Finite Group Databasebeta

← #355467

Subgroups — #355467

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C2×#1899C212yesmaximal
3#13795C211yesmaximal, characteristic, complemented
4#13776C211yesmaximal, characteristic, complemented
5#10960C212yesmaximal
6#10912C212yesmaximal, complemented
7#10801C212yesmaximal, complemented
8#10761C212yesmaximal
9#10736C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2212yes—
13C23×C4⋊C4C2211yescharacteristic
14C22×#99C2214yes—
15#2322—24no—
16#2322—24no—
17C2×#389C412yescomplemented
18C2×#388C412yescomplemented
19C2×#387C412yescomplemented
20C2×#386C2211yescharacteristic
21C2×#385C412yescomplemented
22C2×#385C2211yescharacteristic
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×C4×C22⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1899C4116yescomplemented
28C2×#342C2214yes—
29C2×#342C2214yes—
30C2×#342C2214yes—
31#1436—24no—
32C4⋊#95—24no—
33C22⋊#95C2212yes—
34C22⋊#95C2212yes—
35C22⋊#95C2212yes—
36C23×D4C2311yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C2×#123—22no—
42C2×#123—22no—
43C2×#123—24no—
44C2×#123C2×C418yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—44no—
48C2×#117—24no—
49C2×#117—44no—
50C2×#117—44no—
51C2×#117C2×C418yescomplemented
52C22×C4⋊C4—22no—
53C22×C4⋊C4—44no—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4D414yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—22no—
70C22×C42—22no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C412yes—
73C2×#99—44no—
74C2×#99—44no—
75C2×#99C2×C4116yes—
76#389—28no—
77#388—28no—
78#387—28no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—28no—
84#381—22no—
85#381—22no—
86#381—24no—
87#381C2×C418yescomplemented
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—44no—
91C2×#95—24no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101#342—44no—
102#342—44no—
103#342C2×C4116yescomplemented
104C25C22×C411yescharacteristic
105C25C2411yescharacteristic
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—22no—
109C22×D4—24no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C22×C414yes—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C414yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4#4114yes—
142#123—216no—
143#123—28no—
144#123—28no—
145C4×D4—48no—
146C4×D4—48no—
147#117—416no—
148#117—416no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C4218yescomplemented
165C2×C4⋊C4C22⋊C418yes—
166C2×C4⋊C4C22⋊C418yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4C4218yescomplemented
194C2×C22⋊C4C22×C418yes—
195C2×C22⋊C4C22×C418yes—
196C2×C22⋊C4C22×C414yes—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—24no—
200C2×C42—42no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42—84no—
205C2×C42—82no—
206#99—416no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—216no—
213#95—44no—
214#95—84no—
215#95—84no—
216#95—84no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24C2×C4212yes—
226C24C23×C411yescharacteristic
227C24C23×C412yes—
228C24C22×D411yescharacteristic
229C24C22×D411yescharacteristic
230C24C2×#4112yes—
231C24H(5,𝔽2)14yes—
232C24H(5,𝔽2)12yes—
233C242−1+412yes—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—28no—
237C2×D4—28no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—44no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—84no—
303C22×C4—88no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4C2×C4212yes—
307C22×C4C2×C4214yes—
308C22×C4C2×C4214yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C414yes—
313C22×C4#11714yes—
314C22×C4C4×D418yes—
315C22×C4C4×D414yes—
316C22×C4C4×D414yes—
317C4⋊C4—216no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—416no—
321C4⋊C4—44no—
322C4⋊C4—84no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—216no—
326C22⋊C4—28no—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—84no—
341C42—416no—
342C42—44no—
343C42—84no—
344C42—88no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—82no—
367C23—84no—
368C23—82no—
369C23C22×C4211yescharacteristic
370C23C22×C22⋊C411yescharacteristic
371C23C2×#11711yescharacteristic
372C23C2×C4×D412yes—
373C23C2×C4×D412yes—
374C23#51812yes—
375C23#51814yes—
376C23#52012yes—
377C23#52014yes—
378C23#53412yes—
379C23#53412yes—
380C23#53511yescharacteristic
381C23#53611yescharacteristic
382C23#53712yes—
383C23#53814yes—
384C23#53914yes—
385D4—44no—
386D4—44no—
387D4—48no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—216no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—416no—
419C2×C4—416no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—88no—
434C2×C4—84no—
435C2×C4—88no—
436C2×C4—88no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4C4×C22⋊C418yes—
444C22—22no—
445C22—24no—
446C22—24no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—84no—
465C22—84no—
466C22—82no—
467C22—81no—
468C22—81no—
469C22C2×C4×C22⋊C411yescharacteristic
470C22#227512yes—
471C22#230811yescharacteristic
472C22#231111yescharacteristic
473C22#231812yes—
474C22#232112yes—
475C22#232212yes—
476C22#232514yes—
477C22#288312yes—
478C22#288412yes—
479C4—28no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—44no—
484C4—42no—
485C4—88no—
486C4—88no—
487C4—82no—
488C4—81no—
489C4—81no—
490C2—22no—
491C2—24no—
492C2—22no—
493C2—22no—
494C2—22no—
495C2—82no—
496C2#1378411yescharacteristic
497C2#2039312yes—
498C2#2174911yescharacteristic
499C2#2175111yescharacteristic
500C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13795C2semidirect
4. #13776C2semidirect
6. #10912C2semidirect
7. #10801C2semidirect
9. #10736C2semidirect
17. C2×#389C4semidirect
18. C2×#388C4semidirect
19. C2×#387C4semidirect
21. C2×#385C4semidirect
27. #1899C4semidirect
51. C2×#117C2×C4semidirect
87. #381C2×C4semidirect
103. #342C2×C4semidirect
164. C2×C4⋊C4C42semidirect
193. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C2×#18992
#137952
#137762
#109602
#109122
#108012
#107612
#107362