Finite Group Databasebeta

← #215608

Subgroups — #215608

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic
3#21902C211yesmaximal, characteristic
4#9880C212yesmaximal, complemented
5#9880C212yesmaximal, complemented
6#9879C212yesmaximal, complemented
7#9878C212yesmaximal, complemented
8#8700C212yesmaximal
9#8698C212yesmaximal
10C22×C22⋊Q8C411yescharacteristic
11#2329—24no—
12#2329—24no—
13C2×#391C411yescharacteristic
14C2×#391C2211yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#387C411yescharacteristic
18C2×#387C411yescharacteristic
19C2×#387C2211yescharacteristic
20C2×#387C2211yescharacteristic
21C2×#385C411yescharacteristic
22C2×#385C411yescharacteristic
23C2×#380C411yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1923—24no—
28#1923—24no—
29#1797C2212yes—
30#1797C2212yes—
31#1779C2212yes—
32#1779C2212yes—
33C4×C22⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35#1280C2214yes—
36#1280C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8C818yescomplemented
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×C4⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8C2314yes—
70C2×C22⋊C8C2314yes—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73#391—22no—
74#391—22no—
75#391—24no—
76#389—22no—
77#389—22no—
78#389—24no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—24no—
84#387—24no—
85#385—24no—
86#385—24no—
87#380—24no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4D414yes—
91C4×C22⋊C4D414yes—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95C2×C411yescharacteristic
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×#95C2311yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112C25C22×C411yescharacteristic
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—81no—
116C22×Q8—81no—
117C22×Q8C2×C814yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—42no—
123C23×C4—81no—
124C23×C4—81no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4C2×C814yes—
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×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C22×C8—24no—
147C22×C8—24no—
148C22⋊Q8—232no—
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—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4—81no—
166C2×C4⋊C4—81no—
167C2×C4⋊C4C2×C814yes—
168C2×C4⋊C4C2×C814yes—
169C2×C4⋊C4C2×C814yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4C22⋊C414yes—
176C2×C22⋊C4C22⋊C414yes—
177C2×C22⋊C4C2×C814yes—
178C2×C22⋊C4C2×C814yes—
179C2×C22⋊C4M1614yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C4⋊C8—82no—
191C4⋊C8—82no—
192C22⋊C8—28no—
193C22⋊C8—28no—
194C22⋊C8—82no—
195C22⋊C8—82no—
196C22⋊C8—82no—
197C22⋊C8—82no—
198C4×C8—82no—
199C4×C8—82no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95C22⋊C414yes—
217#95C22⋊C414yes—
218C24—22no—
219C24—22no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24C2×C22⋊C412yes—
225C24C22×C812yes—
226C24C2×M1612yes—
227C24C23×C411yescharacteristic
228C2×Q8—28no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—84no—
232C2×Q8—84no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
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—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
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×C4—44no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—88no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—88no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C22×C812yes—
302C22×C4C22×C812yes—
303C22×C4C22×C812yes—
304C22×C4C22×C812yes—
305C22×C4C22×C812yes—
306C22×C4C22×C812yes—
307C22×C4C2×M1612yes—
308C22×C4C2×M1612yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C42−1+414yes—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—28no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—28no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4C22⋊C8116yes—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—82no—
336C42—82no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23C2×C22⋊C814yes—
358C23C22×C22⋊C411yescharacteristic
359C23#51912yes—
360C23C23×C811yescharacteristic
361C23C22×M1611yescharacteristic
362C23#56812yes—
363Q8—88no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—88no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4C2×C22⋊C814yes—
430C2×C4C2×C22⋊C814yes—
431C2×C4#53612yes—
432C2×C4#53612yes—
433C8—84no—
434C8—84no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—48no—
447C22—48no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22#231011yescharacteristic
453C22C22×C22⋊C811yescharacteristic
454C22#287911yescharacteristic
455C22#288311yescharacteristic
456C22#288311yescharacteristic
457C22#288811yescharacteristic
458C22#288911yescharacteristic
459C22#298212yes—
460C22#298312yes—
461C4—22no—
462C4—22no—
463C4—24no—
464C4—24no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C2—22no—
477C2—22no—
478C2—44no—
479C2#1735011yescharacteristic
480C2#1735111yescharacteristic
481C2#1740511yescharacteristic
482C2#1740611yescharacteristic
483C2#2181811yescharacteristic
484C2#2182711yescharacteristic
485C2#2183211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9880C2semidirect
5. #9880C2semidirect
6. #9879C2semidirect
7. #9878C2semidirect
45. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#219022
#98802
#98792
#98782
#87002
#86982