Finite Group Databasebeta

← #467941

Subgroups — #467941

Order 512.

1098 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2043C212yesmaximal
2#15151C211yesmaximal, characteristic
3#14155C211yesmaximal, characteristic
4#11202C211yesmaximal, characteristic
5#11190C212yesmaximal
6#11190C212yesmaximal
7#11077C212yesmaximal
8#11050C212yesmaximal
9#10884C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C2×#398—21no—
17C2×#398—21no—
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2043—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1445—21no—
33#1445—21no—
34#1445—21no—
35#1445—21no—
36C4⋊#95—21no—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C23×Q8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8D414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—41no—
64C2×C4⋊C8—42no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77#398—44no—
78#398—44no—
79#391—24no—
80#391—24no—
81#391—42no—
82#391—42no—
83#389—24no—
84#389—24no—
85#384—41no—
86#384—41no—
87C4×C4⋊C4—41no—
88C4×C4⋊C4—41no—
89C4×C4⋊C4—41no—
90C4×C4⋊C4—41no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—41no—
109#336—42no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
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×Q8C2×D414yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
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—
149C4⋊Q8—44no—
150C4⋊Q8—44no—
151C4⋊Q8—44no—
152C4⋊Q8—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
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—24no—
162C2×C4⋊C4—24no—
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—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4SD1614yes—
190C2×C4⋊C4#4114yes—
191C2×C4⋊C4#4114yes—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×D412yes—
208C2×C42#4112yes—
209C4⋊C8—44no—
210#103—48no—
211#103—48no—
212#103—48no—
213#103—48no—
214#95—41no—
215#95—41no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—42no—
221#95—42no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—44no—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
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—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
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#12112yes—
309C22×C4#12112yes—
310C22×C4C2×SD1612yes—
311C22×C4C2×SD1612yes—
312C22×C4C2×SD1612yes—
313C22×C4AGL(1,ℤ/8)12yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)12yes—
324C22×C42−1+412yes—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
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—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—88no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—42no—
363C42—42no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—84no—
369C23C2×#12111yescharacteristic
370C23#53411yescharacteristic
371C23#53511yescharacteristic
372C23#54612yes—
373C23#54812yes—
374C23C22×SD1611yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376C23C2×#13711yescharacteristic
377C23#57511yescharacteristic
378C23#57812yes—
379C23#57812yes—
380Q8—44no—
381Q8—44no—
382Q8—44no—
383Q8—44no—
384Q8—82no—
385Q8—82no—
386Q8—82no—
387Q8—82no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
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#46014yes—
448C2×C4#52912yes—
449C2×C4#53411yescharacteristic
450C2×C4#53411yescharacteristic
451C2×C4#53811yescharacteristic
452C2×C4#53811yescharacteristic
453C2×C4#53912yes—
454C2×C4#54012yes—
455C2×C4#54012yes—
456C2×C4#54212yes—
457C2×C4#54212yes—
458C2×C4#54612yes—
459C2×C4#54712yes—
460C2×C4#55312yes—
461C2×C4#55511yescharacteristic
462C2×C4#55911yescharacteristic
463C2×C4#56011yescharacteristic
464C2×C4#56212yes—
465C2×C4#56311yescharacteristic
466C8—82no—
467C8—82no—
468C22#242712yes—
469C22#261711yescharacteristic
470C22#270911yescharacteristic
471C22#271112yes—
472C22#273111yescharacteristic
473C22C2×#46011yescharacteristic
474C22#303711yescharacteristic
475C22#304412yes—
476C22#318711yescharacteristic
477C22#319311yescharacteristic
478C22#320211yescharacteristic
479C22#320211yescharacteristic
480C22#320611yescharacteristic
481C22#321311yescharacteristic
482C22#322312yes—
483C22#329812yes—
484C22#331212yes—
485C22#331512yes—
486C22#333012yes—
487C22#335912yes—
488C22#336712yes—
489C22#337412yes—
490C22#337712yes—
491C4—24no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—84no—
498C4—84no—
499C4—84no—
500C4—162no—
501C4#259311yescharacteristic
502C4#265112yes—
503C4#269911yescharacteristic
504C4#270112yes—
505C4#272411yescharacteristic
506C4#283011yescharacteristic
507C2#1659611yescharacteristic
508C2#2532012yes—
509C2#2533312yes—
510C2#2971311yescharacteristic
511C2#2972411yescharacteristic
512C2#3202711yescharacteristic
513C2#3203911yescharacteristic
514C2#3208212yes—
515C2#3210312yes—
516C2#3263011yescharacteristic
517C2#3264411yescharacteristic
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×#20432
#151512
#141552
#112022
#111902
#110772
#110502
#108842