Finite Group Databasebeta

← #315733

Subgroups — #315733

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33171C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic, complemented
3#27909C211yesmaximal, characteristic, complemented
4#27909C211yesmaximal, characteristic, complemented
5#27906C211yesmaximal, characteristic, complemented
6C2×#1834C211yesmaximal, characteristic, complemented
7#10672C212yesmaximal, complemented
8#10671C212yesmaximal, complemented
9#9832C212yesmaximal, complemented
10#8820C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#481—22no—
14C2×#480—22no—
15C2×#480—22no—
16C2×#466—22no—
17C22×C4×C8C2211yescharacteristic
18#2856C2211yescharacteristic
19#2856C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2094—22no—
23#2093—22no—
24#2093—22no—
25#2071—22no—
26#1970—22no—
27#1957C2212yes—
28#1957C2212yes—
29#1957C2212yes—
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1874—22no—
34#1845C2212yes—
35#1845C2212yes—
36#1845C2211yescharacteristic
37#1844C2212yes—
38#1834—22no—
39#1834C2212yes—
40#1834C2212yes—
41#1834C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×#322C2212yes—
46#1273C2212yes—
47C22×D8—22no—
48C23×C8D412yes—
49C2×#127C2311yescharacteristic
50C2×#127C2311yescharacteristic
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42C2311yescharacteristic
64#493—24no—
65#481—48no—
66#480—48no—
67#480—48no—
68#466—48no—
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—22no—
72C2×#106—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—24no—
79C2×C22⋊C8D412yes—
80C2×C22⋊C8D412yes—
81C2×C4×C8—24no—
82C2×C4×C8D412yes—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2312yes—
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87#390—21no—
88#390—21no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2312yes—
102#384C2312yes—
103#340—24no—
104#340—24no—
105#340—24no—
106#336D412yes—
107#336D412yes—
108#322—24no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—41no—
116C22×D4—41no—
117C23×C4—41no—
118C23×C4—41no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×D8—44no—
123C2×D8—44no—
124C2×D8—44no—
125C22×C8—24no—
126C22×C8—24no—
127C22×C8—42no—
128C22×C8C2×D412yes—
129C22×C8C2×D414yes—
130C22×C8C2×D412yes—
131C22×C8C2×D414yes—
132#127—22no—
133#127—22no—
134#127—22no—
135#127—22no—
136#121—44no—
137#121—44no—
138#121—44no—
139#121—44no—
140#121—44no—
141#121—44no—
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—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—82no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2411yescharacteristic
178#107—44no—
179#107—48no—
180#107—48no—
181#106—48no—
182#102—44no—
183#102—44no—
184#102—48no—
185#102—48no—
186#102—48no—
187#102—84no—
188C22⋊C8—24no—
189C22⋊C8D818yescomplemented
190C4×C8—24no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8#4114yes—
197C24—41no—
198C24—41no—
199C24#12711yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—82no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C2×D4—82no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4#12711yescharacteristic
244C22×C4#12711yescharacteristic
245C22×C4#12711yescharacteristic
246C22×C4C2×D814yes—
247C22×C4AGL(1,ℤ/8)14yes—
248C22×C4AGL(1,ℤ/8)14yes—
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252D8—88no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—48no—
264C2×C8#12314yes—
265C2×C8C2×D818yes—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C22⋊C4—82no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42C2×#4112yes—
295C422−1+412yes—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23#49312yes—
315C23#49612yes—
316C23#49612yes—
317C23C2×#12711yescharacteristic
318D4—84no—
319D4—84no—
320D4—84no—
321D4—84no—
322D4—82no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—164no—
327D4—164no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—164no—
351C2×C4—164no—
352C2×C4#49312yes—
353C2×C4#49612yes—
354C2×C4#49612yes—
355C2×C4C2×#12311yescharacteristic
356C2×C4#53611yescharacteristic
357C2×C4#54712yes—
358C2×C4#54712yes—
359C2×C4C22×D812yes—
360C2×C4C2×AGL(1,ℤ/8)12yes—
361C2×C4C2×AGL(1,ℤ/8)12yes—
362C2×C4#57512yes—
363C2×C4#57712yes—
364C2×C4#57812yes—
365C8—28no—
366C8—24no—
367C8—44no—
368C8—42no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22—161no—
388C22—161no—
389C22#172114yes—
390C22#250111yescharacteristic
391C22C2×#49311yescharacteristic
392C22C2×#49611yescharacteristic
393C22C2×#49611yescharacteristic
394C22#315311yescharacteristic
395C22#315411yescharacteristic
396C22#315611yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—162no—
401C4—161no—
402C4—161no—
403C4—162no—
404C4#309412yes—
405C4#323412yes—
406C4#333414yes—
407C4#334514yes—
408C2—24no—
409C2—321no—
410C2—321no—
411C2C2×#172111yescharacteristic
412C2#1963111yescharacteristic
413C2#1967411yescharacteristic
414C2#1970011yescharacteristic
415C2#2791811yescharacteristic
416C2#2792911yescharacteristic
417C2#2793411yescharacteristic
subgroupG/H
center, socleC23C2×#127
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. #27973C2semidirect
3. #27909C2semidirect
4. #27909C2semidirect
5. #27906C2semidirect
6. C2×#1834C2semidirect
7. #10672C2semidirect
8. #10671C2semidirect
9. #9832C2semidirect
10. #8820C2semidirect
11. #8145C2semidirect
189. C22⋊C8D8semidirect

Maximal subgroups

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

subgroupindex
#331712
#279732
#279092
#279062
C2×#18342
#106722
#106712
#98322
#88202
#81452