Finite Group Databasebeta

← #6458285

Subgroups — #6458285

Order 512.

2028 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#27956C214yesmaximal, complemented
4#27955C214yesmaximal, complemented
5#27954C214yesmaximal, complemented
6#27953C214yesmaximal, complemented
7#23656C214yesmaximal, complemented
8#21321C212yesmaximal, complemented
9#20843C214yesmaximal
10#20799C212yesmaximal, complemented
11C2×#1749C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15C2×#427C2218yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2855—22no—
21#2851C2212yes—
22#2851C2212yes—
23#2848—22no—
24C4×C22⋊Q8—22no—
25C4×C22⋊Q8—22no—
26#2296C2212yes—
27#2285—22no—
28#2285—22no—
29#2285C2211yescharacteristic
30#2285C2212yes—
31#2285C2212yes—
32#2285C2212yes—
33#2285C2212yes—
34C2×#384C2212yes—
35C2×#382C2212yes—
36#2280C2212yes—
37C2×#380—22no—
38C2×C4×C4⋊C4—22no—
39#2085C22116yescomplemented
40#2084C2218yescomplemented
41#2083C2218yescomplemented
42#1993C2218yescomplemented
43#1992C2218yescomplemented
44#1854C22116yes—
45#1853C2218yes—
46#1852C2218yes—
47#1847C2218yescomplemented
48#1847C2218yescomplemented
49#1749C2212yes—
50#1749C2212yes—
51#1749C2212yes—
52#1749C2212yes—
53#1735C2218yes—
54C2×#336C2212yes—
55C24×C4D412yes—
56C24×C4C2311yescharacteristic
57C23×C8C2314yes—
58#533—28no—
59C2×C4⋊Q8C2312yes—
60C2×#125C2312yes—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C4×Q8—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—48no—
68C2×#117C2312yes—
69C2×#117C2314yes—
70C2×#117C2312yes—
71C2×#117C2314yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2314yes—
75C22×C22⋊C4—22no—
76C22×C42—28no—
77C22×C42—22no—
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#427—416no—
88C2×#107C23116yes—
89C2×#106C23116yes—
90C2×C22⋊C8Q8116yescomplemented
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C2×#97—28no—
94C2×#97C2314yes—
95C2×#97C2314yes—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389C2314yes—
101#389C2314yes—
102#387—24no—
103#387—24no—
104#385—24no—
105#385—24no—
106#385—24no—
107#384—22no—
108#384—22no—
109#384C2314yes—
110#384C2312yes—
111#384C2314yes—
112#382—24no—
113#382C2312yes—
114#382C2314yes—
115#382C2314yes—
116#381—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—24no—
121#380—24no—
122#380—24no—
123#380—24no—
124#380C2314yes—
125#380C2314yes—
126#379—22no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—24no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95—24no—
136#340C23116yes—
137#340C23116yes—
138#336C2314yes—
139#336C2314yes—
140C25C2×D411yescharacteristic
141C22×Q8—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—28no—
145C23×C4—24no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×Q818yes—
158C23×C4C2411yescharacteristic
159C23×C4C2411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—28no—
162C22×C8—28no—
163C22×C8C2×Q8116yes—
164C22×C8C2418yes—
165C4⋊Q8—28no—
166#125—28no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C22⋊Q8—48no—
170C4×Q8—48no—
171#117—28no—
172#117—216no—
173#117—28no—
174#117—48no—
175#117—48no—
176#117—48no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4C2418yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—48no—
201C2×C42—28no—
202C2×C42—24no—
203C2×C42—28no—
204C2×C42—28no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—48no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42#4118yes—
221C2×C42C2411yescharacteristic
222C2×C42C2411yescharacteristic
223C2×C42C2411yescharacteristic
224C2×C42C2411yescharacteristic
225#107—416no—
226#106—416no—
227C22⋊C8C2×Q8116yes—
228C22⋊C8C2×Q8116yes—
229#97—28no—
230#97—28no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238C24—22no—
239C24—22no—
240C24—24no—
241C24—42no—
242C24C4⋊Q814yes—
243C24C22×D411yescharacteristic
244C24C22×D411yescharacteristic
245C24C22×D411yescharacteristic
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—232no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—432no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4#12318yes—
285C22×C4C4⋊Q814yes—
286C22×C4C4⋊Q814yes—
287C22×C4C4⋊Q814yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×Q814yes—
296C22×C4C22×Q814yes—
297C22×C4C22×Q814yes—
298C22×C4C2×#4114yes—
299C22×C4H(5,𝔽2)14yes—
300C22×C4C2511yescharacteristic
301C2×C8—232no—
302C2×C8—28no—
303C2×C8—28no—
304C2×C8C22×Q8116yes—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—216no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—44no—
322C22⋊C4—48no—
323C22⋊C4—44no—
324C22⋊C4—48no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—48no—
333C42—44no—
334C42—48no—
335C42—44no—
336C42#12318yes—
337C42C2×#4114yes—
338C42C2×#4114yes—
339C23—21no—
340C23—21no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—28no—
345C23—21no—
346C23—21no—
347C23—28no—
348C23—22no—
349C23—22no—
350C23—44no—
351C23—48no—
352C23—48no—
353C23C2×#12312yes—
354C23C2×C4⋊Q812yes—
355C23C2×C4⋊Q812yes—
356C23C2×C4⋊Q812yes—
357C23#53512yes—
358C23#57514yes—
359C23C23×D411yescharacteristic
360Q8—88no—
361C2×C4—216no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—28no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—816no—
395C2×C4C2×#12314yes—
396C2×C4C2×#12314yes—
397C2×C4C2×#12312yes—
398C2×C4C2×C4⋊Q812yes—
399C2×C4C2×C4⋊Q812yes—
400C2×C4C2×C4⋊Q812yes—
401C2×C4#53512yes—
402C2×C4#554116yes—
403C2×C4#57514yes—
404C2×C4C23×Q812yes—
405C2×C4C22×#4112yes—
406C2×C4C2×H(5,𝔽2)12yes—
407C8—216no—
408C22—24no—
409C22—24no—
410C22—24no—
411C22—24no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—416no—
420C22—42no—
421C22—42no—
422C22#250214yes—
423C22#316214yes—
424C22C22×#12311yescharacteristic
425C22C22×C4⋊Q811yescharacteristic
426C22C2×#53511yescharacteristic
427C22C2×#57512yes—
428C22C2×#57512yes—
429C4—24no—
430C4—24no—
431C4—24no—
432C4—22no—
433C4—22no—
434C4—42no—
435C4—84no—
436C4—84no—
437C4—88no—
438C4#250214yes—
439C4C2×#55414yes—
440C2—24no—
441C2—24no—
442C2—42no—
443C2C2×#250211yescharacteristic
444C2#6051112yes—
445C2C2×#316211yescharacteristic
446C2#6066311yescharacteristic
447C2#6079012yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27956C2semidirect
4. #27955C2semidirect
5. #27954C2semidirect
6. #27953C2semidirect
7. #23656C2semidirect
8. #21321C2semidirect
10. #20799C2semidirect
39. #2085C22semidirect
40. #2084C22semidirect
41. #2083C22semidirect
42. #1993C22semidirect
43. #1992C22semidirect
47. #1847C22semidirect
48. #1847C22semidirect
90. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22852
#279562
#279552
#279542
#279532
#236562
#213212
#208432
#207992
C2×#17492