Finite Group Databasebeta

← #409661

Subgroups — #409661

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14707C211yesmaximal, characteristic
2#14703C211yesmaximal, characteristic
3#11292C212yesmaximal
4#11208C211yesmaximal, characteristic
5#11208C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11052C211yesmaximal, characteristic
8#11052C211yesmaximal, characteristic
9#10926C212yesmaximal
10#10820C212yesmaximal
11C4×#336C211yesmaximal, characteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×C4×C8C2212yes—
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C43C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32#1436—22no—
33#1436—22no—
34#1436—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38#1434—22no—
39C23×Q8C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
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—
60C2×C4×C8—24no—
61C2×C4×C8—24no—
62C2×C4×C8—42no—
63#391—22no—
64#391—22no—
65#391—22no—
66#391—42no—
67#391Q814yes—
68#389—22no—
69#389—22no—
70#389—42no—
71#389—42no—
72#382—24no—
73#382—42no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—21no—
88C43—42no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—42no—
93#340—42no—
94#340—42no—
95#336—24no—
96#336—22no—
97#336—22no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8C2×Q812yes—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2411yescharacteristic
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—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8#4114yes—
132C22×C8#4114yes—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4C2×Q812yes—
163C2×C4⋊C4#4114yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C2×Q812yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182#103—48no—
183#103—48no—
184C4×C8—48no—
185#95—24no—
186#95—24no—
187#95—28no—
188#95—42no—
189#95—42no—
190#95—44no—
191#95—44no—
192#95—42no—
193#95—42no—
194#95—42no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199C24C22×D411yescharacteristic
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C22⋊Q814yes—
271C22×C4#12412yes—
272C22×C4#12412yes—
273C22×C4#13514yes—
274C22×C4#13514yes—
275C22×C4#13714yes—
276C22×C4C22×Q811yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C4H(5,𝔽2)11yescharacteristic
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—84no—
317C4⋊C4—88no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—48no—
323C42—48no—
324C42—42no—
325C42—42no—
326C23C2×C22⋊Q811yescharacteristic
327C23#52511yescharacteristic
328C23C2×#12411yescharacteristic
329C23#53211yescharacteristic
330C23#53211yescharacteristic
331C23#53411yescharacteristic
332C23#53511yescharacteristic
333C23C2×#13512yes—
334C23C2×#13512yes—
335C23C2×#13712yes—
336C23#57512yes—
337Q8—44no—
338Q8—44no—
339Q8—82no—
340Q8—82no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#47918yes—
401C2×C4#52912yes—
402C2×C4#52912yes—
403C2×C4#52912yes—
404C2×C4#53312yes—
405C2×C4#53812yes—
406C2×C4#54012yes—
407C2×C4#55112yes—
408C2×C4#55112yes—
409C2×C4#55312yes—
410C2×C4#55312yes—
411C2×C4#55312yes—
412C2×C4#55411yescharacteristic
413C2×C4#55412yes—
414C2×C4#55411yescharacteristic
415C2×C4#55511yescharacteristic
416C2×C4#55511yescharacteristic
417C8—44no—
418C8—44no—
419C22#239611yescharacteristic
420C22#256311yescharacteristic
421C22#256411yescharacteristic
422C22#259111yescharacteristic
423C22#260311yescharacteristic
424C22#266111yescharacteristic
425C22#286111yescharacteristic
426C22C2×#47912yes—
427C22#307612yes—
428C22#310212yes—
429C22#310712yes—
430C22#313111yescharacteristic
431C22#313111yescharacteristic
432C22#313911yescharacteristic
433C22#314011yescharacteristic
434C22#316812yes—
435C22#316812yes—
436C22#317212yes—
437C22#318112yes—
438C22#319012yes—
439C22#319112yes—
440C22#321012yes—
441C22#321112yes—
442C4—28no—
443C4—44no—
444C4—42no—
445C4—42no—
446C4—84no—
447C4—82no—
448C4—82no—
449C4—162no—
450C4—161no—
451C4—161no—
452C4#251812yes—
453C4#252212yes—
454C4#252612yes—
455C4#258712yes—
456C2#1504711yescharacteristic
457C2#2377912yes—
458C2#2914211yescharacteristic
459C2#2914411yescharacteristic
460C2#2915812yes—
461C2#2933211yescharacteristic
462C2#2933311yescharacteristic
463C2#2949612yes—
464C2#3032112yes—
465C2#3327911yescharacteristic
466C2#3328111yescharacteristic
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
#147072
#147032
#112922
#112082
#110672
#110522
#109262
#108202
C4×#3362