Finite Group Databasebeta

← #465036

Subgroups — #465036

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal
2C2×#2043C212yesmaximal
3C2×#1927C212yesmaximal
4#15180C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11202C211yesmaximal, characteristic
8#11067C212yesmaximal
9#11050C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#106C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#384—21no—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#2066—28no—
26#2043—28no—
27#1927—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34#1444—21no—
35#1444—21no—
36#1442—21no—
37#1442—21no—
38#1441—21no—
39#1441—21no—
40C4⋊#95—21no—
41C4⋊#95C2211yescharacteristic
42C23×Q8C2312yes—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—42no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—41no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103D418yes—
81#391—24no—
82#391—24no—
83#391—24no—
84#391—42no—
85#389—24no—
86#389—24no—
87#389—42no—
88#389—42no—
89#384—41no—
90#384—41no—
91#384—44no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—41no—
94C4×C4⋊C4—41no—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—42no—
110#336—24no—
111#336—24no—
112#336—41no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8C2×D414yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C2×D414yes—
152C4⋊Q8—44no—
153C4⋊Q8—48no—
154C4⋊Q8—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4SD1614yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4#4114yes—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C2×D412yes—
210C2×C42#4112yes—
211#106—48no—
212C4⋊C8—44no—
213#103—216no—
214#103—48no—
215#103—48no—
216#103—48no—
217#95—24no—
218#95—41no—
219#95—41no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—44no—
226#95—44no—
227#95—42no—
228C24C22×D411yescharacteristic
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4C2×SD1612yes—
313C22×C4C2×SD1612yes—
314C22×C4C2×SD1612yes—
315C22×C4#13712yes—
316C22×C4#13712yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D412yes—
319C22×C4C2×#4112yes—
320C22×C4C2×#4111yescharacteristic
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+412yes—
326C22×C42−1+411yescharacteristic
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—88no—
357C4⋊C4—88no—
358C4⋊C4—84no—
359C4⋊C4—84no—
360C4⋊C4#12118yes—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—42no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—48no—
372C23C2×#12111yescharacteristic
373C23#53411yescharacteristic
374C23#53611yescharacteristic
375C23D4212yes—
376C23#54812yes—
377C23C22×SD1611yescharacteristic
378C23C2×#13711yescharacteristic
379C23C2×#13711yescharacteristic
380C23#57511yescharacteristic
381C23#57712yes—
382C23#57812yes—
383Q8—44no—
384Q8—44no—
385Q8—44no—
386Q8—44no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—28no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4#46114yes—
451C2×C4C2×#12112yes—
452C2×C4#53512yes—
453C2×C4#53612yes—
454C2×C4#53712yes—
455C2×C4#53711yescharacteristic
456C2×C4#53711yescharacteristic
457C2×C4#53911yescharacteristic
458C2×C4#53911yescharacteristic
459C2×C4#54212yes—
460C2×C4#54212yes—
461C2×C4D4211yescharacteristic
462C2×C4#54712yes—
463C2×C4#54712yes—
464C2×C4#55312yes—
465C2×C4#55511yescharacteristic
466C2×C4#56011yescharacteristic
467C2×C4#56311yescharacteristic
468C2×C4#56312yes—
469C8—44no—
470C8—82no—
471C22#241612yes—
472C22#261811yescharacteristic
473C22#266911yescharacteristic
474C22#267412yes—
475C22#273111yescharacteristic
476C22C2×#46111yescharacteristic
477C22#303711yescharacteristic
478C22#304612yes—
479C22#318711yescharacteristic
480C22#319511yescharacteristic
481C22#320011yescharacteristic
482C22#320211yescharacteristic
483C22#322911yescharacteristic
484C22#323611yescharacteristic
485C22#324212yes—
486C22#327012yes—
487C22#327112yes—
488C22#327612yes—
489C22D4×SD1612yes—
490C22#335912yes—
491C22#336912yes—
492C22#337212yes—
493C22#337712yes—
494C4—24no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—48no—
500C4—81no—
501C4—84no—
502C4—162no—
503C4—162no—
504C4#253711yescharacteristic
505C4#262012yes—
506C4#268011yescharacteristic
507C4#271812yes—
508C4#279511yescharacteristic
509C4#282311yescharacteristic
510C2#1651411yescharacteristic
511C2#2473012yes—
512C2#2474112yes—
513C2#2975611yescharacteristic
514C2#2976611yescharacteristic
515C2#3050611yescharacteristic
516C2#3051711yescharacteristic
517C2#3080812yes—
518C2#3083412yes—
519C2#3263011yescharacteristic
520C2#3264511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20662
C2×#20432
C2×#19272
#151802
#139102
#112202
#112022
#110672
#110502