Finite Group Databasebeta

← #409230

Subgroups — #409230

Order 512.

782 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21159C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#21020C211yesmaximal, characteristic
4#11331C211yesmaximal, characteristic
5#11331C211yesmaximal, characteristic
6#11290C212yesmaximal
7#11270C211yesmaximal, characteristic
8#11270C211yesmaximal, characteristic
9#11216C212yesmaximal
10#10913C212yesmaximal
11C4×#341C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×C4×M16C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×#389C2212yes—
21#1944—22no—
22#1943—22no—
23#1942—22no—
24#1942—22no—
25#1942—22no—
26#1941—22no—
27#1916—22no—
28#1916—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#341C2212yes—
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36#1736C2212yes—
37#1736C2212yes—
38C2×#340C2212yes—
39#1729C2211yescharacteristic
40#1729C2211yescharacteristic
41#1729C2211yescharacteristic
42#1728C2211yescharacteristic
43C22×M16C2311yescharacteristic
44C22×M16C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55#428—24no—
56#428—24no—
57#428—24no—
58#428—42no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—24no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—42no—
65C2×#106—22no—
66C2×#106—22no—
67#424—42no—
68C2×C4⋊C8—22no—
69C4×M16—24no—
70C4×M16—42no—
71C2×#97—22no—
72C2×C4×C8—22no—
73C2×C4×C8—42no—
74C2×C4×C8D412yes—
75C2×C4×C8D412yes—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389D411yescharacteristic
81#389D411yescharacteristic
82#389D411yescharacteristic
83#389D411yescharacteristic
84#385—21no—
85#385—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C2×#95—22no—
89#343—24no—
90#343—24no—
91#343—24no—
92#343—42no—
93#341—24no—
94#341Q814yes—
95#341Q814yes—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—21no—
105#336—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4C2411yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—21no—
127C2×M16C2×Q814yes—
128C2×M16C2×Q814yes—
129C2×M16#4114yes—
130C2×M16#4114yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C22×C8C2×Q812yes—
144C22×C8C2×Q812yes—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4#4114yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182#107—28no—
183#107—28no—
184#107—44no—
185#107—44no—
186#106—28no—
187#106—44no—
188C4⋊C8—44no—
189#97—44no—
190C4×C8—42no—
191C4×C8—42no—
192C4×C8—44no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—42no—
198#95—42no—
199C24C2×#4111yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C22×Q811yescharacteristic
227C22×C4C22×Q811yescharacteristic
228C22×C4C2×#4112yes—
229C22×C4C2×#4112yes—
230C22×C4C2×#4112yes—
231C22×C4H(5,𝔽2)12yes—
232C22×C42−1+411yescharacteristic
233C22×C42−1+411yescharacteristic
234M16—24no—
235M16—24no—
236M16—24no—
237M16—24no—
238M16—24no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16#12518yes—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8C4⋊Q814yes—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×#12111yescharacteristic
303C23#52912yes—
304C23#55312yes—
305C23#55711yescharacteristic
306C23#55711yescharacteristic
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—162no—
340C2×C4—162no—
341C2×C4#47212yes—
342C2×C4#47312yes—
343C2×C4#52511yescharacteristic
344C2×C4#52512yes—
345C2×C4C2×#12512yes—
346C2×C4C2×C4⋊Q811yescharacteristic
347C2×C4#53211yescharacteristic
348C2×C4#53211yescharacteristic
349C2×C4#53312yes—
350C2×C4#53411yescharacteristic
351C2×C4#53611yescharacteristic
352C2×C4#53912yes—
353C2×C4#54212yes—
354C2×C4#54612yes—
355C2×C4D4×Q811yescharacteristic
356C2×C4D4×Q811yescharacteristic
357C2×C4#55212yes—
358C2×C4#55412yes—
359C2×C4#55412yes—
360C2×C4#55612yes—
361C8—24no—
362C8—28no—
363C8—42no—
364C8—44no—
365C8—81no—
366C8—81no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22#239211yescharacteristic
376C22#250912yes—
377C22#254311yescharacteristic
378C22#254611yescharacteristic
379C22#258512yes—
380C22C2×#47211yescharacteristic
381C22C2×#47311yescharacteristic
382C22#306711yescharacteristic
383C22#306711yescharacteristic
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—84no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—162no—
394C4#246512yes—
395C4#247312yes—
396C4#258311yescharacteristic
397C4#260411yescharacteristic
398C4#266311yescharacteristic
399C4#285911yescharacteristic
400C2—22no—
401C2—22no—
402C2#1501211yescharacteristic
403C2#2371611yescharacteristic
404C2#2371711yescharacteristic
405C2#2859111yescharacteristic
406C2#2859211yescharacteristic
407C2#2869911yescharacteristic
408C2#2870211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#211592
C2×#19422
#210202
#113312
#112902
#112702
#112162
#109132
C4×#3412