Finite Group Databasebeta

← #244057

Subgroups — #244057

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21984C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic
5#21833C211yesmaximal, characteristic
6#21804C211yesmaximal, characteristic
7#10686C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10682C211yesmaximal, characteristic, complemented
10#10681C211yesmaximal, characteristic, complemented
11#10681C211yesmaximal, characteristic, complemented
12#10680C211yesmaximal, characteristic, complemented
13#10675C211yesmaximal, characteristic, complemented
14#10674C211yesmaximal, characteristic, complemented
15#7851C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1988—22no—
32#1987—22no—
33#1981—22no—
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1949—22no—
39#1949—22no—
40#1949—22no—
41#1854—22no—
42#1854—22no—
43#1854—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1824—22no—
57#1823—22no—
58#1787—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C23×Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8Q814yes—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8D412yes—
93C2×C4⋊C8D412yes—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8D412yes—
100C2×C22⋊C8D412yes—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#382—21no—
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142C22×Q8—24no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—42no—
146C22×Q8—41no—
147C22×Q8—82no—
148C22×Q8—82no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8C2×Q812yes—
172C4⋊Q8C2×Q812yes—
173#117—82no—
174#117—82no—
175#117—82no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#107—82no—
219#107—82no—
220#107—82no—
221#106—82no—
222#106—82no—
223#106—82no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227#103—82no—
228#103—82no—
229#103—82no—
230#103—82no—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—24no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24#12711yescharacteristic
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—44no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—88no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—88no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C22⋊Q812yes—
284C22×C4C22⋊Q812yes—
285C22×C4C22⋊Q812yes—
286C22×C4#12312yes—
287C22×C4#12312yes—
288C22×C4#12312yes—
289C22×C4#12711yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4#12711yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—82no—
345C42—82no—
346C42—82no—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42C22⋊Q812yes—
350C42#12312yes—
351C42#12312yes—
352C42#12312yes—
353C42C22×Q811yescharacteristic
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C42C2×#4111yescharacteristic
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23#39912yes—
364C23C2×#12711yescharacteristic
365Q8—44no—
366Q8—44no—
367Q8—88no—
368Q8—82no—
369Q8—82no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—88no—
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—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#39912yes—
408C2×C4#39912yes—
409C2×C4#39912yes—
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4C2×#12311yescharacteristic
414C2×C4C2×#12311yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C2×C4#57612yes—
417C2×C4#57712yes—
418C2×C4#57712yes—
419C2×C4#57712yes—
420C2×C4#57812yes—
421C2×C4#57812yes—
422C8—42no—
423C8—42no—
424C8—42no—
425C8—42no—
426C8—42no—
427C8—42no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22C2×#39911yescharacteristic
436C22#315411yescharacteristic
437C22#315411yescharacteristic
438C22#315411yescharacteristic
439C22#315511yescharacteristic
440C22#315611yescharacteristic
441C22#315611yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—82no—
446C4—82no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4#308112yes—
454C4#308212yes—
455C4#308212yes—
456C4#309612yes—
457C4#309712yes—
458C4#309712yes—
459C2—42no—
460C2#1973811yescharacteristic
461C2#1973911yescharacteristic
462C2#1974211yescharacteristic
463C2#1974411yescharacteristic
464C2#2373511yescharacteristic
465C2#2373611yescharacteristic
466C2#2373611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10686C2semidirect
8. #10684C2semidirect
9. #10682C2semidirect
10. #10681C2semidirect
11. #10681C2semidirect
12. #10680C2semidirect
13. #10675C2semidirect
14. #10674C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#218342
#218332
#218042
#106862
#106842
#106822
#106812
#106802
#106752
#106742
#78512