Finite Group Databasebeta

← #206932

Subgroups — #206932

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic
3#20534C211yesmaximal, characteristic
4#9882C212yesmaximal
5#9881C212yesmaximal
6#9880C212yesmaximal
7#9878C212yesmaximal
8#7912C212yesmaximal
9#7910C212yesmaximal
10C22×C4⋊Q8C411yescharacteristic
11C2×#432C2212yes—
12C2×#432C2212yes—
13#2331—24no—
14#2331—24no—
15C2×#391C411yescharacteristic
16C2×#391C411yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C411yescharacteristic
20C2×#389C411yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C411yescharacteristic
23C2×#387C2211yescharacteristic
24C2×#387C2211yescharacteristic
25C2×#382C411yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1940—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1771—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1280C2214yes—
36#1280C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#432—24no—
62#432—28no—
63#432Q818yescomplemented
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—22no—
74#391—22no—
75#391—22no—
76#391—22no—
77#391—22no—
78#391—24no—
79#391—24no—
80#389—24no—
81#389—24no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—24no—
87#387—24no—
88#382—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4D414yes—
92C4×C4⋊C4D414yes—
93#376—22no—
94#376—22no—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2311yescharacteristic
108C2×#95C2311yescharacteristic
109C2×#95C2311yescharacteristic
110C2×#95C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—81no—
121C22×Q8—81no—
122C22×Q8—81no—
123C22×Q8—81no—
124C22×Q8M1614yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—24no—
141C22×C8—24no—
142C4⋊Q8—232no—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—81no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22⋊C414yes—
166C2×C4⋊C4M1614yes—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—42no—
174C2×C42—81no—
175C2×C42—81no—
176C2×C42—81no—
177C2×C42—81no—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×Q814yes—
185C2×C42#4114yes—
186C4⋊C8—28no—
187C4⋊C8—44no—
188C4⋊C8—82no—
189C4⋊C8—82no—
190C22⋊C8—24no—
191C22⋊C8—28no—
192C22⋊C8C2×Q818yes—
193#97—28no—
194#97—44no—
195#97—82no—
196C4×C8—82no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95C22⋊C414yes—
216#95C22⋊C414yes—
217C24C23×C411yescharacteristic
218C2×Q8—28no—
219C2×Q8—28no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—84no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
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×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C4×Q814yes—
286C22×C4C4×Q814yes—
287C22×C4C2×M1612yes—
288C22×C4C2×M1612yes—
289C22×C4C2×M1612yes—
290C22×C4C2×M1612yes—
291C22×C4C2×M1612yes—
292C22×C4C2×M1612yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×Q812yes—
296C22×C4C2×#4112yes—
297C2×C8—28no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C42—28no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—44no—
319C42—82no—
320C42—82no—
321C42C22⋊Q814yes—
322C42C22⋊Q814yes—
323C42#12414yes—
324C42C4⋊Q814yes—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23C22×C22⋊C411yescharacteristic
330C23C2×C4×Q812yes—
331C23C22×M1611yescharacteristic
332C23C22×M1611yescharacteristic
333C23#56812yes—
334Q8—88no—
335Q8—88no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—44no—
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—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#39114yes—
403C2×C4#39114yes—
404C2×C4#40714yes—
405C2×C4C2×C22⋊Q812yes—
406C2×C4C2×#12411yescharacteristic
407C2×C4C2×C4⋊Q811yescharacteristic
408C8—48no—
409C8—84no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22C2×#39111yescharacteristic
419C22C2×#40711yescharacteristic
420C22#287911yescharacteristic
421C22#288311yescharacteristic
422C22#288411yescharacteristic
423C22#288811yescharacteristic
424C22#288911yescharacteristic
425C22Q8×M1612yes—
426C22Q8×M1612yes—
427C4—24no—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—24no—
432C4—24no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C2—22no—
445C2—22no—
446C2#1736711yescharacteristic
447C2#1737111yescharacteristic
448C2#1740711yescharacteristic
449C2#1740911yescharacteristic
450C2#2130311yescharacteristic
451C2#2133411yescharacteristic
452C2#2133811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ 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
63. #432Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
#219552
#205342
#98822
#98812
#98802
#98782
#79122
#79102