811yescharacteristic
421C2×C4#55011yescharacteristic
422C2×C4#55211yescharacteristic
423C2×C4#55311yescharacteristic
424C2×C4#55611yescharacteristic
425C2×C4#55711yescharacteristic
426C2×C4#55711yescharacteristic
427C2×C4Q8211yescharacteristic
428C8—24no—
429C8—22no—
430C8—22no—
431C8—42no—
432C8—82no—
433C8#38714yes—
434C8#38714yes—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22#231411yescharacteristic
448C22D4×C4⋊C411yescharacteristic
449C22#235511yescharacteristic
450C22#244511yescharacteristic
451C22#245211yescharacteristic
452C22#250711yescharacteristic
453C22#251111yescharacteristic
454C22#329011yescharacteristic
455C22#329011yescharacteristic
456C22#329211yescharacteristic
457C22#329311yescharacteristic
458C4—44no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—84no—
464C4—82no—
465C4—82no—
466C4—162no—
467C4#228511yescharacteristic
468C4C2×#38711yescharacteristic
469C4Q8×C4⋊C411yescharacteristic
470C4#235211yescharacteristic
471C4#244611yescharacteristic
472C4#245111yescharacteristic
473C4#250611yescharacteristic
474C4#251211yescharacteristic
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2#1388411yescharacteristic
480C2#2269111yescharacteristic
481C2#2269211yescharacteristic
482C2#2627111yescharacteristic
483C2#2627211yescharacteristic
484C2#2650111yescharacteristic
485C2#2650211yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
34. #1772C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#210392
#208032
#204552
C2×#17722
#112762
#112492
#108992
#108702
#108622
#108592