Finite Group Databasebeta

← #194748

Subgroups — #194748

Order 512.

1480 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×#2580C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#20741C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9024C212yesmaximal, complemented
6#9021C212yesmaximal, complemented
7#9016C212yesmaximal, complemented
8#9013C212yesmaximal, complemented
9#8017C212yesmaximal, complemented
10C22×C4×Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#432C2212yes—
13#2580—24no—
14#2580C418yescomplemented
15C2×#399C2211yescharacteristic
16C2×#399C2211yescharacteristic
17C2×#393C2211yescharacteristic
18C2×#393C2211yescharacteristic
19#2318—24no—
20#2315C2212yes—
21#2313—21no—
22#2312C2212yes—
23#2308—21no—
24C2×#390—21no—
25C2×#389—21no—
26#2290—24no—
27C2×#385—21no—
28C2×#385—21no—
29#2283C2212yes—
30C2×#382C2211yescharacteristic
31#2280—21no—
32C2×C4×C22⋊C4—21no—
33#1824C2214yes—
34#1805—24no—
35#1787C2214yes—
36#1763—24no—
37#1466—24no—
38C2×#323C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8C2311yescharacteristic
43C23×D4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—24no—
50C2×C4×Q8C2×C414yescomplemented
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×#117—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73#432—24no—
74#432—24no—
75C2×C4⋊C8—22no—
76C2×#103—28no—
77C2×C22⋊C8C2314yes—
78C2×#97—22no—
79#399—22no—
80#399—22no—
81#399C2×C414yes—
82#399C2×C414yescomplemented
83#393—22no—
84#393—22no—
85#393C2×C414yes—
86#393C2×C414yescomplemented
87#390—28no—
88#389—28no—
89#388—28no—
90#387—28no—
91#386—28no—
92#385—28no—
93#385—28no—
94#385—28no—
95#383—22no—
96#382—22no—
97#382C2×C414yescomplemented
98#382D414yes—
99#381—22no—
100#381—22no—
101#380—22no—
102#380—22no—
103#380C2314yes—
104#379—22no—
105C4×C4⋊C4D414yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111#376—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2311yescharacteristic
117C2×#95C2311yescharacteristic
118#340—28no—
119#323—24no—
120#323—24no—
121C25C2×D411yescharacteristic
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8C22×C412yes—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—42no—
132C22×D4C22⋊C414yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C22⋊C414yes—
147C23×C4C22⋊C414yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—24no—
154C4×Q8—28no—
155C4×Q8—28no—
156C4×Q8—44no—
157C4×Q8—44no—
158C4×D4—28no—
159C4×D4—28no—
160C4×D4—28no—
161C4×D4—28no—
162#117—84no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C22⋊C414yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D418yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4C22×C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—84no—
210C2×C42C22×C412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42#4114yes—
215C4⋊C8—44no—
216#103—416no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219#97—44no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—21no—
229#95—21no—
230#95—24no—
231#95C22×C412yes—
232#95C22×C412yes—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—21no—
237C24—22no—
238C24—21no—
239C24—22no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24C2×C22⋊C412yes—
245C24C2×C22⋊C412yes—
246C24#13714yes—
247C24C22×D411yescharacteristic
248C2×Q8—28no—
249C2×Q8—24no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—48no—
254C2×Q8—42no—
255C2×D4—28no—
256C2×D4—28no—
257C2×D4—44no—
258C2×D4—48no—
259C2×D4—48no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—28no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4#12012yes—
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D412yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4112yes—
317C2×C8—48no—
318C2×C8—44no—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C42—48no—
343C42—48no—
344C42—84no—
345C42C4×D414yes—
346C42C4×D414yes—
347C42#12114yes—
348C42#12114yes—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—22no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—48no—
363C23—48no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—84no—
375C23—84no—
376C23#41914yes—
377C23C22×C22⋊C411yescharacteristic
378C23C2×#12011yescharacteristic
379C23C2×#12311yescharacteristic
380C23C2×#13712yes—
381C23#57512yes—
382Q8—28no—
383Q8—48no—
384Q8—44no—
385D4—416no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—42no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—816no—
431C2×C4—82no—
432C2×C4#38614yes—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4C2×C4×D412yes—
436C2×C4C2×#12112yes—
437C8—84no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—24no—
444C22—24no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—48no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—82no—
454C22—84no—
455C22—82no—
456C22—84no—
457C22—84no—
458C22C2×#38611yescharacteristic
459C22C2×#41911yescharacteristic
460C22#289611yescharacteristic
461C22#300712yes—
462C22#301912yes—
463C22#302512yes—
464C22#309512yes—
465C4—24no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—44no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—88no—
477C2—21no—
478C2—21no—
479C2—22no—
480C2—44no—
481C2—82no—
482C2#1815914yes—
483C2#2091111yescharacteristic
484C2#2095311yescharacteristic
485C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20741C2semidirect
5. #9024C2semidirect
6. #9021C2semidirect
7. #9016C2semidirect
8. #9013C2semidirect
9. #8017C2semidirect
14. #2580C4semidirect
50. C2×C4×Q8C2×C4semidirect
82. #399C2×C4semidirect
86. #393C2×C4semidirect
97. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
#340572
#207412
C2×#14662
#90242
#90212
#90162
#90132
#80172