Finite Group Databasebeta

← #417030

Subgroups — #417030

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14348C211yesmaximal, characteristic
2#14277C211yesmaximal, characteristic
3#11316C212yesmaximal
4#11315C212yesmaximal
5#11293C212yesmaximal
6#11292C212yesmaximal
7#10843C211yesmaximal, characteristic
8#10843C211yesmaximal, characteristic
9#10842C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#97C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#376C2211yescharacteristic
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1444—22no—
32#1442—22no—
33#1442—22no—
34#1442—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#97—22no—
55C2×#97—22no—
56C2×#97—22no—
57C2×C4×C8—22no—
58C2×C4×C8—22no—
59C2×C4×C8—22no—
60#389—42no—
61#389—42no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384—22no—
66#382—22no—
67#382—22no—
68#382—22no—
69#382Q814yes—
70#376—22no—
71#376—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D412yes—
83C2×#95D412yes—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#336—24no—
93#336—24no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2411yescharacteristic
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8#4114yes—
118C22×C8#4114yes—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
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—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
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—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C2×Q812yes—
156C2×C4⋊C4C2×Q812yes—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42C2×Q812yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168C2×C42#4112yes—
169C2×C42#4112yes—
170C2×C42#4112yes—
171#97—44no—
172C4×C8—44no—
173#95—28no—
174#95—24no—
175#95—24no—
176#95—42no—
177#95—42no—
178#95—44no—
179#95—82no—
180#95—82no—
181#95—82no—
182#95—82no—
183#95—82no—
184#95—82no—
185C24C22×D411yescharacteristic
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
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—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
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×C4C22⋊Q814yes—
242C22×C4#12314yes—
243C22×C4#12414yes—
244C22×C4C22×Q811yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4112yes—
251C22×C4C2×#4112yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+412yes—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
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—88no—
285C4⋊C4—88no—
286C4⋊C4—84no—
287C4⋊C4—84no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42#12514yes—
293C42#12514yes—
294C42#12614yes—
295C42#12614yes—
296C23C2×C22⋊Q811yescharacteristic
297C23C2×#12311yescharacteristic
298C23C2×#12411yescharacteristic
299C23#53211yescharacteristic
300C23#53211yescharacteristic
301C23#53511yescharacteristic
302C23#53611yescharacteristic
303C23#57612yes—
304C23#57712yes—
305C23#57712yes—
306C23#57812yes—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4#40018yes—
365C2×C4C2×#12511yescharacteristic
366C2×C4C2×#12511yescharacteristic
367C2×C4C2×#12611yescharacteristic
368C2×C4C2×#12611yescharacteristic
369C2×C4#52912yes—
370C2×C4#52912yes—
371C2×C4#54212yes—
372C2×C4#54212yes—
373C2×C4#55212yes—
374C2×C4#55212yes—
375C2×C4#55312yes—
376C2×C4#55312yes—
377C2×C4#55412yes—
378C2×C4#55512yes—
379C2×C4#55612yes—
380C2×C4#55712yes—
381C8—44no—
382C8—44no—
383C22C2×#40011yescharacteristic
384C22#249911yescharacteristic
385C22#250011yescharacteristic
386C22#256511yescharacteristic
387C22#256611yescharacteristic
388C22#264011yescharacteristic
389C22#286111yescharacteristic
390C22#308312yes—
391C22#308412yes—
392C22#309612yes—
393C22#309712yes—
394C22#313612yes—
395C22#313712yes—
396C22#317312yes—
397C22#317512yes—
398C22#317812yes—
399C22#317912yes—
400C22#322011yescharacteristic
401C22#322111yescharacteristic
402C22#322211yescharacteristic
403C22#322311yescharacteristic
404C22#324112yes—
405C22#324212yes—
406C4—28no—
407C4—82no—
408C4—82no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4—162no—
413C4—162no—
414C4—161no—
415C4—161no—
416C4#249412yes—
417C4#249512yes—
418C4#250812yes—
419C4#250912yes—
420C2#1523911yescharacteristic
421C2#2374812yes—
422C2#2788612yes—
423C2#2790211yescharacteristic
424C2#2790311yescharacteristic
425C2#2918812yes—
426C2#2919911yescharacteristic
427C2#2920011yescharacteristic
428C2#3020911yescharacteristic
429C2#3021011yescharacteristic
430C2#3328312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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
#143482
#142772
#113162
#113152
#112932
#112922
#108432
#108422
#108412
#106942