Finite Group Databasebeta

← #494447

Subgroups — #494447

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3058C211yesmaximal, characteristic
2C2×#2913C211yesmaximal, characteristic, complemented
3C2×#2140C211yesmaximal, characteristic, complemented
4C22×#357C211yesmaximal, characteristic, complemented
5C2×#2113C211yesmaximal, characteristic, complemented
6#12992C214yesmaximal, complemented
7#12991C214yesmaximal, complemented
8#12557C211yesmaximal, characteristic, complemented
9#12556C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C22×#131C2211yescharacteristic
12C2×#535—21no—
13C2×#520—21no—
14C22×M32C2211yescharacteristic
15C23×C16C2211yescharacteristic
16#3058—24no—
17#3058C418yescomplemented
18C2×#469—22no—
19C2×#466C2214yes—
20#2913—24no—
21#2913—24no—
22C2×#428—21no—
23C2×#428—22no—
24C2×#427C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2212yescomplemented
27C22×#107C2211yescharacteristic
28C22×#107C2212yescomplemented
29C2×#418—21no—
30#2140—24no—
31C2×#357—24no—
32C2×#357C2214yes—
33C2×#357C2212yes—
34C2×#357C2212yes—
35#2113—24no—
36C2×C22⋊C16C2214yescomplemented
37#1380C2212yescomplemented
38#1380C2212yescomplemented
39#1380C2214yescomplemented
40C22×#41D411yescharacteristic
41C23×D4—21no—
42C22×D8—22no—
43C22×D8—22no—
44C22×D8—21no—
45C22×D8—22no—
46C22×D8—22no—
47C22×D8—22no—
48C22×D8C2×C414yescomplemented
49C22×D8C2312yes—
50C2×#131—22no—
51C2×#131—22no—
52C2×#131—22no—
53C2×#131—22no—
54C2×#131C2×C414yes—
55C2×#131D414yes—
56C22×M16D411yescharacteristic
57C22×M16D412yes—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60#535—28no—
61#535—44no—
62C2×#127—41no—
63C2×#124—41no—
64C2×#121—24no—
65C2×#120—42no—
66#520—28no—
67#520—44no—
68C2×C4×Q8—41no—
69C2×C4×D4—42no—
70C2×C4×D4—41no—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C2×M32—22no—
77C2×M32—22no—
78C2×M32—22no—
79C22×C16—22no—
80C22×C16—22no—
81C22×C16—22no—
82C22×C16C2312yes—
83C22×C16C2312yes—
84#469—216no—
85#466—28no—
86#466C2×C4116yescomplemented
87#428—28no—
88#428—216no—
89#427—22no—
90#427—22no—
91#427—24no—
92#427—24no—
93#427C2×C414yescomplemented
94#427D414yescomplemented
95C2×#107—21no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#107D414yescomplemented
101C2×#107C2312yes—
102C2×#107C2312yes—
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#106—21no—
106C2×#106—21no—
107C2×#106—22no—
108#418—28no—
109C2×#102—24no—
110#357—28no—
111#357—24no—
112#357—24no—
113C22⋊C16—28no—
114C25—41no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41C22⋊C414yes—
120C2×#41SD1614yes—
121C22×Q8D811yescharacteristic
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—44no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—41no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4D812yes—
132C22×D4D811yescharacteristic
133C23×C4—42no—
134C23×C4—41no—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C2×D8—24no—
138C2×D8—28no—
139C2×D8—28no—
140C2×D8—22no—
141C2×D8—22no—
142C2×D8—22no—
143C2×D8—22no—
144C2×D8—22no—
145C2×D8—42no—
146C2×D8—44no—
147C2×D8—44no—
148C2×D8—42no—
149C2×D8C22×C414yes—
150#131—28no—
151#131—28no—
152#131—28no—
153#131—28no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16C22⋊C414yes—
159C2×M16C22⋊C418yes—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C22×C414yes—
165C22×C8C22×C412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D414yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8#4112yes—
173C22×C8#4112yes—
174C22×C8C2411yescharacteristic
175#127—48no—
176#124—48no—
177#121—216no—
178#121—48no—
179#120—416no—
180C4×Q8—48no—
181C4×D4—416no—
182C4×D4—48no—
183#117—28no—
184#117—24no—
185#117—24no—
186#117—44no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—44no—
203C2×C42—42no—
204C2×C42—41no—
205C2×C42—41no—
206M32—44no—
207C2×C16—22no—
208C2×C16—22no—
209C2×C16—22no—
210C2×C16—22no—
211C2×C16—22no—
212C2×C16—22no—
213C2×C16—44no—
214#107—28no—
215#107—24no—
216#107—22no—
217#107—21no—
218#107—21no—
219#107—22no—
220#107C22×C414yes—
221#107C2×D412yes—
222#107C2×D414yes—
223#107C2×D412yes—
224#106—28no—
225#106—24no—
226#106—24no—
227#102—216no—
228C24—42no—
229C24—44no—
230C24—82no—
231C24—82no—
232C24—162no—
233C24—162no—
234C24C2×D812yes—
235C24C2×D811yescharacteristic
236#41—28no—
237#41—28no—
238#41—28no—
239#41—28no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8#10214yes—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—416no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—48no—
255C2×D4—44no—
256C2×D4—48no—
257C2×D4—82no—
258C2×D4—84no—
259C2×D4—82no—
260C2×D4—84no—
261C2×D4—164no—
262C2×D4—164no—
263C2×D4#10218yes—
264C2×D4#10214yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—162no—
276C22×C4—164no—
277C22×C4C2×C22⋊C414yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4#12012yes—
280C22×C4#12312yes—
281C22×C4C2×D812yes—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×SD1614yes—
284C22×C4C2×SD1612yes—
285C22×C4AGL(1,ℤ/8)12yes—
286C22×C4#13712yes—
287C22×C4C22×D411yescharacteristic
288D8—24no—
289D8—24no—
290D8—42no—
291D8—42no—
292D8—48no—
293M16—28no—
294M16—216no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8C2×C22⋊C414yes—
302C2×C8C2×C22⋊C412yes—
303C2×C8C4×D414yes—
304C2×C8C4×D414yes—
305C2×C8#12012yes—
306C2×C8#12114yes—
307C2×C8#12114yes—
308C2×C8#12312yes—
309C2×C8C23×C411yescharacteristic
310C2×C8C22×D411yescharacteristic
311C2×C8C22×D411yescharacteristic
312C2×C8C22×D411yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C22⋊C4—48no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—416no—
329C42—48no—
330C42—44no—
331C42—44no—
332C16—42no—
333C16—42no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—24no—
338C23—24no—
339C23—22no—
340C23—44no—
341C23—82no—
342C23—88no—
343C23—84no—
344C23—84no—
345C23—82no—
346C23—168no—
347C23—168no—
348C23—164no—
349C23—164no—
350C23C2×#10214yes—
351C23C2×#10212yes—
352C23#44712yes—
353C23#48012yes—
354C23C22×D811yescharacteristic
355Q8—28no—
356D4—216no—
357D4—28no—
358D4—44no—
359D4—44no—
360D4—82no—
361D4—88no—
362D4—82no—
363D4—1616no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—164no—
380C2×C4—168no—
381C2×C4#38614yes—
382C2×C4C2×#10214yes—
383C2×C4C2×#10212yes—
384C2×C4#41714yes—
385C2×C4#44712yes—
386C2×C4#44814yes—
387C2×C4#48012yes—
388C2×C4#48114yes—
389C2×C4C22×C22⋊C411yescharacteristic
390C2×C4C2×#12011yescharacteristic
391C2×C4C2×#12311yescharacteristic
392C2×C4C22×SD1611yescharacteristic
393C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C8—24no—
396C8#38614yes—
397C8C2×C4×D411yescharacteristic
398C8C2×C4×D411yescharacteristic
399C8C2×#12111yescharacteristic
400C8C2×#12111yescharacteristic
401C22—24no—
402C22—24no—
403C22—22no—
404C22—22no—
405C22—28no—
406C22—24no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—88no—
411C22—84no—
412C22—1616no—
413C22—164no—
414C22—162no—
415C22—162no—
416C22#187414yes—
417C22C22×#10211yescharacteristic
418C22C2×#44711yescharacteristic
419C22C2×#48011yescharacteristic
420C22#341512yes—
421C22#341612yes—
422C4—24no—
423C4—84no—
424C4—84no—
425C4—82no—
426C4—82no—
427C4#187414yes—
428C4C2×#38611yescharacteristic
429C4C2×#41711yescharacteristic
430C4C2×#44811yescharacteristic
431C4C2×#48111yescharacteristic
432C2—24no—
433C2—44no—
434C2—164no—
435C2C2×#187411yescharacteristic
436C2#3338112yes—
437C2#3352211yescharacteristic
438C2#3352411yescharacteristic
439C2#3365012yes—
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3058C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2913C2semidirect
3. C2×#2140C2semidirect
4. C22×#357C2semidirect
5. C2×#2113C2semidirect
6. #12992C2semidirect
7. #12991C2semidirect
8. #12557C2semidirect
9. #12556C2semidirect
17. #3058C4semidirect
26. C2×#427C22semidirect
28. C22×#107C22semidirect
36. C2×C22⋊C16C22semidirect
37. #1380C22semidirect
38. #1380C22semidirect
39. #1380C22semidirect
48. C22×D8C2×C4semidirect
86. #466C2×C4semidirect
93. #427C2×C4semidirect
94. #427D4semidirect
100. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30582
C2×#29132
C2×#21402
C22×#3572
C2×#21132
#129922
#129912
#125572
#125562