Finite Group Databasebeta

← #411240

Subgroups — #411240

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14707C211yesmaximal, characteristic
2C4×#382C211yesmaximal, characteristic
3#11372C212yesmaximal
4#11358C211yesmaximal, characteristic
5#11358C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11268C211yesmaximal, characteristic
8#11268C212yesmaximal
9#10885C212yesmaximal
10#10884C212yesmaximal
11#10795C211yesmaximal, characteristic
12C2×#389C2211yescharacteristic
13C2×#389C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4C2212yes—
19C2×C43C2211yescharacteristic
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1436—22no—
33#1436—22no—
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37#1434—22no—
38C4×#95—22no—
39C23×C8C2312yes—
40C23×C8C2312yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2311yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C42—22no—
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55#389—24no—
56#389—24no—
57#382—22no—
58#382—22no—
59#382—24no—
60#382—22no—
61#382—24no—
62#382Q814yes—
63C4×C4⋊C4—42no—
64C4×C4⋊C4—42no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C43—24no—
77#340—24no—
78#340—24no—
79#340—24no—
80#340—24no—
81#340—42no—
82#340—42no—
83#340—42no—
84#340—42no—
85#336—22no—
86#336—22no—
87#336—24no—
88#336—24no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
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—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
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—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C2×Q814yes—
143C2×C4⋊C4#4112yes—
144C2×C4⋊C4#4112yes—
145C2×C4⋊C4#4114yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—44no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42C2×Q812yes—
164C2×C42#4112yes—
165C2×C42#4114yes—
166C2×C42#4114yes—
167#95—28no—
168#95—28no—
169#95—28no—
170#95—44no—
171#95—44no—
172#95—44no—
173#95—44no—
174#95—44no—
175#95—44no—
176#95—42no—
177#95—42no—
178C24C22×D411yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4C22⋊Q814yes—
233C22×C4#12314yes—
234C22×C4#12412yes—
235C22×C4#12412yes—
236C22×C4#13514yes—
237C22×C4C22×Q811yescharacteristic
238C22×C4C2×#4112yes—
239C22×C4C2×#4111yescharacteristic
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4112yes—
242C22×C4C2×#4111yescharacteristic
243C22×C4C2×#4112yes—
244C22×C4H(5,𝔽2)11yescharacteristic
245C22×C4H(5,𝔽2)11yescharacteristic
246C22×C42−1+412yes—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C42—24no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—48no—
286C42—48no—
287C42—88no—
288C23C2×C22⋊Q811yescharacteristic
289C23C2×#12311yescharacteristic
290C23#52511yescharacteristic
291C23#52511yescharacteristic
292C23C2×#12411yescharacteristic
293C23#53511yescharacteristic
294C23#53611yescharacteristic
295C23C2×#13512yes—
296C23#57512yes—
297C23#57512yes—
298C23#57512yes—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—82no—
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—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4#48814yes—
357C2×C4#52912yes—
358C2×C4#52912yes—
359C2×C4#52912yes—
360C2×C4#53312yes—
361C2×C4#53912yes—
362C2×C4#53912yes—
363C2×C4#55212yes—
364C2×C4#55212yes—
365C2×C4#55312yes—
366C2×C4#55312yes—
367C2×C4#55411yescharacteristic
368C2×C4#55411yescharacteristic
369C2×C4#55612yes—
370C2×C4#55712yes—
371C2×C4#55911yescharacteristic
372C2×C4#55911yescharacteristic
373C8—82no—
374C8—82no—
375C22#239811yescharacteristic
376C22#248411yescharacteristic
377C22#250411yescharacteristic
378C22#256811yescharacteristic
379C22#258011yescharacteristic
380C22#263311yescharacteristic
381C22#285511yescharacteristic
382C22#307612yes—
383C22#308012yes—
384C22#309112yes—
385C22#309312yes—
386C22#310212yes—
387C22#310212yes—
388C22#311112yes—
389C22#311112yes—
390C22C2×#48811yescharacteristic
391C22#313511yescharacteristic
392C22#314011yescharacteristic
393C22#314211yescharacteristic
394C22#321112yes—
395C22#321912yes—
396C22#323312yes—
397C22#324012yes—
398C4—24no—
399C4—42no—
400C4—44no—
401C4—44no—
402C4—82no—
403C4—82no—
404C4—84no—
405C4—84no—
406C4—161no—
407C4—161no—
408C4#258512yes—
409C4#258512yes—
410C4#259612yes—
411C4#259612yes—
412C2#1511111yescharacteristic
413C2#2381811yescharacteristic
414C2#2382611yescharacteristic
415C2#2761812yes—
416C2#2797711yescharacteristic
417C2#2797911yescharacteristic
418C2#2922211yescharacteristic
419C2#2922711yescharacteristic
420C2#2926112yes—
421C2#3008112yes—
422C2#3315212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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
#147072
C4×#3822
#113722
#113582
#112682
#108852
#108842
#107952