Finite Group Databasebeta

← #403161

Subgroups — #403161

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C211yesmaximal, characteristic
2C2×#1919C211yesmaximal, characteristic
3#15197C211yesmaximal, characteristic
4#13818C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11255C211yesmaximal, characteristic
7#11227C211yesmaximal, characteristic
8#11219C212yesmaximal
9#11219C212yesmaximal
10#10908C212yesmaximal
11#10806C212yesmaximal
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20#1947—24no—
21#1919—24no—
22C2×#340C2211yescharacteristic
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1442—22no—
34#1441—21no—
35#1441—21no—
36#1436—22no—
37C4⋊#95—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42—21no—
57C22×C42—22no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#107—24no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106Q814yes—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68#389—24no—
69#389—24no—
70#389—22no—
71#389—22no—
72#384—22no—
73#384—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—42no—
77C4×C4⋊C4D414yes—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95D411yescharacteristic
89C2×#95D411yescharacteristic
90C2×#95D412yes—
91#340—24no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#336—24no—
100#336—22no—
101#336—42no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2411yescharacteristic
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8C2×Q812yes—
129C22×C8#4114yes—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
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—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×Q812yes—
169C2×C4⋊C4C2×Q812yes—
170C2×C4⋊C4#4114yes—
171C2×C4⋊C4#4112yes—
172C2×C4⋊C4#4114yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—82no—
185C2×C42—82no—
186C2×C42C2×D412yes—
187C2×C42#4112yes—
188#107—48no—
189#106—28no—
190C4⋊C8—28no—
191#95—28no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—44no—
197#95—42no—
198#95—42no—
199#95—82no—
200#95—81no—
201#95—82no—
202#95—82no—
203#95—82no—
204C24C22×D411yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
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—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4#13512yes—
263C22×C4#13714yes—
264C22×C4#13712yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×Q811yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4112yes—
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C42−1+412yes—
274C22×C42−1+411yescharacteristic
275C2×C8—22no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—41no—
281C2×C8—42no—
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—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—88no—
315C4⋊C4—84no—
316C4⋊C4C22⋊Q814yes—
317C4⋊C4#12414yes—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—84no—
326C42—88no—
327C23C2×#12111yescharacteristic
328C23#52512yes—
329C23#54611yescharacteristic
330C23#54712yes—
331C23D4×Q811yescharacteristic
332C23C2×#13511yescharacteristic
333C23C2×#13711yescharacteristic
334C23C2×#13712yes—
335C23#57612yes—
336C23#57611yescharacteristic
337C23#57711yescharacteristic
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4#46414yes—
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4#52511yescharacteristic
400C2×C4C2×#12411yescharacteristic
401C2×C4#52912yes—
402C2×C4#52912yes—
403C2×C4#53211yescharacteristic
404C2×C4#53312yes—
405C2×C4#53611yescharacteristic
406C2×C4#53711yescharacteristic
407C2×C4#53912yes—
408C2×C4#54112yes—
409C2×C4#55212yes—
410C2×C4#55212yes—
411C2×C4#55211yescharacteristic
412C2×C4#55312yes—
413C2×C4#55411yescharacteristic
414C2×C4#55612yes—
415C2×C4#55712yes—
416C8—44no—
417C8—42no—
418C8—81no—
419C22#241311yescharacteristic
420C22#241711yescharacteristic
421C22#246212yes—
422C22#246612yes—
423C22#285011yescharacteristic
424C22C2×#46411yescharacteristic
425C22#303911yescharacteristic
426C22#304111yescharacteristic
427C22#304511yescharacteristic
428C22#310712yes—
429C22#311312yes—
430C22#311912yes—
431C22#312212yes—
432C22#330511yescharacteristic
433C22#331412yes—
434C22#332911yescharacteristic
435C22#333912yes—
436C22#334412yes—
437C22#334612yes—
438C22#335712yes—
439C22#338111yescharacteristic
440C22#338512yes—
441C22#338511yescharacteristic
442C4—24no—
443C4—44no—
444C4—44no—
445C4—42no—
446C4—44no—
447C4—82no—
448C4—84no—
449C4—82no—
450C4—161no—
451C4—162no—
452C4#239611yescharacteristic
453C4#253312yes—
454C4#256211yescharacteristic
455C4#256811yescharacteristic
456C4#259012yes—
457C4#266411yescharacteristic
458C2#1486911yescharacteristic
459C2#2458311yescharacteristic
460C2#2459811yescharacteristic
461C2#2481411yescharacteristic
462C2#2483311yescharacteristic
463C2#2686712yes—
464C2#2687912yes—
465C2#2704112yes—
466C2#2705312yes—
467C2#3303311yescharacteristic
468C2#3303811yescharacteristic
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
#151972
#138182
#113572
#112552
#112272
#112192
#109082
#108062