Finite Group Databasebeta

← #379953

Subgroups — #379953

Order 512.

1588 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic, complemented
3#21261C212yesmaximal
4C2×#1801C211yesmaximal, characteristic, complemented
5#11055C212yesmaximal, complemented
6#11054C212yesmaximal
7#10978C212yesmaximal, complemented
8#10976C212yesmaximal
9#10852C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#124C2211yescharacteristic
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#423C2212yes—
15C2×#419C2214yes—
16C2×#418C2214yes—
17C22×#101C2212yes—
18#2545—24no—
19#2545C418yescomplemented
20C2×#396—22no—
21#2320—24no—
22#2316C2212yes—
23#2313—22no—
24#2310C2212yes—
25#2295—24no—
26C2×#391C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#388—22no—
29#2283—22no—
30C4×#117C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1983—24no—
33#1983—24no—
34#1801—24no—
35C2×#343C2212yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×Q8C2311yescharacteristic
39C23×Q8C2311yescharacteristic
40C23×D4C2311yescharacteristic
41C22×M16C2314yes—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—44no—
44C2×C4⋊Q8C2×C414yes—
45C2×#124—22no—
46C2×#124—22no—
47C2×#124—22no—
48C2×#124—22no—
49C2×#124—24no—
50C2×#124—44no—
51C2×#124C2×C414yes—
52C2×#124C2×C414yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8D414yes—
58C2×#117C2314yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#423—24no—
72#423—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75#419—44no—
76#419—44no—
77#418—44no—
78#418—44no—
79C2×#103—28no—
80C2×#102—28no—
81C2×#101—24no—
82C2×#101—44no—
83#396—216no—
84#391—28no—
85#391—22no—
86#391C2×C414yes—
87#390—22no—
88#390C2×C414yes—
89#389—28no—
90#388—216no—
91#387—216no—
92#383—24no—
93#382—24no—
94#381—24no—
95#380—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4D414yes—
100C4×C4⋊C4D414yes—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103#376—24no—
104#376—22no—
105C2×#95—22no—
106C2×#95—21no—
107C43—22no—
108#343—24no—
109#339—44no—
110#339—44no—
111C25—21no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22×C412yes—
124C22×Q8C22×C412yes—
125C22×Q8C2×D412yes—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C22×C412yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—24no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—24no—
147C22×C8—24no—
148C4⋊Q8—48no—
149C4⋊Q8—48no—
150#124—28no—
151#124—28no—
152#124—48no—
153#124—48no—
154#124—44no—
155#124—44no—
156C4×Q8—28no—
157C4×Q8—28no—
158C4×Q8—28no—
159C4×Q8—28no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4C22⋊C418yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D418yes—
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—48no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42C22⋊C418yes—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42#4114yes—
207C4⋊C8—44no—
208C4⋊C8—44no—
209#103—416no—
210#102—416no—
211#101—48no—
212#95—28no—
213#95—28no—
214#95—28no—
215#95—24no—
216#95—28no—
217#95—24no—
218#95—24no—
219C24—22no—
220C24—44no—
221C24—44no—
222C24—41no—
223C24—41no—
224C24—42no—
225C24—42no—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4#12011yescharacteristic
288C22×C4#12012yes—
289C22×C4#12011yescharacteristic
290C22×C4#12312yes—
291C22×C4#12312yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)14yes—
297C22×C4H(5,𝔽2)12yes—
298C22×C42−1+412yes—
299M16—44no—
300M16—44no—
301M16—44no—
302M16—44no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—216no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—48no—
312C4⋊C4—416no—
313C4⋊C4—88no—
314C22⋊C4—416no—
315C22⋊C4—416no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—48no—
321C42—28no—
322C42—28no—
323C42—24no—
324C42—24no—
325C42—48no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42—84no—
331C42C4×D414yes—
332C42C4×D414yes—
333C42#12114yes—
334C42#12114yes—
335C23—22no—
336C23—22no—
337C23—44no—
338C23—44no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—82no—
343C23—82no—
344C23—88no—
345C23—88no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23C22×C22⋊C411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23#53512yes—
354C23#53712yes—
355Q8—48no—
356Q8—44no—
357Q8—44no—
358Q8—48no—
359Q8—82no—
360Q8—82no—
361D4—48no—
362D4—82no—
363D4—82no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#38614yes—
399C2×C4C2×C4×D412yes—
400C2×C4#51812yes—
401C2×C4#52012yes—
402C2×C4C2×#12112yes—
403C2×C4#53514yes—
404C2×C4#53511yescharacteristic
405C2×C4#53512yes—
406C2×C4#53611yescharacteristic
407C2×C4#53714yes—
408C2×C4#54014yes—
409C2×C4#54214yes—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—24no—
417C22—84no—
418C22—84no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—164no—
423C22—164no—
424C22C2×#38611yescharacteristic
425C22#230911yescharacteristic
426C22#231111yescharacteristic
427C22#261512yes—
428C22#262412yes—
429C22#301812yes—
430C22#302012yes—
431C4—24no—
432C4—22no—
433C4—22no—
434C4—44no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—82no—
439C4—88no—
440C4#232414yes—
441C4#261912yes—
442C4#262012yes—
443C2—21no—
444C2—21no—
445C2—22no—
446C2—162no—
447C2#1425711yescharacteristic
448C2#2095211yescharacteristic
449C2#2095411yescharacteristic
450C2#2968314yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34059C2semidirect
4. C2×#1801C2semidirect
5. #11055C2semidirect
7. #10978C2semidirect
9. #10852C2semidirect
19. #2545C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
#340592
#212612
C2×#18012
#110552
#110542
#109782
#109762
#108522