Finite Group Databasebeta

← #475243

Subgroups — #475243

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15133C212yesmaximal
2#11357C212yesmaximal
3#11328C212yesmaximal
4#11232C211yesmaximal, characteristic
5#11232C211yesmaximal, characteristic
6#11219C212yesmaximal
7#11207C212yesmaximal
8#10835C211yesmaximal, characteristic
9#10821C212yesmaximal
10C22×#107C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C2×#398—22no—
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#384C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1441—22no—
30#1441—22no—
31#1441—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C23×C8C2312yes—
36C23×C8C2311yescharacteristic
37C23×C8C2311yescharacteristic
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C42C2312yes—
49C22×C42C2311yescharacteristic
50C2×#107—24no—
51C2×#107—41no—
52C2×#107—42no—
53C2×#107—41no—
54C2×C4⋊C8—41no—
55C2×C4⋊C8—41no—
56C2×C4⋊C8Q814yes—
57C2×#97—41no—
58C2×#97—41no—
59C2×#97Q814yes—
60#398—48no—
61#389—24no—
62#389—42no—
63#389—42no—
64#389—42no—
65#384—24no—
66#384—42no—
67C4×C4⋊C4—41no—
68C4×C4⋊C4—41no—
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×#95D412yes—
78#340—24no—
79#340—24no—
80#340—24no—
81#340—41no—
82#340—42no—
83#340—42no—
84#340—41no—
85#340—42no—
86#336—42no—
87#336—42no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2411yescharacteristic
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8C2×Q814yes—
112C22×C8C2×Q812yes—
113C22×C8C2×Q812yes—
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—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
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—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—82no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4#4114yes—
153C2×C4⋊C4#4114yes—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42C2×Q812yes—
165C2×C42C2×Q812yes—
166#107—44no—
167#107—44no—
168C4⋊C8—44no—
169#97—44no—
170#95—42no—
171#95—44no—
172#95—44no—
173#95—44no—
174#95—44no—
175#95—44no—
176#95—82no—
177#95—82no—
178#95—82no—
179C24C22×D411yescharacteristic
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
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×C4C4⋊Q812yes—
231C22×C4C4⋊Q812yes—
232C22×C4AGL(1,ℤ/8)14yes—
233C22×C4#13714yes—
234C22×C4C22×Q811yescharacteristic
235C22×C4C22×Q811yescharacteristic
236C22×C4C2×#4112yes—
237C22×C4C2×#4112yes—
238C22×C4H(5,𝔽2)11yescharacteristic
239C22×C4H(5,𝔽2)11yescharacteristic
240C22×C4H(5,𝔽2)12yes—
241C22×C4H(5,𝔽2)11yescharacteristic
242C22×C42−1+412yes—
243C22×C42−1+411yescharacteristic
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—44no—
266C4⋊C4—44no—
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—42no—
278C4⋊C4—42no—
279C4⋊C4—88no—
280C4⋊C4—88no—
281C4⋊C4—88no—
282C4⋊C4—88no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C23C2×C4⋊Q811yescharacteristic
292C23#53411yescharacteristic
293C23#53711yescharacteristic
294C23#54612yes—
295C23#54812yes—
296C23C2×AGL(1,ℤ/8)12yes—
297C23C2×#13712yes—
298C23#57611yescharacteristic
299C23#57611yescharacteristic
300C23#57811yescharacteristic
301C23#57811yescharacteristic
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
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#53312yes—
357C2×C4#53912yes—
358C2×C4#53912yes—
359C2×C4#54012yes—
360C2×C4#54012yes—
361C2×C4#54311yescharacteristic
362C2×C4#54311yescharacteristic
363C2×C4#54411yescharacteristic
364C2×C4#54412yes—
365C2×C4#54411yescharacteristic
366C2×C4#55111yescharacteristic
367C2×C4#55212yes—
368C2×C4#55412yes—
369C2×C4#55511yescharacteristic
370C2×C4#55712yes—
371C2×C4#56011yescharacteristic
372C2×C4#56312yes—
373C2×C4#56311yescharacteristic
374C8—44no—
375C8—42no—
376C8—42no—
377C22#255012yes—
378C22#265911yescharacteristic
379C22#269911yescharacteristic
380C22#270712yes—
381C22#272811yescharacteristic
382C22#316411yescharacteristic
383C22#316411yescharacteristic
384C22#316512yes—
385C22#319211yescharacteristic
386C22#319511yescharacteristic
387C22#320311yescharacteristic
388C22#320311yescharacteristic
389C22#325412yes—
390C22#325911yescharacteristic
391C22#326111yescharacteristic
392C22#330812yes—
393C22#331212yes—
394C22#331412yes—
395C22#331812yes—
396C22#336612yes—
397C22#336712yes—
398C22#337412yes—
399C22#337512yes—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—44no—
404C4—84no—
405C4—84no—
406C4—162no—
407C4—162no—
408C4#261211yescharacteristic
409C4#266712yes—
410C4#274811yescharacteristic
411C4#279112yes—
412C4#280911yescharacteristic
413C4#284311yescharacteristic
414C2#1703611yescharacteristic
415C2#2880112yes—
416C2#2880412yes—
417C2#3029611yescharacteristic
418C2#3030111yescharacteristic
419C2#3173211yescharacteristic
420C2#3173611yescharacteristic
421C2#3197812yes—
422C2#3198412yes—
423C2#3255911yescharacteristic
424C2#3256311yescharacteristic
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
#151332
#113572
#113282
#112322
#112192
#112072
#108352
#108212