Finite Group Databasebeta

← #189661

Subgroups — #189661

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2334C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3#21951C211yesmaximal, characteristic, complemented
4C2×#1473C212yesmaximal
5#8562C212yesmaximal, complemented
6#8561C212yesmaximal
7#8560C212yesmaximal, complemented
8#8557C212yesmaximal
9#7914C212yesmaximal, complemented
10C22×#126C2211yescharacteristic
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2334—24no—
14#2334—24no—
15C2×#389C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#388C411yescharacteristic
18C2×#388C411yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C411yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#385C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1940—24no—
30#1939—24no—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1473—28no—
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C24×C4C2×C411yescharacteristic
41C2×#126—22no—
42C2×#126—22no—
43C2×#126—22no—
44C2×#126—22no—
45C2×#126D414yes—
46C2×#126Q814yes—
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×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#432—24no—
70#432—24no—
71#431—24no—
72#431—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82C2×C4×C8—22no—
83#389—24no—
84#389—24no—
85#388—24no—
86#388—22no—
87#388—22no—
88#388—24no—
89#387—24no—
90#387—22no—
91#387—22no—
92#387—24no—
93#385—24no—
94#385—24no—
95#382—22no—
96#382—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C4×C4⋊C4D414yes—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2311yescharacteristic
114C2×#95C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#324—28no—
120#323—28no—
121C25C22×C411yescharacteristic
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—81no—
125C23×C4—81no—
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—
142#126—28no—
143#126—28no—
144#126—28no—
145#126—28no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4—81no—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C4⋊C414yes—
169C2×C4⋊C4C4⋊C414yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—81no—
185C2×C22⋊C4—81no—
186C2×C22⋊C4C4⋊C414yes—
187C2×C22⋊C4C4⋊C414yes—
188C2×C22⋊C4C2×D412yes—
189C2×C22⋊C4C2×Q812yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—42no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42—81no—
197C2×C42—81no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C4⋊C8—82no—
211C4⋊C8—82no—
212C4⋊C8—82no—
213C4⋊C8—82no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218#97—44no—
219C4×C8—44no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C22⋊C414yes—
233#95C22⋊C414yes—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24C2×C4⋊C412yes—
243C24C23×C411yescharacteristic
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
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×C4—41no—
280C22×C4—41no—
281C22×C4—44no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—88no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C4⋊C412yes—
300C22×C4C2×C4⋊C412yes—
301C22×C4C2×C4⋊C412yes—
302C22×C4C2×C4⋊C412yes—
303C22×C4C2×C4⋊C412yes—
304C22×C4#11712yes—
305C22×C4#11712yes—
306C22×C4#13114yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q811yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—28no—
337C22⋊C4—28no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—44no—
346C42—82no—
347C42—82no—
348C42#12114yes—
349C42#12114yes—
350C42C22⋊Q814yes—
351C42C22⋊Q814yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—24no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23C22×C22⋊C411yescharacteristic
374C23C22×C4⋊C411yescharacteristic
375C23C2×#11711yescharacteristic
376C23C2×#13112yes—
377C23#56812yes—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—88no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4#38014yes—
442C2×C4#38014yes—
443C2×C4#40814yes—
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C8—84no—
449C8—84no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—84no—
462C22—84no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22C2×#38011yescharacteristic
468C22C2×#40811yescharacteristic
469C22#287911yescharacteristic
470C22#288311yescharacteristic
471C22#288411yescharacteristic
472C22#288811yescharacteristic
473C22#288911yescharacteristic
474C22#290612yes—
475C22#292512yes—
476C4—24no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—24no—
482C4—24no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C2—22no—
494C2—22no—
495C2—82no—
496C2#1738611yescharacteristic
497C2#1739111yescharacteristic
498C2#1740611yescharacteristic
499C2#1740911yescharacteristic
500C2#2062911yescharacteristic
501C2#2063411yescharacteristic
502C2#2064011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
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
2. #21955C2semidirect
3. #21951C2semidirect
5. #8562C2semidirect
7. #8560C2semidirect
9. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23342
#219552
#219512
C2×#14732
#85622
#85612
#85602
#85572
#79142