Finite Group Databasebeta

← #405010

Subgroups — #405010

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1948C211yesmaximal, characteristic
2C2×#1918C211yesmaximal, characteristic
3#15197C211yesmaximal, characteristic
4#13810C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11256C212yesmaximal
8#11255C211yesmaximal, characteristic
9#11227C211yesmaximal, characteristic
10#10908C212yesmaximal
11#10908C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#376—21no—
20#1948—24no—
21#1918—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—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—22no—
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⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—22no—
56C22×C42—21no—
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67#389—22no—
68#389—22no—
69#384—22no—
70#384—22no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4D414yes—
75#376—44no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D411yescharacteristic
88C2×#95D412yes—
89C2×#95D411yescharacteristic
90#340—24no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#336—24no—
99#336—24no—
100#336—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4C2×D412yes—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—21no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8#4112yes—
128C22×C8#4114yes—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
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—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
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⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4#4114yes—
167C2×C4⋊C4#4112yes—
168C2×C4⋊C4#4114yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—28no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—44no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—44no—
183C2×C42C2×D412yes—
184C2×C42#4112yes—
185#107—28no—
186#106—48no—
187C4⋊C8—28no—
188#95—28no—
189#95—24no—
190#95—22no—
191#95—22no—
192#95—24no—
193#95—44no—
194#95—42no—
195#95—42no—
196#95—81no—
197#95—82no—
198#95—82no—
199#95—82no—
200#95—82no—
201#95—82no—
202C24C22×D411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
264C22×C4#13714yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4112yes—
272C22×C42−1+412yes—
273C22×C42−1+411yescharacteristic
274C22×C42−1+411yescharacteristic
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
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—42no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4#12414yes—
317C42—22no—
318C42—22no—
319C42—24no—
320C42—416no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—48no—
326C23C2×#12111yescharacteristic
327C23#52512yes—
328C23#54712yes—
329C23#54711yescharacteristic
330C23#54811yescharacteristic
331C23C2×#13511yescharacteristic
332C23C2×AGL(1,ℤ/8)11yescharacteristic
333C23C2×#13712yes—
334C23#57611yescharacteristic
335C23#57712yes—
336C23#57711yescharacteristic
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—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
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—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4#46314yes—
399C2×C4#52511yescharacteristic
400C2×C4#52511yescharacteristic
401C2×C4C2×#12411yescharacteristic
402C2×C4#52912yes—
403C2×C4#52912yes—
404C2×C4#52912yes—
405C2×C4#53211yescharacteristic
406C2×C4#53611yescharacteristic
407C2×C4#53611yescharacteristic
408C2×C4#54112yes—
409C2×C4#54112yes—
410C2×C4#55212yes—
411C2×C4#55212yes—
412C2×C4#55212yes—
413C2×C4#55212yes—
414C2×C4#55211yescharacteristic
415C2×C4#55612yes—
416C2×C4#55611yescharacteristic
417C8—42no—
418C8—44no—
419C8—81no—
420C22#241311yescharacteristic
421C22#241511yescharacteristic
422C22#247512yes—
423C22#247812yes—
424C22#285011yescharacteristic
425C22C2×#46311yescharacteristic
426C22#303911yescharacteristic
427C22#304311yescharacteristic
428C22#304511yescharacteristic
429C22#310712yes—
430C22#311012yes—
431C22#311912yes—
432C22#312612yes—
433C22#333712yes—
434C22#334111yescharacteristic
435C22#334412yes—
436C22#334412yes—
437C22#334512yes—
438C22#335411yescharacteristic
439C22#335712yes—
440C22#336011yescharacteristic
441C22#336611yescharacteristic
442C22#336912yes—
443C4—24no—
444C4—44no—
445C4—42no—
446C4—44no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—84no—
451C4—161no—
452C4—162no—
453C4#239811yescharacteristic
454C4#252812yes—
455C4#253312yes—
456C4#260411yescharacteristic
457C4#261111yescharacteristic
458C4#264211yescharacteristic
459C2#1490811yescharacteristic
460C2#2458111yescharacteristic
461C2#2459811yescharacteristic
462C2#2467711yescharacteristic
463C2#2469811yescharacteristic
464C2#2737312yes—
465C2#2738112yes—
466C2#2746912yes—
467C2#2747712yes—
468C2#3303211yescharacteristic
469C2#3303911yescharacteristic
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×#19482
C2×#19182
#151972
#138102
#113572
#113162
#112562
#112552
#112272
#109082