Finite Group Databasebeta

← #189871

Subgroups — #189871

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21738C211yesmaximal, characteristic
3#13940C211yesmaximal, characteristic
4#8717C212yesmaximal, complemented
5#8716C212yesmaximal, complemented
6#8715C214yesmaximal, complemented
7#8024C212yesmaximal, complemented
8#8012C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2357—22no—
12#2357—22no—
13#2354C414yescomplemented
14#2354C414yescomplemented
15#2312—21no—
16#2312—21no—
17#2309—21no—
18#2285C2211yescharacteristic
19#2285C2211yescharacteristic
20#2280C2212yes—
21#2280C2212yes—
22C2×#381C2211yescharacteristic
23#2274—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×#117—21no—
50C2×#117—21no—
51C2×#117C2×C414yes—
52C2×#117C2×C414yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8C2314yes—
66#390—24no—
67#390—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#384C2×C412yes—
73#384C2×C412yes—
74#382—21no—
75#382—21no—
76#381—22no—
77#381—22no—
78#381—24no—
79#380—21no—
80#380—21no—
81#380C2×C414yes—
82#380C2×C414yes—
83#380C2312yes—
84#380C2312yes—
85#379—22no—
86#379—22no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95C2312yes—
103C2×#95C2312yes—
104#336—22no—
105#336—22no—
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—82no—
119C23×C4—82no—
120C23×C4C22×C412yes—
121C23×C4C22×C412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4#4114yes—
129C23×C4#4114yes—
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C22×C8—44no—
133C22×C8—84no—
134C4×D4—28no—
135C4×D4—28no—
136C4×D4—24no—
137C4×D4—24no—
138C4×D4—24no—
139C4×D4—24no—
140#117—44no—
141#117—44no—
142#117—84no—
143#117—84no—
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⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—82no—
170C2×C22⋊C4—82no—
171C2×C22⋊C4C22⋊C412yes—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C412yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—42no—
182C2×C42—82no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C22⋊C8—24no—
188C22⋊C8—24no—
189C22⋊C8—44no—
190C22⋊C8—84no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—44no—
200#95—42no—
201#95—44no—
202#95—82no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—21no—
207C24—21no—
208C24—22no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24C2×C22⋊C412yes—
214C24C2×C22⋊C412yes—
215C24#12312yes—
216C24#12312yes—
217C24C22×D411yescharacteristic
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—48no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
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—84no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4#11718yes—
271C22×C4#12314yes—
272C22×C4#12314yes—
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4#13514yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C2×C8—48no—
281C2×C8—44no—
282C2×C8—88no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—48no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C22⋊C4—88no—
303C22⋊C4—84no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—44no—
309C42—44no—
310C42—84no—
311C42—84no—
312C42—84no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—22no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—42no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—82no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—81no—
343C23—81no—
344C23#38114yes—
345C23C22×C22⋊C411yescharacteristic
346C23C2×#12311yescharacteristic
347C23C2×#12311yescharacteristic
348C23C2×#13512yes—
349C23#57512yes—
350D4—48no—
351D4—48no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#38114yes—
402C2×C4#41612yes—
403C2×C4#41612yes—
404C2×C4C2×#11712yes—
405C2×C4C2×#12312yes—
406C8—84no—
407C22—22no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—24no—
417C22—24no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—42no—
422C22—42no—
423C22—44no—
424C22—42no—
425C22—84no—
426C22—84no—
427C22—84no—
428C22—84no—
429C22C2×#38111yescharacteristic
430C22#288312yes—
431C22C2×#41611yescharacteristic
432C22#289611yescharacteristic
433C22#290112yes—
434C22#309112yes—
435C22#309112yes—
436C4—22no—
437C4—22no—
438C4—24no—
439C4—22no—
440C4—22no—
441C4—42no—
442C4—41no—
443C4—41no—
444C4—84no—
445C4—84no—
446C4—84no—
447C4—162no—
448C2—22no—
449C2—22no—
450C2—44no—
451C2—82no—
452C2#1761712yes—
453C2#1762912yes—
454C2#2068911yescharacteristic
455C2#2069911yescharacteristic
456C2#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. #8717C2semidirect
5. #8716C2semidirect
6. #8715C2semidirect
7. #8024C2semidirect
8. #8012C2semidirect
13. #2354C4semidirect
14. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#217382
#139402
#87172
#87162
#87152
#80242
#80122