Finite Group Databasebeta

← #475953

Subgroups — #475953

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2101C212yesmaximal
2#14707C211yesmaximal, characteristic
3#14707C211yesmaximal, characteristic
4#11358C211yesmaximal, characteristic
5#11358C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11268C211yesmaximal, characteristic
8#11228C212yesmaximal
9#11227C212yesmaximal
10#11208C212yesmaximal
11#10832C211yesmaximal, characteristic
12C22×C4⋊C8C2212yes—
13C2×#389C2212yes—
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C43C2211yescharacteristic
21#2101—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1436—22no—
34#1436—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—24no—
57C2×C4⋊C8—42no—
58C2×C4⋊C8—42no—
59#389—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#382—22no—
64#382—22no—
65#382—42no—
66#382—42no—
67#382D414yes—
68#382Q814yes—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—21no—
80C43—42no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—42no—
89#336—24no—
90#336—22no—
91#336—22no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2411yescharacteristic
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—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—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—21no—
133C2×C4⋊C4—21no—
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—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×Q814yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—24no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42C2×D412yes—
166C2×C42C2×Q812yes—
167C2×C42#4112yes—
168C2×C42#4112yes—
169C2×C42#4114yes—
170C4⋊C8—48no—
171#95—28no—
172#95—28no—
173#95—44no—
174#95—44no—
175#95—44no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—42no—
180#95—42no—
181#95—44no—
182C24C22×D411yescharacteristic
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
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—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
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—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
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#12114yes—
237C22×C4C22⋊Q814yes—
238C22×C4#12412yes—
239C22×C4#12412yes—
240C22×C4#13514yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×Q811yescharacteristic
243C22×C4C2×#4112yes—
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C22×C4H(5,𝔽2)12yes—
247C22×C4H(5,𝔽2)11yescharacteristic
248C22×C42−1+412yes—
249C22×C42−1+411yescharacteristic
250C22×C42−1+412yes—
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—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
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—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—84no—
282C4⋊C4—88no—
283C4⋊C4—84no—
284C42—24no—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—24no—
290C42—48no—
291C42—48no—
292C23C2×#12111yescharacteristic
293C23C2×C22⋊Q811yescharacteristic
294C23C2×#12411yescharacteristic
295C23#53411yescharacteristic
296C23#53611yescharacteristic
297C23#53611yescharacteristic
298C23#53711yescharacteristic
299C23C2×#13512yes—
300C23#57512yes—
301C23#57512yes—
302C23#57512yes—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—42no—
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—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#48814yes—
361C2×C4#53712yes—
362C2×C4#53812yes—
363C2×C4#53912yes—
364C2×C4#53912yes—
365C2×C4#53912yes—
366C2×C4#54412yes—
367C2×C4#54712yes—
368C2×C4D4×Q811yescharacteristic
369C2×C4#55011yescharacteristic
370C2×C4#55411yescharacteristic
371C2×C4#55612yes—
372C2×C4#55612yes—
373C2×C4#55611yescharacteristic
374C2×C4#55712yes—
375C2×C4#56312yes—
376C2×C4#56412yes—
377C8—82no—
378C8—82no—
379C22#253811yescharacteristic
380C22#254711yescharacteristic
381C22#256811yescharacteristic
382C22#261811yescharacteristic
383C22#262711yescharacteristic
384C22#263911yescharacteristic
385C22#282211yescharacteristic
386C22#303512yes—
387C22#304012yes—
388C22#307612yes—
389C22#308012yes—
390C22C2×#48811yescharacteristic
391C22#313511yescharacteristic
392C22#314011yescharacteristic
393C22#314211yescharacteristic
394C22#319112yes—
395C22#319712yes—
396C22#323312yes—
397C22#323312yes—
398C22#324012yes—
399C22#324012yes—
400C22#325012yes—
401C22#325612yes—
402C4—24no—
403C4—44no—
404C4—44no—
405C4—42no—
406C4—82no—
407C4—82no—
408C4—84no—
409C4—162no—
410C4—161no—
411C4—161no—
412C4#272612yes—
413C4#272712yes—
414C4#280512yes—
415C4#281012yes—
416C2#1707411yescharacteristic
417C2#2828512yes—
418C2#2872111yescharacteristic
419C2#2872711yescharacteristic
420C2#2922211yescharacteristic
421C2#2922711yescharacteristic
422C2#2976212yes—
423C2#2997412yes—
424C2#3018611yescharacteristic
425C2#3019811yescharacteristic
426C2#3285112yes—
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
C2×#21012
#147072
#113582
#112682
#112282
#112272
#112082
#108322