Finite Group Databasebeta

← #472421

Subgroups — #472421

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic
2C2×#1916C211yesmaximal, characteristic
3#15221C212yesmaximal
4#11377C211yesmaximal, characteristic
5#11377C211yesmaximal, characteristic
6#11328C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11256C211yesmaximal, characteristic
9#11227C212yesmaximal
10#11208C212yesmaximal
11#10842C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20#1942—24no—
21#1916—24no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1445—22no—
35#1444—22no—
36#1442—22no—
37#1442—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—42no—
60C2×#107Q814yes—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—42no—
63C2×C4×C8—42no—
64C2×C4×C8D414yes—
65#389—24no—
66#389—24no—
67#389—24no—
68#389—42no—
69#382—24no—
70#382—42no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95D411yescharacteristic
84C2×#95D411yescharacteristic
85#340—24no—
86#340—22no—
87#340—22no—
88#340—24no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8C2×D412yes—
125C22×C8C2×D412yes—
126C22×C8C2×Q812yes—
127C22×C8#4112yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
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—42no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C2×Q814yes—
162C2×C4⋊C4#4114yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C2×D412yes—
174C2×C42#4112yes—
175#107—28no—
176#107—44no—
177C4⋊C8—44no—
178C4×C8—44no—
179#95—22no—
180#95—22no—
181#95—44no—
182#95—44no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—82no—
189#95—82no—
190#95—82no—
191C24C22×D411yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
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—42no—
246C22×C4#12112yes—
247C22×C4#12112yes—
248C22×C4#13512yes—
249C22×C4#13512yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×Q811yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4112yes—
254C22×C4C2×#4111yescharacteristic
255C22×C4H(5,𝔽2)12yes—
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C42−1+412yes—
258C22×C42−1+412yes—
259C22×C42−1+411yescharacteristic
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8C22⋊Q814yes—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—88no—
299C4⋊C4—88no—
300C42—22no—
301C42—22no—
302C42—24no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—42no—
308C42—42no—
309C23C2×#12111yescharacteristic
310C23#53612yes—
311C23#54612yes—
312C23D4×Q811yescharacteristic
313C23#55011yescharacteristic
314C23C2×#13511yescharacteristic
315C23C2×#13511yescharacteristic
316C23#57512yes—
317C23#57612yes—
318C23#57811yescharacteristic
319C23#57811yescharacteristic
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4#53211yescharacteristic
380C2×C4#53411yescharacteristic
381C2×C4#53611yescharacteristic
382C2×C4#53611yescharacteristic
383C2×C4#53711yescharacteristic
384C2×C4#53912yes—
385C2×C4#53912yes—
386C2×C4#54112yes—
387C2×C4#54112yes—
388C2×C4#54412yes—
389C2×C4#55212yes—
390C2×C4#55312yes—
391C2×C4#55411yescharacteristic
392C2×C4#55611yescharacteristic
393C2×C4#55712yes—
394C2×C4#56212yes—
395C2×C4#56312yes—
396C8—24no—
397C8—42no—
398C8—81no—
399C8—81no—
400C22#253811yescharacteristic
401C22#254311yescharacteristic
402C22#262111yescharacteristic
403C22#269412yes—
404C22#271312yes—
405C22#305011yescharacteristic
406C22#305411yescharacteristic
407C22#305811yescharacteristic
408C22#305911yescharacteristic
409C22#323212yes—
410C22#323312yes—
411C22#324312yes—
412C22#324312yes—
413C22#329712yes—
414C22#330512yes—
415C22#331312yes—
416C22#331512yes—
417C22#338311yescharacteristic
418C22#338311yescharacteristic
419C22#338912yes—
420C22#339311yescharacteristic
421C22#339311yescharacteristic
422C22#340312yes—
423C4—42no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—84no—
428C4—84no—
429C4—81no—
430C4—81no—
431C4—162no—
432C4—162no—
433C4#256511yescharacteristic
434C4#262711yescharacteristic
435C4#266411yescharacteristic
436C4#277612yes—
437C4#280212yes—
438C4#281411yescharacteristic
439C2#1682011yescharacteristic
440C2#2826811yescharacteristic
441C2#2826911yescharacteristic
442C2#2855211yescharacteristic
443C2#2856411yescharacteristic
444C2#2987911yescharacteristic
445C2#2988011yescharacteristic
446C2#3157912yes—
447C2#3158212yes—
448C2#3214012yes—
449C2#3214512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#19422
C2×#19162
#152212
#113772
#113282
#113152
#112562
#112272
#112082
#108422