Finite Group Databasebeta

← #355654

Subgroups — #355654

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C212yesmaximal
2#13761C211yesmaximal, characteristic
3#10922C212yesmaximal
4#10877C212yesmaximal, complemented
5#10833C211yesmaximal, characteristic, complemented
6#10797C212yesmaximal, complemented
7#10796C211yesmaximal, characteristic, complemented
8#10713C211yesmaximal, characteristic
9#10713C211yesmaximal, characteristic
10#10693C211yesmaximal, characteristic, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2212yes—
17C22×C4×C8C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20C2×#376C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#1923C4116yescomplemented
23C2×#336C2211yescharacteristic
24C2×#336C2211yescharacteristic
25C2×#336C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1437C414yes—
35#1436C414yes—
36C4×#95C2211yescharacteristic
37C4×#95C2211yescharacteristic
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—42no—
46C2×#117—42no—
47C22×C4⋊C4C2×C412yes—
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C42C2×C414yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—42no—
59C2×C4⋊C8—42no—
60C2×C4⋊C8C2×C418yes—
61C2×C22⋊C8—42no—
62C2×C22⋊C8—42no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8—42no—
65C2×C22⋊C8C2×C418yes—
66C2×C22⋊C8C2×C418yes—
67C2×#97—24no—
68C2×#97—42no—
69C2×#97—42no—
70C2×C4×C8—42no—
71C2×C4×C8—42no—
72C2×C4×C8C2×C418yes—
73C4×C22⋊C4—24no—
74C4×C22⋊C4C2×C418yes—
75#376—42no—
76C2×#95C2×C414yes—
77C2×#95C2×C412yes—
78C2×#95C2×C414yes—
79C2×#95C2×C414yes—
80C2×#95C2×C412yes—
81C43—42no—
82#336—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86#336—24no—
87#336—42no—
88#336—42no—
89#336—42no—
90#336—42no—
91#336C2×C418yes—
92#336C2×C418yes—
93C25C22×C411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C22×C412yes—
104C23×C4C22×C412yes—
105C23×C4C22×C412yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C414yes—
108C23×C4C22×C412yes—
109C23×C4C22×C412yes—
110C23×C4C22×C412yes—
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8C22×C414yes—
137C22×C8C22×C414yes—
138C22×C8C22×C414yes—
139C22×C8C22×C414yes—
140#117—48no—
141#117—48no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4C4218yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—28no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—44no—
173C2×C42—44no—
174C2×C42C22×C414yes—
175C2×C42C22×C414yes—
176C2×C42#4112yes—
177C2×C42#4112yes—
178C2×C42#4114yes—
179C4⋊C8—48no—
180C22⋊C8—48no—
181C22⋊C8—48no—
182#97—48no—
183C4×C8—48no—
184#95—24no—
185#95—28no—
186#95—44no—
187#95—44no—
188#95C4218yes—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—24no—
196C24—24no—
197C24C2×C4212yes—
198C24C23×C411yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4C2×C4212yes—
247C22×C4C2×C4214yes—
248C22×C4C2×C4214yes—
249C22×C4#11714yes—
250C22×C4#11714yes—
251C22×C4#13118yes—
252C22×C4C23×C412yes—
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4112yes—
256C22×C4H(5,𝔽2)11yescharacteristic
257C22×C4H(5,𝔽2)11yescharacteristic
258C22×C4H(5,𝔽2)12yes—
259C22×C4H(5,𝔽2)12yes—
260C22×C42−1+411yescharacteristic
261C22×C42−1+411yescharacteristic
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—48no—
291C22⋊C4—216no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—416no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—42no—
307C23—24no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23C22×C4211yescharacteristic
325C23C2×#11711yescharacteristic
326C23C2×#11711yescharacteristic
327C23#51811yescharacteristic
328C23#51811yescharacteristic
329C23#52011yescharacteristic
330C23#52011yescharacteristic
331C23C2×#13114yes—
332C23#56814yes—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4#40518yes—
374C2×C4C4×#4112yes—
375C2×C4C4×#4112yes—
376C2×C4#51812yes—
377C2×C4#51812yes—
378C2×C4#52012yes—
379C2×C4#52012yes—
380C2×C4#53812yes—
381C2×C4#53812yes—
382C2×C4#53812yes—
383C2×C4#53912yes—
384C2×C4#54011yescharacteristic
385C2×C4#54012yes—
386C2×C4#54111yescharacteristic
387C2×C4#54212yes—
388C2×C4#54211yescharacteristic
389C2×C4#54211yescharacteristic
390C8—44no—
391C8—44no—
392C8—44no—
393C8—44no—
394C22—44no—
395C22—44no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22C4×#11711yescharacteristic
404C22#227511yescharacteristic
405C22#227511yescharacteristic
406C22#231211yescharacteristic
407C22#231411yescharacteristic
408C22#231811yescharacteristic
409C22#231911yescharacteristic
410C22C2×#40512yes—
411C22#287512yes—
412C22#292112yes—
413C22#292212yes—
414C22#292212yes—
415C22#292312yes—
416C22#297614yes—
417C22#298114yes—
418C22#299214yes—
419C22#299314yes—
420C4—28no—
421C4—44no—
422C4—44no—
423C4—42no—
424C4—44no—
425C4—82no—
426C4—82no—
427C4#233312yes—
428C4#233412yes—
429C4#233512yes—
430C4#233712yes—
431C2—82no—
432C2#1378811yescharacteristic
433C2#2046211yescharacteristic
434C2#2046511yescharacteristic
435C2#2056012yes—
436C2#2056512yes—
437C2#2190512yes—
438C2#2192512yes—
439C2#2199912yes—
440C2#2201112yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10877C2semidirect
5. #10833C2semidirect
6. #10797C2semidirect
7. #10796C2semidirect
10. #10693C2semidirect
11. C4×#336C2semidirect
22. #1923C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
#137612
#109222
#108772
#108332
#107972
#107962
#107132
#106932
C4×#3362