Finite Group Databasebeta

← #470169

Subgroups — #470169

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 463 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#13936C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11256C212yesmaximal
9#11232C212yesmaximal
10#11231C212yesmaximal
11#11227C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1947—24no—
20#1919—24no—
21C2×#340C2211yescharacteristic
22C2×#340C2211yescharacteristic
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31#1443—22no—
32#1442—22no—
33#1441—21no—
34#1441—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42—21no—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—42no—
60C2×#107—42no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—42no—
69#389—22no—
70#389—22no—
71#384—22no—
72#384—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4D414yes—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89C2×#95D412yes—
90#340—24no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#336—22no—
99#336—42no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8#4114yes—
127C22×C8#4112yes—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
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—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
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—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—81no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4#4112yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—82no—
178C2×C42C2×D412yes—
179C2×C42#4112yes—
180#107—48no—
181#106—28no—
182C4⋊C8—28no—
183C4⋊C8—48no—
184#95—24no—
185#95—22no—
186#95—24no—
187#95—22no—
188#95—42no—
189#95—42no—
190#95—44no—
191#95—82no—
192#95—82no—
193#95—81no—
194#95—82no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24C22×D411yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
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—22no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4#12112yes—
256C22×C4#12112yes—
257C22×C4#13512yes—
258C22×C4#13714yes—
259C22×C4#13712yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4H(5,𝔽2)12yes—
266C22×C42−1+412yes—
267C22×C42−1+412yes—
268C22×C42−1+411yescharacteristic
269C22×C42−1+411yescharacteristic
270C2×C8—24no—
271C2×C8—22no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C42—22no—
313C42—22no—
314C42—24no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—84no—
320C23C2×#12111yescharacteristic
321C23#53712yes—
322C23#54712yes—
323C23#54711yescharacteristic
324C23#54811yescharacteristic
325C23C2×#13511yescharacteristic
326C23C2×#13712yes—
327C23C2×#13711yescharacteristic
328C23#57612yes—
329C23#57611yescharacteristic
330C23#57711yescharacteristic
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
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—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4#46414yes—
391C2×C4#52511yescharacteristic
392C2×C4#52511yescharacteristic
393C2×C4#53511yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4#53611yescharacteristic
397C2×C4#54112yes—
398C2×C4#54112yes—
399C2×C4#54112yes—
400C2×C4#54112yes—
401C2×C4#54212yes—
402C2×C4#55112yes—
403C2×C4#55212yes—
404C2×C4#55211yescharacteristic
405C2×C4#55312yes—
406C2×C4#55611yescharacteristic
407C2×C4#56312yes—
408C2×C4#56312yes—
409C8—44no—
410C8—42no—
411C8—81no—
412C22#241311yescharacteristic
413C22#241511yescharacteristic
414C22#262011yescharacteristic
415C22#272212yes—
416C22#272512yes—
417C22C2×#46411yescharacteristic
418C22#303911yescharacteristic
419C22#304111yescharacteristic
420C22#304511yescharacteristic
421C22#324812yes—
422C22#325512yes—
423C22#325712yes—
424C22#325912yes—
425C22#333912yes—
426C22#334011yescharacteristic
427C22#334412yes—
428C22#334612yes—
429C22#334612yes—
430C22#335712yes—
431C22#335711yescharacteristic
432C22#336111yescharacteristic
433C22#337211yescharacteristic
434C22#337512yes—
435C4—24no—
436C4—42no—
437C4—44no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—161no—
443C4—162no—
444C4—162no—
445C4#259811yescharacteristic
446C4#263411yescharacteristic
447C4#263411yescharacteristic
448C4#275812yes—
449C4#276612yes—
450C4#282111yescharacteristic
451C2#1667811yescharacteristic
452C2#2458311yescharacteristic
453C2#2459811yescharacteristic
454C2#2467911yescharacteristic
455C2#2469611yescharacteristic
456C2#2983111yescharacteristic
457C2#2983411yescharacteristic
458C2#3241512yes—
459C2#3242012yes—
460C2#3248512yes—
461C2#3249212yes—
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
#139362
#113572
#113222
#113152
#112562
#112322
#112312
#112272