Finite Group Databasebeta

← #410542

Subgroups — #410542

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C212yesmaximal
2C2×#1933C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4#15192C211yesmaximal, characteristic
5#15149C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11122C212yesmaximal
9#11077C211yesmaximal, characteristic
10#10861C211yesmaximal, characteristic
11#10820C212yesmaximal
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#398—21no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1933—28no—
26#1933—24no—
27C8×C4⋊C4—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1443—22no—
37#1442—22no—
38#1442—21no—
39#1441—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#106—24no—
60C2×#106—24no—
61C2×#106—42no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—22no—
64C2×#103—24no—
65C2×#103—22no—
66C2×#103—24no—
67C2×#103—22no—
68C2×#103—24no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103D414yes—
72C2×C4×C8—24no—
73C2×C4×C8—24no—
74C2×C4×C8—24no—
75C2×C4×C8—24no—
76C2×C4×C8D414yes—
77#398—44no—
78#391—24no—
79#391—22no—
80#391—22no—
81#391—42no—
82#389—24no—
83#389—22no—
84#389—22no—
85#389—42no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95D412yes—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—24no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8C2×D412yes—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8#4114yes—
146C22×C8#4112yes—
147C22×C8#4114yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
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—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4SD1614yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4#4114yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C2×D412yes—
194C2×C42#4112yes—
195#106—48no—
196C4⋊C8—28no—
197#103—216no—
198#103—28no—
199#103—44no—
200C4×C8—28no—
201C4×C8—48no—
202#95—24no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—42no—
207#95—42no—
208#95—42no—
209#95—42no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—42no—
214#95—81no—
215#95—81no—
216#95—82no—
217C24C22×D411yescharacteristic
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C2×SD1614yes—
293C22×C4C2×SD1612yes—
294C22×C4#13514yes—
295C22×C4#13512yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4C2×#4111yescharacteristic
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C42−1+412yes—
305C22×C42−1+411yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8#12114yes—
322C2×C8#12314yes—
323C4⋊C4—28no—
324C4⋊C4—28no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—88no—
344C4⋊C4—84no—
345C42—22no—
346C42—22no—
347C42—24no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C23C2×#12111yescharacteristic
353C23#53212yes—
354C23D4211yescharacteristic
355C23#54711yescharacteristic
356C23#54812yes—
357C23C22×SD1611yescharacteristic
358C23C2×#13512yes—
359C23C2×#13511yescharacteristic
360C23#57711yescharacteristic
361C23#57711yescharacteristic
362C23#57812yes—
363Q8—42no—
364Q8—44no—
365Q8—42no—
366Q8—81no—
367Q8—81no—
368Q8—82no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4#46514yes—
429C2×C4C2×#12111yescharacteristic
430C2×C4C2×#12311yescharacteristic
431C2×C4#52511yescharacteristic
432C2×C4#52512yes—
433C2×C4#52912yes—
434C2×C4#52912yes—
435C2×C4#53211yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#53711yescharacteristic
438C2×C4#54112yes—
439C2×C4#54212yes—
440C2×C4#54711yescharacteristic
441C2×C4#54712yes—
442C2×C4#55112yes—
443C2×C4#55212yes—
444C2×C4#55212yes—
445C2×C4#55511yescharacteristic
446C2×C4#55612yes—
447C8—24no—
448C8—24no—
449C8—44no—
450C22#246112yes—
451C22#251912yes—
452C22#253711yescharacteristic
453C22#254511yescharacteristic
454C22#285411yescharacteristic
455C22C2×#46511yescharacteristic
456C22#305011yescharacteristic
457C22#305911yescharacteristic
458C22#306011yescharacteristic
459C22#316712yes—
460C22#316812yes—
461C22#317812yes—
462C22#317812yes—
463C22D4×SD1611yescharacteristic
464C22#328411yescharacteristic
465C22#328512yes—
466C22#333511yescharacteristic
467C22#333612yes—
468C22#334011yescharacteristic
469C22#335912yes—
470C22#336012yes—
471C22#336412yes—
472C22#336512yes—
473C4—24no—
474C4—44no—
475C4—44no—
476C4—42no—
477C4—82no—
478C4—84no—
479C4—82no—
480C4—161no—
481C4—162no—
482C4—161no—
483C4#239211yescharacteristic
484C4#247812yes—
485C4#248611yescharacteristic
486C4#249811yescharacteristic
487C4#253212yes—
488C4#263711yescharacteristic
489C2#1508111yescharacteristic
490C2#2682012yes—
491C2#2682312yes—
492C2#2803012yes—
493C2#2803312yes—
494C2#2822311yescharacteristic
495C2#2822611yescharacteristic
496C2#2863911yescharacteristic
497C2#2864211yescharacteristic
498C2#3312811yescharacteristic
499C2#3313311yescharacteristic
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×#19332
C2×C8×C4⋊C42
#151922
#151492
#112272
#112202
#111222
#110772
#108612
#108202