Finite Group Databasebeta

← #404789

Subgroups — #404789

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C211yesmaximal, characteristic
2C2×#1919C211yesmaximal, characteristic
3#15242C211yesmaximal, characteristic
4#13812C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic
6#11325C212yesmaximal
7#11322C211yesmaximal, characteristic
8#11232C212yesmaximal
9#11227C211yesmaximal, characteristic
10#10850C212yesmaximal
11#10806C212yesmaximal
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21#1947—24no—
22#1919—24no—
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1445—22no—
34#1441—21no—
35#1441—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39#1434—21no—
40#1434—22no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—21no—
56C22×C42—22no—
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59C2×#107—24no—
60C2×#107—42no—
61C2×#107—42no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—42no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384—22no—
74#382—24no—
75#382—42no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4D414yes—
79C4×C4⋊C4Q818yes—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D411yescharacteristic
92C2×#95D411yescharacteristic
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—42no—
101#340—42no—
102#336—22no—
103#336—24no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8#4112yes—
131C22×C8#4114yes—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×Q814yes—
169C2×C4⋊C4C2×Q814yes—
170C2×C4⋊C4#4112yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—28no—
179C2×C42—82no—
180C2×C42—82no—
181C2×C42—82no—
182C2×C42C2×D412yes—
183C2×C42C2×Q814yes—
184C2×C42#4112yes—
185C2×C42#4114yes—
186#107—48no—
187#106—28no—
188C4⋊C8—28no—
189C4⋊C8—48no—
190#95—24no—
191#95—28no—
192#95—22no—
193#95—22no—
194#95—42no—
195#95—44no—
196#95—44no—
197#95—44no—
198#95—42no—
199#95—42no—
200#95—82no—
201#95—82no—
202#95—82no—
203#95—81no—
204C24C22×D411yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4#12112yes—
260C22×C4#12112yes—
261C22×C4C22⋊Q814yes—
262C22×C4C22⋊Q814yes—
263C22×C4#13512yes—
264C22×C4#13712yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q812yes—
267C22×C4C2×#4112yes—
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C42−1+412yes—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4C4⋊Q814yes—
317C42—22no—
318C42—24no—
319C42—22no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—84no—
325C42—88no—
326C23C2×#12111yescharacteristic
327C23C2×C22⋊Q812yes—
328C23#54611yescharacteristic
329C23#54712yes—
330C23#55011yescharacteristic
331C23C2×#13511yescharacteristic
332C23C2×#13711yescharacteristic
333C23#57512yes—
334C23#57612yes—
335C23#57711yescharacteristic
336C23#57811yescharacteristic
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
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—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#46414yes—
397C2×C4#52511yescharacteristic
398C2×C4#52511yescharacteristic
399C2×C4C2×C4⋊Q811yescharacteristic
400C2×C4#53211yescharacteristic
401C2×C4#53312yes—
402C2×C4#53312yes—
403C2×C4#53312yes—
404C2×C4#53511yescharacteristic
405C2×C4#53611yescharacteristic
406C2×C4#54412yes—
407C2×C4#54412yes—
408C2×C4#55112yes—
409C2×C4#55212yes—
410C2×C4#55212yes—
411C2×C4#55212yes—
412C2×C4#55312yes—
413C2×C4#55311yescharacteristic
414C2×C4#55511yescharacteristic
415C8—44no—
416C8—42no—
417C8—81no—
418C22#241311yescharacteristic
419C22#241411yescharacteristic
420C22#243812yes—
421C22#244112yes—
422C22#285711yescharacteristic
423C22C2×#46411yescharacteristic
424C22#304011yescharacteristic
425C22#304111yescharacteristic
426C22#304611yescharacteristic
427C22#307612yes—
428C22#307812yes—
429C22#308112yes—
430C22#308312yes—
431C22#330211yescharacteristic
432C22#330912yes—
433C22#333111yescharacteristic
434C22#333912yes—
435C22#334612yes—
436C22#334912yes—
437C22#335112yes—
438C22#339411yescharacteristic
439C22#339811yescharacteristic
440C22#340312yes—
441C4—24no—
442C4—42no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4—82no—
447C4—82no—
448C4—84no—
449C4—161no—
450C4—162no—
451C4#239911yescharacteristic
452C4#260311yescharacteristic
453C4#260612yes—
454C4#261111yescharacteristic
455C4#261212yes—
456C4#266011yescharacteristic
457C2#1490311yescharacteristic
458C2#2458311yescharacteristic
459C2#2459311yescharacteristic
460C2#2462911yescharacteristic
461C2#2464711yescharacteristic
462C2#2599712yes—
463C2#2601812yes—
464C2#2609612yes—
465C2#2610112yes—
466C2#3320211yescharacteristic
467C2#3320311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#19472
C2×#19192
#152422
#138122
#113712
#113252
#113222
#112322
#112272
#108502
#108062