Finite Group Databasebeta

← #242547

Subgroups — #242547

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic, complemented
2#21954C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21809C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic, complemented
6C2×#1553C211yesmaximal, characteristic
7#10602C212yesmaximal, complemented
8#10599C212yesmaximal, complemented
9#10598C212yesmaximal, complemented
10#10595C212yesmaximal, complemented
11#8040C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2285C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#1990C2211yescharacteristic
20#1990C2211yescharacteristic
21#1982C2212yes—
22#1981C2212yes—
23#1959—21no—
24#1959—21no—
25#1957C2211yescharacteristic
26#1957C2211yescharacteristic
27#1948C2212yes—
28#1948C2212yes—
29#1940—22no—
30#1939C2212yes—
31#1939C2212yes—
32#1874—22no—
33#1853—22no—
34#1844—21no—
35#1834—21no—
36#1802—22no—
37#1798—22no—
38#1788—22no—
39#1572—24no—
40C2×#335C2212yes—
41C2×#334C2212yes—
42#1553—24no—
43C2×#332C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C22×#41—21no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532Q814yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×#127C2311yescharacteristic
54C2×#124—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×#117—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42C2311yescharacteristic
64#432—22no—
65#432—22no—
66#432—22no—
67#432—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431D414yes—
72C2×#106—22no—
73C2×#106—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8D412yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8D412yes—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#384—21no—
96#384—21no—
97#384—21no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#382—21no—
103#380—21no—
104#380D412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2312yes—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#335—24no—
114#334—24no—
115#332—24no—
116#331—24no—
117C2×#41—24no—
118C2×#41—42no—
119C2×#41—82no—
120C2×#41—82no—
121C22×Q8—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—41no—
126C23×C4—41no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8C2×Q812yes—
140#127—21no—
141#127—21no—
142#127—21no—
143#127C2×Q812yes—
144#124—24no—
145#124—42no—
146#121—24no—
147#121—24no—
148#121—42no—
149#121—42no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—42no—
153C4×D4—42no—
154#117—82no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187#106—82no—
188#106—82no—
189C4⋊C8—24no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—82no—
196C4⋊C8—82no—
197C4⋊C8C2×D414yes—
198#103—82no—
199#103—82no—
200#102—82no—
201#102—82no—
202#102—82no—
203#102—82no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—42no—
207C22⋊C8—42no—
208#97—42no—
209#97—42no—
210C4×C8—24no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215#95—22no—
216#95—22no—
217C24—41no—
218C24#12111yescharacteristic
219#41—88no—
220#41—88no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—44no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—81no—
237C2×D4—81no—
238C2×D4—82no—
239C2×D4—82no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—44no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4#12112yes—
258C22×C4#12112yes—
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4C22⋊Q812yes—
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4#12612yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—82no—
308C42—42no—
309C42—42no—
310C42#12112yes—
311C42#12112yes—
312C42C22⋊Q812yes—
313C42C22⋊Q812yes—
314C42C22⋊Q812yes—
315C42#12612yes—
316C42C22×D411yescharacteristic
317C42C22×Q811yescharacteristic
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—42no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23#39612yes—
334C23C2×#12111yescharacteristic
335Q8—44no—
336Q8—82no—
337D4—44no—
338D4—44no—
339D4—82no—
340D4—82no—
341D4—88no—
342D4—82no—
343D4—82no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—88no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—164no—
375C2×C4#39612yes—
376C2×C4#39612yes—
377C2×C4#39612yes—
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×#12111yescharacteristic
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×C22⋊Q811yescharacteristic
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×#12611yescharacteristic
384C2×C4#57612yes—
385C2×C4#57712yes—
386C2×C4#57712yes—
387C2×C4#57712yes—
388C8—44no—
389C8—42no—
390C8—42no—
391C8—82no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22C2×#39611yescharacteristic
408C22#304211yescharacteristic
409C22#304311yescharacteristic
410C22#305711yescharacteristic
411C22#305811yescharacteristic
412C22#306811yescharacteristic
413C22#306911yescharacteristic
414C4—21no—
415C4—21no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4#304512yes—
425C4#306012yes—
426C4#308112yes—
427C4#308412yes—
428C2—42no—
429C2—162no—
430C2#1951311yescharacteristic
431C2#1951411yescharacteristic
432C2#1952111yescharacteristic
433C2#1952211yescharacteristic
434C2#2360411yescharacteristic
435C2#2361011yescharacteristic
436C2#2361211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
1. #21981C2semidirect
2. #21954C2semidirect
5. C2×#1572C2semidirect
7. #10602C2semidirect
8. #10599C2semidirect
9. #10598C2semidirect
10. #10595C2semidirect
11. #8040C2semidirect

Maximal subgroups

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

subgroupindex
#219812
#219542
#218592
#218092
C2×#15722
C2×#15532
#106022
#105992
#105982
#105952
#80402