Finite Group Databasebeta

← #411224

Subgroups — #411224

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14703C211yesmaximal, characteristic
2C4×#382C211yesmaximal, characteristic
3#11316C212yesmaximal
4#11268C211yesmaximal, characteristic
5#11268C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11052C211yesmaximal, characteristic
8#11052C211yesmaximal, characteristic
9#10932C212yesmaximal
10#10884C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C2×#391C2211yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#389C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×C43C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1436—22no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36#1434—22no—
37#1434—22no—
38C4×#95—22no—
39C23×Q8C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C42—22no—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#103—24no—
55C2×#103—24no—
56C2×#103—24no—
57C2×#103—42no—
58C2×#103—42no—
59C2×#103—42no—
60#391—22no—
61#391—22no—
62#391—22no—
63#391—22no—
64#391—24no—
65#389—24no—
66#382—24no—
67#382—24no—
68C4×C4⋊C4—42no—
69C4×C4⋊C4—42no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C43—22no—
83C43—22no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—42no—
88#340—42no—
89#340—42no—
90#336—22no—
91#336—22no—
92#336—24no—
93#336—42no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8#4112yes—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4#4112yes—
153C2×C4⋊C4#4114yes—
154C2×C4⋊C4#4114yes—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42—44no—
175C2×C42#4112yes—
176C2×C42#4112yes—
177C2×C42#4114yes—
178#103—48no—
179#103—48no—
180#95—28no—
181#95—28no—
182#95—28no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—42no—
189#95—42no—
190#95—42no—
191#95—44no—
192C24C22×D411yescharacteristic
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4#12314yes—
265C22×C4#12412yes—
266C22×C4#12412yes—
267C22×C4#13514yes—
268C22×C4#13714yes—
269C22×C4C2×#4112yes—
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)12yes—
278C22×C42−1+411yescharacteristic
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—84no—
308C42—24no—
309C42—24no—
310C42—24no—
311C42—24no—
312C42—48no—
313C42—48no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—88no—
319C23C2×#12311yescharacteristic
320C23#52511yescharacteristic
321C23#52511yescharacteristic
322C23#52511yescharacteristic
323C23C2×#12411yescharacteristic
324C23#53511yescharacteristic
325C23#53711yescharacteristic
326C23C2×#13512yes—
327C23C2×#13712yes—
328C23#57512yes—
329C23#57512yes—
330Q8—44no—
331Q8—44no—
332Q8—82no—
333Q8—82no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4#48814yes—
394C2×C4#52912yes—
395C2×C4#52912yes—
396C2×C4#52912yes—
397C2×C4#52912yes—
398C2×C4#53812yes—
399C2×C4#53912yes—
400C2×C4#55112yes—
401C2×C4#55211yescharacteristic
402C2×C4#55312yes—
403C2×C4#55312yes—
404C2×C4#55312yes—
405C2×C4#55311yescharacteristic
406C2×C4#55511yescharacteristic
407C2×C4#55612yes—
408C2×C4#55611yescharacteristic
409C2×C4#55912yes—
410C8—82no—
411C8—82no—
412C22#239811yescharacteristic
413C22#248511yescharacteristic
414C22#250411yescharacteristic
415C22#259211yescharacteristic
416C22#259811yescharacteristic
417C22#263611yescharacteristic
418C22#285311yescharacteristic
419C22#309112yes—
420C22#309512yes—
421C22#310212yes—
422C22#310212yes—
423C22#310712yes—
424C22#311112yes—
425C22#311612yes—
426C22#311712yes—
427C22C2×#48811yescharacteristic
428C22#313511yescharacteristic
429C22#313911yescharacteristic
430C22#314211yescharacteristic
431C22#321112yes—
432C22#321612yes—
433C22#324812yes—
434C22#325612yes—
435C4—24no—
436C4—42no—
437C4—44no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4—84no—
442C4—84no—
443C4—161no—
444C4—161no—
445C4#257312yes—
446C4#259312yes—
447C4#259412yes—
448C4#259512yes—
449C2#1511011yescharacteristic
450C2#2381811yescharacteristic
451C2#2382511yescharacteristic
452C2#2763512yes—
453C2#2797711yescharacteristic
454C2#2798011yescharacteristic
455C2#2934911yescharacteristic
456C2#2935311yescharacteristic
457C2#2937512yes—
458C2#3013512yes—
459C2#3310112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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
#147032
C4×#3822
#113162
#112682
#111902
#110522
#109322
#108842
#107952