Finite Group Databasebeta

← #390377

Subgroups — #390377

Order 512.

1125 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23498C211yesmaximal, characteristic
2#21071C212yesmaximal, complemented
3#20980C212yesmaximal
4#11370C211yesmaximal, characteristic, complemented
5#11289C212yesmaximal, complemented
6#11114C212yesmaximal, complemented
7#10893C211yesmaximal, characteristic, complemented
8#10790C212yesmaximal, complemented
9#10772C212yesmaximal, complemented
10#3449C2211yescharacteristic
11C2×C4×#41C2212yes—
12C2×#430C2212yes—
13C2×#421C2212yes—
14C22×#104C2212yes—
15C22×#104C2212yes—
16#2896C2212yes—
17#2879C2212yes—
18#2873C2211yescharacteristic
19#2285C2211yescharacteristic
20#2040C412yescomplemented
21#2040C412yescomplemented
22#2039C412yescomplemented
23#2039C412yescomplemented
24#1929—24no—
25#1929—24no—
26#1897—24no—
27#1896—24no—
28C2×#344C2212yes—
29#1742C2211yescharacteristic
30#1740C2211yescharacteristic
31#1733C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36#1728C2212yes—
37#1728C2211yescharacteristic
38C4×#95C2211yescharacteristic
39C22×#41C2312yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C22×M16C2312yes—
43#537C2×C412yes—
44#537C2×C412yes—
45#536—22no—
46#535—22no—
47#534—41no—
48#532—41no—
49#525—24no—
50C4×#41—24no—
51C4×#41—42no—
52C4×#41—42no—
53C4×#41—42no—
54C2×C4×Q8—22no—
55C2×C4×D4—22no—
56C2×C4×D4—42no—
57C2×#117—22no—
58C2×#117C2312yes—
59C2×#117C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2312yes—
64#430—42no—
65#430—42no—
66C2×#108—24no—
67#421—24no—
68#421—42no—
69C2×#104—24no—
70C2×#104—42no—
71C2×#104—42no—
72C2×#104—42no—
73C2×#104—42no—
74C2×#104C2×C414yescomplemented
75C2×#104C2×C414yescomplemented
76C2×#104D414yes—
77C2×#104D414yes—
78#419—22no—
79#418—22no—
80#417C2×C414yescomplemented
81#417C2×C414yescomplemented
82#416—24no—
83#408—24no—
84#408—42no—
85#407—24no—
86#405—24no—
87C4×M16—41no—
88#384C2×C411yescharacteristic, complemented
89#384C2×C411yescharacteristic, complemented
90#382C2×C411yescharacteristic, complemented
91#382C2×C411yescharacteristic, complemented
92#380—41no—
93C4×C4⋊C4—41no—
94C2×#95—41no—
95#344—42no—
96#343C2×C412yescomplemented
97#343C2×C412yescomplemented
98#342—41no—
99#339—24no—
100#339—24no—
101#339—22no—
102#339—22no—
103#339—22no—
104#339—22no—
105#339—42no—
106#336—22no—
107#336—22no—
108#336—41no—
109C2×#41—22no—
110C2×#41—22no—
111C2×#41—22no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41C22×C414yes—
120C2×#41C2×D414yes—
121C22×Q8#4112yes—
122C22×D4—22no—
123C22×D4#4112yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—42no—
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16C22×C414yes—
146C2×M16C2×D414yes—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—41no—
151#127—41no—
152#124—42no—
153#123—44no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156#121—24no—
157#121—24no—
158#121—42no—
159#121—42no—
160#120—44no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×D4—48no—
164C4×D4—44no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—44no—
168C4×D4—48no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—44no—
177#117—44no—
178#117#4114yes—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—42no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42—42no—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209#108—48no—
210#104—44no—
211#104—44no—
212#104—44no—
213#104—44no—
214#104—48no—
215#103—24no—
216#103—24no—
217#102—24no—
218#102—24no—
219C22⋊C8—48no—
220#97—42no—
221C4×C8—42no—
222#95—41no—
223#95—41no—
224#95—44no—
225C24—22no—
226C24H(5,𝔽2)11yescharacteristic
227#41—24no—
228#41—24no—
229#41—24no—
230#41—24no—
231#41—24no—
232#41—24no—
233#41—24no—
234#41—24no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241#41—42no—
242#41—42no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8C4×D414yes—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4C4×D414yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—84no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—84no—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C42−1+412yes—
297M16—24no—
298M16—24no—
299M16—24no—
300M16—24no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—42no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C22⋊C4—24no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—48no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—24no—
343C42—22no—
344C42—22no—
345C42—48no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—41no—
353C42—41no—
354C23—24no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23#51811yescharacteristic
362C23#53812yes—
363C23#54612yes—
364C23H(3,𝔽4)11yescharacteristic
365C23#56311yescharacteristic
366Q8—24no—
367Q8—24no—
368Q8—42no—
369Q8—42no—
370D4—24no—
371D4—24no—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—44no—
376D4—42no—
377D4—42no—
378D4—162no—
379D4—162no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—816no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4C2×C4×D412yes—
415C2×C4C4×#4111yescharacteristic
416C2×C4#51811yescharacteristic
417C2×C4#52012yes—
418C2×C4#53412yes—
419C2×C4#53712yes—
420C2×C4#53812yes—
421C2×C4#53912yes—
422C2×C4#54012yes—
423C2×C4D4211yescharacteristic
424C2×C4#54612yes—
425C2×C4#54712yes—
426C2×C4#55112yes—
427C2×C4#55111yescharacteristic
428C2×C4#55212yes—
429C2×C4#56212yes—
430C8—44no—
431C8—44no—
432C8—82no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—22no—
437C22—42no—
438C22—41no—
439C22—41no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—84no—
444C22—84no—
445C22#233311yescharacteristic
446C22#235811yescharacteristic
447C22#237911yescharacteristic
448C22#268412yes—
449C22#269512yes—
450C4—42no—
451C4—44no—
452C4—44no—
453C4—82no—
454C4—81no—
455C4—84no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4#232512yes—
461C4#235912yes—
462C4#267112yes—
463C4#274312yes—
464C2—22no—
465C2—21no—
466C2—22no—
467C2—21no—
468C2—84no—
469C2#1457111yescharacteristic
470C2#2289811yescharacteristic
471C2#2289911yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2358
prime radical#3449C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21071C2semidirect
4. #11370C2semidirect
5. #11289C2semidirect
6. #11114C2semidirect
7. #10893C2semidirect
8. #10790C2semidirect
9. #10772C2semidirect
20. #2040C4semidirect
21. #2040C4semidirect
22. #2039C4semidirect
23. #2039C4semidirect
74. C2×#104C2×C4semidirect
75. C2×#104C2×C4semidirect
80. #417C2×C4semidirect
81. #417C2×C4semidirect
88. #384C2×C4semidirect
89. #384C2×C4semidirect
90. #382C2×C4semidirect
91. #382C2×C4semidirect
96. #343C2×C4semidirect
97. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#234982
#210712
#209802
#113702
#112892
#111142
#108932
#107902
#107722