Finite Group Databasebeta

← #410713

Subgroups — #410713

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1836C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic
3#15235C211yesmaximal, characteristic
4#15196C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11228C212yesmaximal
9#10880C212yesmaximal
10#10821C212yesmaximal
11#10821C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1836—24no—
21#1770—24no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32#1444—22no—
33#1443—22no—
34#1442—22no—
35#1441—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×#97—24no—
59C2×#97—22no—
60C2×#97—24no—
61C2×#97D414yes—
62C2×C4×C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65#389—24no—
66#389—24no—
67#384—22no—
68#384—22no—
69#382—22no—
70#382—22no—
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×#95D412yes—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—24no—
92#336—22no—
93#336—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2411yescharacteristic
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8C2×D412yes—
119C22×C8C2×D412yes—
120C22×C8#4112yes—
121C22×C8#4112yes—
122C22×C8#4114yes—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
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—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4#4114yes—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42C2×D412yes—
167C2×C42#4112yes—
168C2×C42#4112yes—
169C2×C42#4112yes—
170C4⋊C8—28no—
171#97—28no—
172#97—44no—
173C4×C8—28no—
174#95—24no—
175#95—44no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—44no—
180#95—82no—
181#95—82no—
182#95—82no—
183#95—82no—
184#95—82no—
185C24C22×D411yescharacteristic
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
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—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#12114yes—
235C22×C4#12412yes—
236C22×C4#12412yes—
237C22×C4#13714yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4112yes—
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4111yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4112yes—
245C22×C4C2×#4111yescharacteristic
246C22×C42−1+412yes—
247C22×C42−1+412yes—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8#12314yes—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—48no—
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—42no—
285C4⋊C4—88no—
286C4⋊C4—88no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—24no—
294C42—41no—
295C42—41no—
296C42—42no—
297C23C2×#12111yescharacteristic
298C23C2×#12411yescharacteristic
299C23#53211yescharacteristic
300C23#54712yes—
301C23#54812yes—
302C23C2×#13712yes—
303C23#57512yes—
304C23#57612yes—
305C23#57712yes—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
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—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#49114yes—
358C2×C4C2×#12311yescharacteristic
359C2×C4#52511yescharacteristic
360C2×C4#52511yescharacteristic
361C2×C4#52912yes—
362C2×C4#52912yes—
363C2×C4#52911yescharacteristic
364C2×C4#52911yescharacteristic
365C2×C4#53211yescharacteristic
366C2×C4#53611yescharacteristic
367C2×C4#53611yescharacteristic
368C2×C4#54112yes—
369C2×C4#54112yes—
370C2×C4#55212yes—
371C2×C4#55212yes—
372C2×C4#55212yes—
373C2×C4#55212yes—
374C2×C4#55212yes—
375C2×C4#55612yes—
376C8—24no—
377C8—42no—
378C8—42no—
379C8—42no—
380C22#248611yescharacteristic
381C22#248912yes—
382C22#251912yes—
383C22#254511yescharacteristic
384C22#285011yescharacteristic
385C22#305612yes—
386C22#306212yes—
387C22C2×#49111yescharacteristic
388C22#313511yescharacteristic
389C22#313612yes—
390C22#317411yescharacteristic
391C22#317611yescharacteristic
392C22#318112yes—
393C22#335112yes—
394C22#335512yes—
395C22#335612yes—
396C22#335712yes—
397C22#336912yes—
398C22#337012yes—
399C22#337512yes—
400C22#337612yes—
401C4—24no—
402C4—44no—
403C4—42no—
404C4—82no—
405C4—84no—
406C4—162no—
407C4—162no—
408C4—162no—
409C4#239511yescharacteristic
410C4#248411yescharacteristic
411C4#252412yes—
412C4#252812yes—
413C4#261111yescharacteristic
414C4#263411yescharacteristic
415C2#1508511yescharacteristic
416C2#2767112yes—
417C2#2773312yes—
418C2#2773812yes—
419C2#2805912yes—
420C2#2806112yes—
421C2#2865112yes—
422C2#3305311yescharacteristic
423C2#3305511yescharacteristic
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×#18362
C2×#17702
#152352
#151962
#113732
#113582
#113152
#112282
#108802
#108212