Finite Group Databasebeta

← #189867

Subgroups — #189867

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#21738C211yesmaximal, characteristic
3#13940C211yesmaximal, characteristic
4#8714C214yesmaximal, complemented
5#8713C212yesmaximal, complemented
6#8712C212yesmaximal, complemented
7#8024C212yesmaximal, complemented
8#8012C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2353—22no—
12#2353—22no—
13#2352C414yescomplemented
14#2352C414yescomplemented
15#2313—21no—
16#2312—21no—
17#2308—21no—
18#2283C2212yes—
19#2280C2212yes—
20C2×#381C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#2273—21no—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1869—22no—
26#1869—22no—
27#1817C2212yes—
28#1779C2212yes—
29#1440—21no—
30#1438—21no—
31#1437—21no—
32#1436—21no—
33#1434C2212yes—
34C22⋊#95C2212yes—
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C23×D4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C2×C4×D4—24no—
44C2×C4×D4—24no—
45C2×C4×D4C2×C412yes—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×#117C2×C414yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4D411yescharacteristic
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#388—24no—
69#388—24no—
70#387—24no—
71#387—24no—
72#386—24no—
73#386—24no—
74#383—22no—
75#382C2×C414yes—
76#381—22no—
77#381—22no—
78#381—24no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380C2×C412yes—
84#380C2×C412yes—
85#380C2×C414yes—
86#380C2312yes—
87#380C2312yes—
88#379—22no—
89C4×C4⋊C4C2×C414yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96#376—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95C2312yes—
104C2×#95C2312yes—
105#336—24no—
106C25C2×D411yescharacteristic
107C25C2×D411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—24no—
111C22×D4—42no—
112C22×D4C22⋊C412yes—
113C22×D4C22⋊C412yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—82no—
121C23×C4—82no—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4#4114yes—
133C23×C4#4114yes—
134C23×C4C2411yescharacteristic
135C22×C8—24no—
136C22×C8—44no—
137C22×C8—84no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144#117—84no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22×C414yes—
154C2×C4⋊C4C22×C414yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4C22⋊C412yes—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—42no—
183C2×C42—82no—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C22⋊C8—24no—
187C22⋊C8—24no—
188C22⋊C8—44no—
189C22⋊C8—82no—
190C22⋊C8—82no—
191#95—22no—
192#95—22no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—82no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217C24#12312yes—
218C24#12312yes—
219C24#13514yes—
220C24C22×D411yescharacteristic
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—48no—
228C2×D4—48no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4#11718yes—
283C22×C4#12314yes—
284C22×C4#12314yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—88no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—44no—
306C22⋊C4—48no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C22⋊C4—88no—
313C22⋊C4—84no—
314C42—24no—
315C42—22no—
316C42—22no—
317C42—84no—
318C42—84no—
319C42—84no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—24no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—24no—
338C23—42no—
339C23—42no—
340C23—44no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—81no—
345C23—81no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23—82no—
352C23—81no—
353C23—81no—
354C23#38114yes—
355C23#41612yes—
356C23#41612yes—
357C23C22×C22⋊C411yescharacteristic
358C23C2×#12311yescharacteristic
359C23C2×#12311yescharacteristic
360C23C2×#13512yes—
361C23#57512yes—
362D4—48no—
363D4—48no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—164no—
414C2×C4—164no—
415C2×C4#38114yes—
416C2×C4C2×#11712yes—
417C2×C4C2×#12312yes—
418C8—84no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—22no—
432C22—22no—
433C22—24no—
434C22—44no—
435C22—41no—
436C22—41no—
437C22—42no—
438C22—44no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—84no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22C2×#38111yescharacteristic
447C22#288312yes—
448C22C2×#41611yescharacteristic
449C22#289611yescharacteristic
450C22#290112yes—
451C22#309112yes—
452C22#309112yes—
453C4—22no—
454C4—22no—
455C4—24no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—42no—
460C4—84no—
461C4—84no—
462C4—84no—
463C4—162no—
464C2—22no—
465C2—22no—
466C2—42no—
467C2—42no—
468C2—44no—
469C2#1761512yes—
470C2#1762512yes—
471C2#2068911yescharacteristic
472C2#2069911yescharacteristic
473C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8714C2semidirect
5. #8713C2semidirect
6. #8712C2semidirect
7. #8024C2semidirect
8. #8012C2semidirect
13. #2352C4semidirect
14. #2352C4semidirect

Maximal subgroups

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

subgroupindex
#341852
#217382
#139402
#87142
#87132
#87122
#80242
#80122