Finite Group Databasebeta

← #319665

Subgroups — #319665

Order 512.

820 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29845C211yesmaximal, characteristic
2#27949C211yesmaximal, characteristic
3#27876C211yesmaximal, characteristic
4#25811C211yesmaximal, characteristic
5#25722C211yesmaximal, characteristic
6#21833C211yesmaximal, characteristic
7#10685C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10318C211yesmaximal, characteristic, complemented
10#10318C211yesmaximal, characteristic, complemented
11#9839C211yesmaximal, characteristic, complemented
12#9828C211yesmaximal, characteristic, complemented
13#9825C211yesmaximal, characteristic, complemented
14#9819C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#494C2211yescharacteristic
21C2×#487C2211yescharacteristic
22C2×#451—22no—
23C2×#431C2211yescharacteristic
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2285C2211yescharacteristic
27#2085—22no—
28#2077—22no—
29#2066—22no—
30#2058—22no—
31#2035—22no—
32#2016—22no—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1990C2211yescharacteristic
41#1990C2211yescharacteristic
42#1990C2211yescharacteristic
43#1990C2211yescharacteristic
44#1988—22no—
45#1956C2211yescharacteristic
46#1956C2211yescharacteristic
47#1956C2211yescharacteristic
48#1919—22no—
49#1854—22no—
50#1853—22no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1844C2211yescharacteristic
54#1844C2211yescharacteristic
55#1834C2211yescharacteristic
56#1823—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C23×Q8—21no—
64C22×Q16—22no—
65C22×Q16—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—41no—
72C2×C4⋊Q8—41no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83#498—22no—
84#498—22no—
85#498—41no—
86#498—41no—
87#494—22no—
88#494—22no—
89#494—41no—
90#494—41no—
91#487—41no—
92#487—41no—
93#451—48no—
94#431—41no—
95#431—41no—
96C2×#107—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×#106—22no—
102C2×C4⋊C8D412yes—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×C22⋊C8D412yes—
108C2×C22⋊C8D412yes—
109C2×C22⋊C8D412yes—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391—21no—
117#391—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#336D412yes—
148#336D412yes—
149#322—41no—
150C82—41no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—41no—
158C22×Q8—44no—
159C22×Q8—82no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×Q16—44no—
165C2×Q16—44no—
166C2×Q16—44no—
167C2×Q16—44no—
168C2×Q16—82no—
169C2×Q16—82no—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C4⋊Q8—24no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C4⋊Q8#4112yes—
197C4⋊Q8#4112yes—
198C22⋊Q8—44no—
199C22⋊Q8—82no—
200#117—82no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—41no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2411yescharacteristic
241#107—82no—
242#106—44no—
243#106—44no—
244#106—44no—
245#106—82no—
246#106—82no—
247C4⋊C8—42no—
248#103—44no—
249#103—44no—
250#103—44no—
251#103—82no—
252C22⋊C8—24no—
253C22⋊C8—42no—
254C22⋊C8SD1618yes—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24#12711yescharacteristic
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—44no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—88no—
292C2×Q8—88no—
293C2×Q8—82no—
294C2×Q8—82no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4#12711yescharacteristic
318C22×C4#12711yescharacteristic
319C22×C4#12711yescharacteristic
320C22×C4C2×SD1614yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324Q16—88no—
325Q16—88no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8C2×SD1614yes—
342C2×C8C2×SD1614yes—
343C2×C8#13714yes—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—82no—
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C42H(5,𝔽2)11yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#49212yes—
397C23C2×#12711yescharacteristic
398Q8—44no—
399Q8—44no—
400Q8—88no—
401Q8—82no—
402Q8—82no—
403Q8—82no—
404Q8—82no—
405Q8—82no—
406Q8—82no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—88no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4#49212yes—
447C2×C4#52511yescharacteristic
448C2×C4#53411yescharacteristic
449C2×C4#54611yescharacteristic
450C2×C4#54611yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4#54611yescharacteristic
453C2×C4C22×SD1612yes—
454C2×C4C2×#13712yes—
455C2×C4#57612yes—
456C2×C4#57712yes—
457C2×C4#57812yes—
458C2×C4#57812yes—
459C8—24no—
460C8—24no—
461C8—24no—
462C8—42no—
463C8—42no—
464C8—42no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22#253911yescharacteristic
473C22C2×#49211yescharacteristic
474C22#315211yescharacteristic
475C22#315411yescharacteristic
476C22#315511yescharacteristic
477C22#315611yescharacteristic
478C22#315611yescharacteristic
479C4—22no—
480C4—41no—
481C4—41no—
482C4—82no—
483C4—82no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4#312112yes—
491C4#320112yes—
492C4#329512yes—
493C4#329512yes—
494C4#331212yes—
495C4#332812yes—
496C2—42no—
497C2#1962211yescharacteristic
498C2#1965311yescharacteristic
499C2#1972811yescharacteristic
500C2#1974111yescharacteristic
501C2#2832411yescharacteristic
502C2#2833211yescharacteristic
503C2#2835511yescharacteristic
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. #10685C2semidirect
8. #10684C2semidirect
9. #10318C2semidirect
10. #10318C2semidirect
11. #9839C2semidirect
12. #9828C2semidirect
13. #9825C2semidirect
14. #9819C2semidirect

Maximal subgroups

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

subgroupindex
#298452
#279492
#278762
#258112
#257222
#218332
#106852
#106842
#103182
#98392
#98282
#98252
#98192
#81482