Finite Group Databasebeta

← #6435245

Subgroups — #6435245

Order 512.

2002 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#34012C211yesmaximal, characteristic
3#29926C218yesmaximal, complemented
4#29923C214yesmaximal, complemented
5#29922C214yesmaximal, complemented
6#23732C212yesmaximal, complemented
7#23629C212yesmaximal, complemented
8#23481C214yesmaximal
9#21259C212yesmaximal, complemented
10#21171C212yesmaximal, complemented
11#20626C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C2×#428C2214yes—
16C2×#423C2212yes—
17C2×#419C2214yes—
18C2×#419C2214yes—
19C2×#407C2212yes—
20#2857—22no—
21#2857—22no—
22#2857C2214yes—
23#2857C2214yes—
24C2×#391C2212yes—
25C2×#389—22no—
26#2285C2212yes—
27#2285C2212yes—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#2283—22no—
32#2283—22no—
33#2283C2214yes—
34#2280—22no—
35#2280C2212yes—
36C2×#380—22no—
37C4×#117—22no—
38C4×#117C2212yes—
39#2097C2218yescomplemented
40#2096C2214yescomplemented
41#2095C2214yescomplemented
42#2074C2218yescomplemented
43#2072C2218yescomplemented
44#2036C22116yes—
45#2035C22116yes—
46#1981C2218yescomplemented
47#1981C2218yescomplemented
48#1948C2218yescomplemented
49#1947C2218yescomplemented
50#1797C2214yes—
51#1797C2214yes—
52#1736C2214yes—
53#1728C2212yes—
54C23×Q8C2312yes—
55C24×C4D412yes—
56C24×C4C2311yescharacteristic
57C22×M16C2314yes—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—28no—
61C2×C4⋊Q8C2314yes—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—48no—
64C2×C22⋊Q8C2318yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117C2312yes—
68C2×#117C2312yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C42—28no—
75C22×C42D412yes—
76C22×C42D412yes—
77C22×C42D412yes—
78C22×C42C2312yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#428—48no—
83C2×#107C2318yes—
84C2×#106C2318yes—
85#423—28no—
86C2×C4⋊C8C2318yes—
87#419—48no—
88#419—48no—
89C2×#103C23116yes—
90C2×#103C2318yes—
91C2×#103C2318yes—
92#407—44no—
93C2×C22⋊C8C2318yes—
94#391—24no—
95#391C2314yes—
96#391C2314yes—
97#389—24no—
98#389—24no—
99#389—24no—
100#389—24no—
101#384—21no—
102#384—21no—
103#384C2312yes—
104#384C2314yes—
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#383—24no—
108#382—22no—
109#382—22no—
110#382C2312yes—
111#382C2312yes—
112#381—24no—
113#381—22no—
114#380—24no—
115#380—24no—
116#380—24no—
117#380—24no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380C2314yes—
123#380C2314yes—
124#379—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2314yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134#376—24no—
135#376—22no—
136C2×#95—24no—
137C43—22no—
138#340C23116yes—
139#336C2314yes—
140#336C2314yes—
141C25C2×D411yescharacteristic
142C22×Q8—22no—
143C22×Q8—28no—
144C22×Q8—28no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8C2414yes—
148C23×C4—28no—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C22×C8C2418yes—
171C4⋊Q8—216no—
172C4⋊Q8—44no—
173C4⋊Q8—44no—
174C22⋊Q8—416no—
175C22⋊Q8—416no—
176#117—24no—
177#117—24no—
178#117—24no—
179#117—24no—
180#117—48no—
181#117—48no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C4⋊C4C2412yes—
199C2×C4⋊C4C2412yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—48no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—28no—
209C2×C42—28no—
210C2×C42—28no—
211C2×C42—28no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4118yes—
224C2×C42C2411yescharacteristic
225C2×C42C2411yescharacteristic
226C2×C42C2412yes—
227#107—48no—
228#106—48no—
229C4⋊C8—216no—
230#103—416no—
231#103—48no—
232#103—48no—
233C22⋊C8—48no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242C24—22no—
243C24—24no—
244C24—22no—
245C24—22no—
246C24—42no—
247C24C22×D412yes—
248C24C22×D411yescharacteristic
249C2×Q8—28no—
250C2×Q8—24no—
251C2×Q8—28no—
252C2×Q8—24no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—416no—
257C2×Q8—416no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—432no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4#12318yes—
298C22×C4#13718yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D412yes—
305C22×C4C2×#4114yes—
306C22×C4H(5,𝔽2)14yes—
307C22×C4H(5,𝔽2)12yes—
308C22×C42−1+412yes—
309C22×C4C2511yescharacteristic
310M16—416no—
311C2×C8—28no—
312C2×C8—28no—
313C2×C8—28no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—28no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—416no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—48no—
331C22⋊C4—416no—
332C22⋊C4—48no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—48no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—22no—
342C42—22no—
343C42—44no—
344C42—44no—
345C42—48no—
346C42—48no—
347C42#12314yes—
348C42#12314yes—
349C42C2×#4118yes—
350C23—22no—
351C23—22no—
352C23—28no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—48no—
360C23—48no—
361C23—44no—
362C23—44no—
363C23C2×#12312yes—
364C23#53512yes—
365C23#53712yes—
366C23C2×#13714yes—
367C23C23×D411yescharacteristic
368Q8—48no—
369Q8—48no—
370Q8—88no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—216no—
376C2×C4—216no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—88no—
404C2×C4—88no—
405C2×C4C2×#12314yes—
406C2×C4C2×#12314yes—
407C2×C4C2×#12312yes—
408C2×C4#53512yes—
409C2×C4#53712yes—
410C2×C4#54018yes—
411C2×C4#54218yes—
412C2×C4C2×#13714yes—
413C2×C4C2×#13714yes—
414C2×C4#57514yes—
415C2×C4C22×#4112yes—
416C2×C4C2×H(5,𝔽2)11yescharacteristic
417C2×C4C2×H(5,𝔽2)12yes—
418C2×C4C2×2−1+411yescharacteristic
419C8—48no—
420C22—24no—
421C22—22no—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—416no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22#262414yes—
433C22C22×#12311yescharacteristic
434C22C2×#53511yescharacteristic
435C22C2×#53711yescharacteristic
436C22C22×#13712yes—
437C22C2×#57512yes—
438C4—24no—
439C4—24no—
440C4—42no—
441C4—42no—
442C4—84no—
443C4—82no—
444C4—82no—
445C4—84no—
446C4—84no—
447C4#262414yes—
448C4#309518yes—
449C4C2×#54012yes—
450C4C2×#54212yes—
451C2—24no—
452C2—44no—
453C2C2×#262411yescharacteristic
454C2C2×#309512yes—
455C2#6080011yescharacteristic
456C2#6080211yescharacteristic
457C2#6091812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29926C2semidirect
4. #29923C2semidirect
5. #29922C2semidirect
6. #23732C2semidirect
7. #23629C2semidirect
9. #21259C2semidirect
10. #21171C2semidirect
39. #2097C22semidirect
40. #2096C22semidirect
41. #2095C22semidirect
42. #2074C22semidirect
43. #2072C22semidirect
46. #1981C22semidirect
47. #1981C22semidirect
48. #1948C22semidirect
49. #1947C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#340122
#299262
#299232
#299222
#237322
#236292
#234812
#212592
#211712
#206262