Finite Group Databasebeta

← #471641

Subgroups — #471641

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1956C211yesmaximal, characteristic
2C2×#1916C211yesmaximal, characteristic
3#15180C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11122C212yesmaximal
8#11122C212yesmaximal
9#11077C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C4×C8C2211yescharacteristic
20C2×#400—21no—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1956—24no—
26#1916—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—21no—
35#1442—21no—
36#1442—21no—
37#1442—21no—
38#1441—21no—
39#1441—21no—
40#1441—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×C4×C8—42no—
84C2×C4×C8D414yes—
85#400—44no—
86#391—24no—
87#391—42no—
88#391—42no—
89#391—42no—
90#384—22no—
91#384—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—42no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—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×Q8—22no—
127C22×Q8—22no—
128C22×Q8#4114yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8#4114yes—
160C4⋊Q8—28no—
161C4⋊Q8—48no—
162C4⋊Q8—44no—
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—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4SD1614yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C2×D412yes—
209C2×C42#4112yes—
210#106—48no—
211C4⋊C8—44no—
212#103—48no—
213#103—48no—
214#103—48no—
215#103—48no—
216C4×C8—44no—
217#95—22no—
218#95—22no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—82no—
230#95—82no—
231C24C22×D411yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
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—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—42no—
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—42no—
311C22×C4—42no—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×SD1612yes—
316C22×C4C2×SD1612yes—
317C22×C4#13512yes—
318C22×C4#13712yes—
319C22×C4#13712yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C42−1+411yescharacteristic
329C22×C42−1+411yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8#12414yes—
350C4⋊C4—28no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C42—22no—
369C42—22no—
370C42—24no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—42no—
378C23C2×#12111yescharacteristic
379C23#53511yescharacteristic
380C23#53611yescharacteristic
381C23#54612yes—
382C23#55012yes—
383C23C22×SD1611yescharacteristic
384C23C2×#13511yescharacteristic
385C23C2×#13711yescharacteristic
386C23C2×#13711yescharacteristic
387C23#57712yes—
388C23#57812yes—
389Q8—44no—
390Q8—44no—
391Q8—44no—
392Q8—44no—
393Q8—82no—
394Q8—82no—
395Q8—82no—
396Q8—82no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—82no—
431C2×C4—82no—
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—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#46514yes—
456C2×C4#47014yes—
457C2×C4#52511yescharacteristic
458C2×C4C2×#12411yescharacteristic
459C2×C4#53411yescharacteristic
460C2×C4#53511yescharacteristic
461C2×C4#53711yescharacteristic
462C2×C4#53711yescharacteristic
463C2×C4#53812yes—
464C2×C4#53812yes—
465C2×C4#53912yes—
466C2×C4#53912yes—
467C2×C4#54211yescharacteristic
468C2×C4#54211yescharacteristic
469C2×C4#55112yes—
470C2×C4#55212yes—
471C2×C4#55312yes—
472C2×C4#55912yes—
473C2×C4#56012yes—
474C2×C4#56212yes—
475C8—24no—
476C8—42no—
477C8—81no—
478C8—81no—
479C22#253411yescharacteristic
480C22#254811yescharacteristic
481C22#262011yescharacteristic
482C22#269412yes—
483C22#271412yes—
484C22C2×#46511yescharacteristic
485C22C2×#47011yescharacteristic
486C22#305711yescharacteristic
487C22#305911yescharacteristic
488C22#320611yescharacteristic
489C22#321011yescharacteristic
490C22#322512yes—
491C22#322911yescharacteristic
492C22#323111yescharacteristic
493C22#324212yes—
494C22#329812yes—
495C22#330012yes—
496C22#331612yes—
497C22#332012yes—
498C22#339212yes—
499C22#339412yes—
500C22#339812yes—
501C22#339812yes—
502C4—24no—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—84no—
508C4—81no—
509C4—162no—
510C4—162no—
511C4—162no—
512C4#256211yescharacteristic
513C4#263611yescharacteristic
514C4#263911yescharacteristic
515C4#275112yes—
516C4#279612yes—
517C4#281311yescharacteristic
518C2#1680211yescharacteristic
519C2#2809811yescharacteristic
520C2#2810811yescharacteristic
521C2#2873811yescharacteristic
522C2#2874411yescharacteristic
523C2#2984511yescharacteristic
524C2#2984811yescharacteristic
525C2#3157112yes—
526C2#3157212yes—
527C2#3216612yes—
528C2#3216912yes—
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×#19562
C2×#19162
#151802
#151522
#113572
#111902
#111222
#110772
#110102
#108612