Finite Group Databasebeta

← #207276

Subgroups — #207276

Order 512.

990 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#35167C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic
3#13806C211yesmaximal, characteristic
4#9912C212yesmaximal
5#9911C212yesmaximal
6#9909C212yesmaximal
7#9908C212yesmaximal
8#8022C212yesmaximal
9#8018C212yesmaximal
10C2×#544—21no—
11C2×#432C2212yes—
12#2857C2212yes—
13#2857C2212yes—
14#2659—21no—
15#2659—21no—
16#2552C414yescomplemented
17#2551C414yescomplemented
18#2550—24no—
19#2316—21no—
20#2285C2212yes—
21C2×#383C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1940—24no—
25#1797C2212yes—
26#1797C2212yes—
27#1436—21no—
28#1436—21no—
29#1434—21no—
30#1434C2211yescharacteristic
31#1434C2211yescharacteristic
32C4×#95—21no—
33C4×#95C2211yescharacteristic
34C4×#95C2211yescharacteristic
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39#544—44no—
40#544—44no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×#125—24no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×#117—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56#432—24no—
57#432—24no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63#398—24no—
64#398—24no—
65#395—24no—
66#395—24no—
67#391—22no—
68#391—22no—
69#389—24no—
70#389—24no—
71#389—22no—
72#389—22no—
73#389C2×C414yes—
74#389C2×C414yes—
75#384C2×C412yes—
76#384C2×C412yes—
77#384C2×C412yes—
78#384C2×C412yes—
79#383—22no—
80#383Q814yescomplemented
81#382—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380C2314yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C414yes—
88C4×C4⋊C4C2×C414yes—
89C4×C4⋊C4D414yes—
90#376—22no—
91#376Q814yescomplemented
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—24no—
123C4⋊Q8—416no—
124C4⋊Q8—82no—
125C4⋊Q8—82no—
126#125—416no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131#117—44no—
132#117—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
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⋊C4C22⋊C414yes—
150C2×C4⋊C4C22⋊C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C22⋊C4—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—42no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—81no—
163C2×C42—81no—
164C2×C42—81no—
165C2×C42—81no—
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×Q812yes—
173C2×C42C2×Q812yes—
174C2×C42#4112yes—
175C2×C42#4112yes—
176C4⋊C8—44no—
177C4⋊C8—82no—
178C4⋊C8—82no—
179C22⋊C8—24no—
180C22⋊C8—24no—
181#97—44no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—81no—
198#95—81no—
199#95C22⋊C414yes—
200#95C22⋊C414yes—
201#95C2×Q812yes—
202#95C2×Q812yes—
203#95C2×Q812yes—
204#95C2×Q812yes—
205C24C22×D411yescharacteristic
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—82no—
211C2×Q8—82no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
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—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4C22⋊Q812yes—
271C22×C4C22⋊Q812yes—
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4AGL(1,ℤ/8)12yes—
274C22×C4#13712yes—
275C22×C4#13712yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×Q811yescharacteristic
279C22×C4C22×Q811yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—44no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—48no—
308C42—48no—
309C42—84no—
310C42—84no—
311C42C4×Q814yes—
312C42C4×Q814yes—
313C42#12414yes—
314C42C4⋊Q814yes—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C22×C22⋊C411yescharacteristic
320C23C2×C22⋊Q811yescharacteristic
321C23C2×C22⋊Q811yescharacteristic
322C23C2×AGL(1,ℤ/8)11yescharacteristic
323C23C2×AGL(1,ℤ/8)11yescharacteristic
324C23C2×#13711yescharacteristic
325C23C2×#13711yescharacteristic
326Q8—164no—
327Q8—164no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#39114yes—
399C2×C4#39114yes—
400C2×C4#41814yes—
401C2×C4#41914yes—
402C2×C4C2×C4×Q811yescharacteristic
403C2×C4C2×C4×Q811yescharacteristic
404C2×C4C2×#12411yescharacteristic
405C2×C4C2×C4⋊Q811yescharacteristic
406C8—84no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22C2×#39111yescharacteristic
420C22#288311yescharacteristic
421C22#288411yescharacteristic
422C22C2×#41811yescharacteristic
423C22C2×#41911yescharacteristic
424C22#290111yescharacteristic
425C22#290111yescharacteristic
426C22#307911yescharacteristic
427C22#307911yescharacteristic
428C22#307911yescharacteristic
429C22#307911yescharacteristic
430C4—24no—
431C4—24no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—24no—
437C4—24no—
438C4—44no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—84no—
445C4—84no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#1762011yescharacteristic
451C2#1762111yescharacteristic
452C2#1763111yescharacteristic
453C2#1763211yescharacteristic
454C2#2131311yescharacteristic
455C2#2133011yescharacteristic
456C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2552C4semidirect
17. #2551C4semidirect
80. #383Q8semidirect
91. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219552
#138062
#99122
#99112
#99092
#99082
#80222
#80182