Finite Group Databasebeta

← #351385

Subgroups — #351385

Order 512.

708 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic
2#29925C212yesmaximal, complemented
3#20476C211yesmaximal, characteristic, complemented
4#9734C211yesmaximal, characteristic, complemented
5#9734C211yesmaximal, characteristic, complemented
6#9615C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#9572C212yesmaximal, complemented
9#9571C212yesmaximal, complemented
10#9142C212yesmaximal, complemented
11#8596C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#466—21no—
14C2×#466—21no—
15C2×#432C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2856C2212yes—
18#2283C2211yescharacteristic
19#2092—22no—
20#2073—22no—
21#2034—22no—
22#2033—22no—
23#1982C2212yes—
24#1982C2212yes—
25#1970—21no—
26#1970—21no—
27#1957C2212yes—
28#1948C2212yes—
29#1947C2212yes—
30#1845C2211yescharacteristic
31#1797C2212yes—
32#1770C2211yescharacteristic
33#1770C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1752—21no—
37C8×C22⋊C4—21no—
38#1746—21no—
39#1726—21no—
40C2×#335C2211yescharacteristic
41C2×#335C2211yescharacteristic
42C2×#331C2212yes—
43C2×#331C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1277C2212yes—
47C22×D8—21no—
48C22×D8—21no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51C2×#127C2312yes—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C22×C42C2311yescharacteristic
56#493—22no—
57#493—22no—
58#466—44no—
59#466—44no—
60#432—22no—
61#432—22no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107—22no—
65C2×#106—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—22no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—22no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97—21no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8D411yescharacteristic
86C2×C4×C8D411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#390—22no—
90#390—22no—
91#390—22no—
92#384C2312yes—
93#383—21no—
94#382D411yescharacteristic
95#382D411yescharacteristic
96#381—21no—
97#380D411yescharacteristic
98#380D411yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C4×C22⋊C4—21no—
102#376—21no—
103#340—22no—
104#340—22no—
105#336—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—22no—
110#335—22no—
111#335—22no—
112#331—24no—
113#331—24no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—42no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C2×D8—42no—
122C2×D8—42no—
123C2×D8—42no—
124C2×D8—42no—
125C2×D8—42no—
126C2×D8—42no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—41no—
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×D411yescharacteristic
141#127—22no—
142#127—22no—
143#127—22no—
144#127—22no—
145#121—44no—
146#121—44no—
147#121—82no—
148#121—82no—
149#121—84no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—42no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—42no—
173#107—42no—
174#107—82no—
175#106—82no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8#4112yes—
183C4⋊C8#4112yes—
184#102—44no—
185#102—44no—
186#102—82no—
187#102—82no—
188#102—82no—
189#102—82no—
190C22⋊C8—24no—
191C22⋊C8—82no—
192#97—24no—
193#97—42no—
194#97—42no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—42no—
200C4×C8—42no—
201C4×C8#4112yes—
202C4×C8#4112yes—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211C24—42no—
212C24#12111yescharacteristic
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4—84no—
230C2×D4—84no—
231C2×D4—84no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C2×#4111yescharacteristic
259D8—84no—
260D8—84no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—44no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—44no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8#12312yes—
283C2×C8#12312yes—
284C2×C8#13512yes—
285C2×C8AGL(1,ℤ/8)12yes—
286C2×C8AGL(1,ℤ/8)12yes—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—84no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42C2×#4111yescharacteristic
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C42C2×#4111yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23#47212yes—
333C23C2×#12111yescharacteristic
334D4—82no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—82no—
339D4—82no—
340D4—82no—
341D4—82no—
342D4—162no—
343D4—162no—
344D4—168no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4C2×#12311yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4#52911yescharacteristic
384C2×C4#52911yescharacteristic
385C2×C4#53211yescharacteristic
386C2×C4#53211yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4#57511yescharacteristic
391C2×C4#57611yescharacteristic
392C2×C4#57611yescharacteristic
393C2×C4#57611yescharacteristic
394C2×C4#57811yescharacteristic
395C8—22no—
396C8—22no—
397C8—41no—
398C8—42no—
399C8—42no—
400C8—82no—
401C8—82no—
402C8#48214yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—162no—
411C22—162no—
412C22—162no—
413C22—162no—
414C22—162no—
415C22—162no—
416C22—162no—
417C22#285411yescharacteristic
418C22#303511yescharacteristic
419C22#303811yescharacteristic
420C22#305811yescharacteristic
421C22#306111yescharacteristic
422C22C2×#47211yescharacteristic
423C22#306711yescharacteristic
424C4—21no—
425C4—21no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—162no—
434C4C2×#48211yescharacteristic
435C4#309411yescharacteristic
436C4#310611yescharacteristic
437C4#311511yescharacteristic
438C4#317511yescharacteristic
439C4#317511yescharacteristic
440C4#317611yescharacteristic
441C4#317611yescharacteristic
442C2—42no—
443C2—322no—
444C2#1921911yescharacteristic
445C2#1922211yescharacteristic
446C2#1935311yescharacteristic
447C2#1935611yescharacteristic
448C2#3311311yescharacteristic
449C2#3312311yescharacteristic
450C2#3313811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #29925C2semidirect
3. #20476C2semidirect
4. #9734C2semidirect
5. #9734C2semidirect
6. #9615C2semidirect
7. #9614C2semidirect
8. #9572C2semidirect
9. #9571C2semidirect
10. #9142C2semidirect
11. #8596C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331712
#299252
#204762
#97342
#96152
#96142
#95722
#95712
#91422
#85962