Finite Group Databasebeta

← #471942

Subgroups — #471942

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2100C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3C2×#1956C211yesmaximal, characteristic
4C2×#1933C211yesmaximal, characteristic
5#15199C211yesmaximal, characteristic
6#15188C211yesmaximal, characteristic
7#11050C212yesmaximal
8#11050C212yesmaximal
9#11024C212yesmaximal
10#11024C212yesmaximal
11#10797C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C4×C8C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27#2100—24no—
28#2016—24no—
29#1956—24no—
30#1933—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1444—22no—
39#1444—22no—
40#1442—22no—
41#1441—22no—
42C4×#95C2211yescharacteristic
43C23×Q8C2312yes—
44C23×Q8C2311yescharacteristic
45C22×Q16—24no—
46C22×Q16—24no—
47C22×Q16—24no—
48C22×Q16—42no—
49C22×Q16—42no—
50C22×Q16—42no—
51C22×Q16—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×C4×C8—42no—
86C2×C4×C8D414yes—
87#391—24no—
88#391—24no—
89#391—22no—
90#391—22no—
91#391—42no—
92#391—42no—
93#391—42no—
94#391—42no—
95#389—22no—
96#389—22no—
97#384—22no—
98#384—22no—
99#384—42no—
100#384—42no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95—21no—
104C2×#95—24no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95D412yes—
108C2×#95D412yes—
109#340—24no—
110#340—24no—
111#340—22no—
112#340—22no—
113#336—42no—
114#336—42no—
115#336D414yes—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8#4114yes—
139C22×Q8#4112yes—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4C2×D414yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C2×Q16—48no—
151C2×Q16—48no—
152C2×Q16—48no—
153C2×Q16—48no—
154C2×Q16—48no—
155C2×Q16—48no—
156C2×Q16—48no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8C2×D414yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C4⋊Q8—28no—
175C4⋊Q8—48no—
176C4⋊Q8—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4#4112yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220#103—28no—
221#103—44no—
222#103—48no—
223#103—48no—
224#103—48no—
225C4×C8—44no—
226#95—24no—
227#95—48no—
228#95—48no—
229#95—48no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95Q1618yescomplemented
234C24C22×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12712yes—
322C22×C4#12712yes—
323C22×C4C2×Q1618yes—
324C22×C4C2×Q1614yes—
325C22×C4#13714yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C42−1+411yescharacteristic
336Q16—432no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—48no—
357C4⋊C4—48no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—816no—
373C4⋊C4—84no—
374C42—22no—
375C42—22no—
376C42—24no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C23C2×#12711yescharacteristic
382C23#53512yes—
383C23#54611yescharacteristic
384C23#54612yes—
385C23#54811yescharacteristic
386C23C22×Q1612yes—
387C23C2×#13712yes—
388C23#57712yes—
389C23#57712yes—
390Q8—42no—
391Q8—44no—
392Q8—42no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—44no—
397Q8—82no—
398Q8—81no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402Q8—82no—
403Q8—81no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—81no—
446C2×C4—84no—
447C2×C4#49414yes—
448C2×C4#49714yes—
449C2×C4#52511yescharacteristic
450C2×C4#52511yescharacteristic
451C2×C4#53411yescharacteristic
452C2×C4#53511yescharacteristic
453C2×C4#53511yescharacteristic
454C2×C4#53512yes—
455C2×C4#53711yescharacteristic
456C2×C4#53812yes—
457C2×C4#53812yes—
458C2×C4#53912yes—
459C2×C4#54012yes—
460C2×C4#54611yescharacteristic
461C2×C4#54612yes—
462C2×C4#54811yescharacteristic
463C2×C4#55112yes—
464C2×C4#55312yes—
465C2×C4#56012yes—
466C2×C4#56212yes—
467C8—42no—
468C8—44no—
469C8—42no—
470C22#253911yescharacteristic
471C22#254011yescharacteristic
472C22#265711yescharacteristic
473C22#269612yes—
474C22#270312yes—
475C22C2×#49411yescharacteristic
476C22C2×#49711yescharacteristic
477C22#315411yescharacteristic
478C22#315411yescharacteristic
479C22#320914yes—
480C22#322214yes—
481C22#329912yes—
482C22#329914yes—
483C22#331614yes—
484C22#333112yes—
485C22#336312yes—
486C22#336912yes—
487C4—24no—
488C4—24no—
489C4—48no—
490C4—82no—
491C4—82no—
492C4—84no—
493C4—162no—
494C4—161no—
495C4—161no—
496C4#259811yescharacteristic
497C4#263511yescharacteristic
498C4#263611yescharacteristic
499C4#268512yes—
500C4#271112yes—
501C4#281311yescharacteristic
502C2#1680911yescharacteristic
503C2#2833311yescharacteristic
504C2#2834711yescharacteristic
505C2#2840111yescharacteristic
506C2#2841911yescharacteristic
507C2#3026211yescharacteristic
508C2#3027111yescharacteristic
509C2#3164014yes—
510C2#3184014yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
233. #95Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#21002
C2×#20162
C2×#19562
C2×#19332
#151992
#151882
#110502
#110242
#107972