Finite Group Databasebeta

← #380286

Subgroups — #380286

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2#20846C212yesmaximal
3C2×#1822C211yesmaximal, characteristic
4#13939C211yesmaximal, characteristic
5#11342C212yesmaximal, complemented
6#11342C212yesmaximal, complemented
7#11296C212yesmaximal, complemented
8#11296C212yesmaximal, complemented
9#10733C212yesmaximal
10C22×#117C2212yes—
11C22×#117C2211yescharacteristic
12C2×#428C2214yes—
13C2×#407C2212yes—
14#2317—22no—
15#2317—22no—
16#2316C2212yes—
17#2316C2212yes—
18#2315—22no—
19#2296—24no—
20#2285—22no—
21C2×#380C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1854—24no—
24#1853—22no—
25#1852—22no—
26#1822C418yescomplemented
27C2×#343C2212yes—
28C2×#343C2212yes—
29#1736C2214yes—
30C2×#339C2214yes—
31C2×#339C2214yes—
32#1440—22no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37#1434—22no—
38C24×C4C2311yescharacteristic
39C22×M16C2314yes—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—24no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—44no—
46C2×#117—44no—
47C2×#117C2×C414yes—
48C2×#117D414yes—
49C2×#117Q814yes—
50C2×#117C2314yes—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4D412yes—
61C22×C42—22no—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#428—44no—
66#428—44no—
67C2×#107—24no—
68C2×#106—24no—
69#407—24no—
70#407C2×C418yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—24no—
77#385—28no—
78#385—28no—
79#384—22no—
80#384—22no—
81#382—24no—
82#380—24no—
83#380—24no—
84#380—22no—
85#380—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C22⋊C4—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95#343C2×C418yescomplemented
96#343C2×C418yescomplemented
97#340—24no—
98#340—24no—
99#339—44no—
100#339—44no—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4C22×C412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×Q812yes—
119C23×C4#4114yes—
120C23×C4C2411yescharacteristic
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16C22×C418yes—
127C22×C8—24no—
128#117—28no—
129#117—28no—
130#117—28no—
131#117—28no—
132#117—24no—
133#117—24no—
134#117—24no—
135#117—24no—
136#117—44no—
137#117—44no—
138#117—44no—
139#117—44no—
140#117—48no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×Q812yes—
164C2×C4⋊C4C2×Q812yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4C4⋊C418yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—48no—
189C2×C42C4⋊C418yes—
190#107—48no—
191#106—48no—
192C22⋊C8—28no—
193C22⋊C8—28no—
194#95—28no—
195#95—28no—
196#95—44no—
197#95—44no—
198#95—41no—
199#95—41no—
200#95—42no—
201#95—82no—
202#95—82no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24C2×C4⋊C412yes—
212C24#12012yes—
213C24#12312yes—
214C24C22×D411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—28no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C2×C4⋊C412yes—
270C22×C4C2×C4⋊C414yes—
271C22×C4C2×C4⋊C414yes—
272C22×C4C4×D414yes—
273C22×C4C4×D414yes—
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q814yes—
277C22×C4C22⋊Q814yes—
278C22×C4#12312yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C22×Q811yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286C22×C42−1+412yes—
287M16—28no—
288M16—28no—
289M16—44no—
290M16—44no—
291C2×C8—28no—
292C2×C8—44no—
293C2×C8—44no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C22⋊C4—216no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—84no—
316C42—28no—
317C42—28no—
318C42—48no—
319C42—44no—
320C42—44no—
321C42—44no—
322C42—44no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23#38514yes—
350C23C22×C4⋊C411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23#53511yescharacteristic
354C23#53511yescharacteristic
355C23#53611yescharacteristic
356C23#53611yescharacteristic
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—88no—
398C2×C4—88no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4#38512yes—
412C2×C4#38512yes—
413C2×C4C2×C4×D412yes—
414C2×C4#52012yes—
415C2×C4#52012yes—
416C2×C4C2×C22⋊Q812yes—
417C2×C4#53414yes—
418C2×C4#53614yes—
419C2×C4#53914yes—
420C2×C4#53914yes—
421C2×C4#54412yes—
422C2×C4#54412yes—
423C8—48no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22C2×#38511yescharacteristic
448C22#231711yescharacteristic
449C22#231711yescharacteristic
450C22#261311yescharacteristic
451C22#261411yescharacteristic
452C22#262311yescharacteristic
453C22#262311yescharacteristic
454C22#301811yescharacteristic
455C22#301811yescharacteristic
456C22#302111yescharacteristic
457C22#302211yescharacteristic
458C4—22no—
459C4—22no—
460C4—44no—
461C4—44no—
462C4—82no—
463C4—82no—
464C4—84no—
465C4—84no—
466C4—161no—
467C4—161no—
468C4#232214yes—
469C4#262714yes—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—44no—
475C2#1426311yescharacteristic
476C2#2085511yescharacteristic
477C2#2085811yescharacteristic
478C2#2962011yescharacteristic
479C2#2962111yescharacteristic
480C2#2965411yescharacteristic
481C2#2965511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11342C2semidirect
6. #11342C2semidirect
7. #11296C2semidirect
8. #11296C2semidirect
26. #1822C4semidirect
95. #343C2×C4semidirect
96. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340522
#208462
C2×#18222
#139392
#113422
#112962
#107332