Finite Group Databasebeta

← #500555

Subgroups — #500555

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic, complemented
2C2×#2224C211yesmaximal, characteristic, complemented
3C2×#2206C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic, complemented
5#13380C212yesmaximal, complemented
6#13304C212yesmaximal, complemented
7#13275C211yesmaximal, characteristic, complemented
8#13252C211yesmaximal, characteristic, complemented
9#13012C212yesmaximal, complemented
10#12935C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#532—21no—
13C22×D16C2212yes—
14#3169—24no—
15#3169—24no—
16C2×#500C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#486—21no—
19C2×#466—21no—
20C2×#466C2211yescharacteristic
21C2×#466C2212yes—
22C2×C4×D8—21no—
23C2×C4×D8C2211yescharacteristic
24C2×C4×D8C2212yes—
25C22×C4×C8C2211yescharacteristic
26C4×#121—21no—
27#2224—24no—
28#2206—24no—
29C2×#366C2212yes—
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2212yes—
33C2×#357C2212yes—
34C2×#357C2212yes—
35C2×#357C2212yes—
36C2×C22⋊C16C2212yes—
37#1970C2211yescharacteristic
38#1942C2211yescharacteristic
39#1934—21no—
40#1893—21no—
41#1808C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#102—21no—
44#1380C2212yes—
45#1380C2212yes—
46#1374C2212yes—
47C22×#41—41no—
48C22×D8—21no—
49C22×D8C2312yes—
50C22×D8C2311yescharacteristic
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53#532—28no—
54#532—44no—
55C2×C4⋊Q8—21no—
56C2×#127—21no—
57C2×#124—41no—
58C2×#121—21no—
59C2×#121—22no—
60C2×#121—41no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—41no—
64C2×C4×D4—41no—
65C22×C42D411yescharacteristic
66C2×D16—24no—
67C2×D16—24no—
68C2×D16—42no—
69C22×C16C2312yes—
70C22×C16C2312yes—
71#500—22no—
72#500D414yescomplemented
73#493—22no—
74#493D414yescomplemented
75#486—28no—
76#466—28no—
77#466—24no—
78#466—22no—
79#466—24no—
80#466—22no—
81C4×D8—28no—
82C4×D8—24no—
83C4×D8—24no—
84C4×D8—22no—
85C4×D8—22no—
86C2×#107—21no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2312yes—
90C2×#107C2311yescharacteristic
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#390—21no—
103#389—21no—
104#388—41no—
105#386—41no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C22⋊C4—41no—
109#366—24no—
110C4⋊C16—22no—
111C4⋊C16—22no—
112#357—24no—
113#357—24no—
114#357—24no—
115#357—42no—
116C22⋊C16—42no—
117#340—21no—
118#340—21no—
119#340—21no—
120C2×#41—48no—
121C2×#41—84no—
122C2×#41—84no—
123C22×Q8—41no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—41no—
127C22×D4—42no—
128C22×D4—41no—
129C22×D4—41no—
130C23×C4—42no—
131C23×C4—41no—
132C23×C4D811yescharacteristic
133C23×C4D811yescharacteristic
134C23×C4C2×D411yescharacteristic
135C2×D8—24no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—22no—
139C2×D8—22no—
140C2×D8—22no—
141C2×D8—21no—
142C2×D8—22no—
143C2×D8—21no—
144C2×D8—22no—
145C2×D8—21no—
146C2×D8—42no—
147C2×D8—42no—
148C2×D8C2×D414yes—
149C2×D8#4112yescomplemented
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—41no—
158C22×C8—41no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163C4⋊Q8—24no—
164C4⋊Q8—42no—
165#127—24no—
166#127—42no—
167#124—48no—
168#121—28no—
169#121—24no—
170#121—42no—
171#121—44no—
172#121—48no—
173C4×D4—28no—
174C4×D4—44no—
175C4×D4—48no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—84no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42D811yescharacteristic
196C2×C42D811yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199D16—48no—
200C2×C16—22no—
201C2×C16—22no—
202C2×C16—22no—
203C2×C16—22no—
204C2×C16—22no—
205C2×C16—22no—
206#107—24no—
207#107—22no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107C2×D414yes—
214#107#4112yescomplemented
215#102—28no—
216#102—28no—
217#102—24no—
218#102—42no—
219#102—42no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8C2×D412yes—
225C4×C8C2×D412yes—
226#95—41no—
227#95—41no—
228#95—41no—
229C24—41no—
230C24—42no—
231C24C2×D811yescharacteristic
232#41—832no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—48no—
239C2×D4—44no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4—84no—
256C2×D4—82no—
257C2×D4—84no—
258C2×D4—82no—
259C2×D4—82no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—48no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4D1614yescomplemented
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C22×D411yescharacteristic
288D8—22no—
289D8—24no—
290D8—24no—
291D8—22no—
292D8—48no—
293D8—41no—
294D8—42no—
295D8—41no—
296D8—42no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—44no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8#12012yes—
311C2×C8C22×D411yescharacteristic
312C2×C8C22×D411yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C2×C82−1+411yescharacteristic
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—44no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—82no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C42—21no—
335C42—21no—
336C42—22no—
337C42—22no—
338C42—82no—
339C42#12012yes—
340C42AGL(1,ℤ/8)12yes—
341C16—42no—
342C16—42no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—42no—
350C23—44no—
351C23—82no—
352C23—81no—
353C23—82no—
354C23—82no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—82no—
359C23—82no—
360C23—81no—
361C23—82no—
362C23—81no—
363C23C2×D1612yes—
364C23C22×D811yescharacteristic
365Q8—88no—
366D4—42no—
367D4—42no—
368D4—44no—
369D4—44no—
370D4—88no—
371D4—88no—
372D4—816no—
373D4—82no—
374D4—88no—
375D4—81no—
376D4—82no—
377D4—81no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—28no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—816no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—164no—
408C2×C4#44712yes—
409C2×C4#44712yes—
410C2×C4C2×D1612yes—
411C2×C4C2×D1612yes—
412C2×C4#50912yes—
413C2×C4C2×#12011yescharacteristic
414C2×C4#52511yescharacteristic
415C2×C4#53711yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
417C2×C4#57511yescharacteristic
418C2×C4#57511yescharacteristic
419C8—22no—
420C8—41no—
421C8—41no—
422C8#54611yescharacteristic
423C8#54711yescharacteristic
424C8#54811yescharacteristic
425C8#55011yescharacteristic
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—82no—
434C22—84no—
435C22—82no—
436C22—82no—
437C22—84no—
438C22—84no—
439C22—161no—
440C22—162no—
441C22—162no—
442C22—161no—
443C22—162no—
444C22—162no—
445C22—161no—
446C22—161no—
447C22C2×#44711yescharacteristic
448C22#310011yescharacteristic
449C22#324511yescharacteristic
450C22C22×D1611yescharacteristic
451C22C2×#50911yescharacteristic
452C22#341411yescharacteristic
453C22#341411yescharacteristic
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—82no—
459C4—84no—
460C4—161no—
461C4—161no—
462C4#218414yes—
463C4#240711yescharacteristic
464C4#300511yescharacteristic
465C4#311511yescharacteristic
466C4#325111yescharacteristic
467C2—42no—
468C2—162no—
469C2—164no—
470C2#2415011yescharacteristic
471C2C2×#218411yescharacteristic
472C2#3351911yescharacteristic
473C2#3366011yescharacteristic
474C2#3367511yescharacteristic
475C2#3381111yescharacteristic
476C2#3381711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3169C2semidirect
2. C2×#2224C2semidirect
3. C2×#2206C2semidirect
4. C4×#466C2semidirect
5. #13380C2semidirect
6. #13304C2semidirect
7. #13275C2semidirect
8. #13252C2semidirect
9. #13012C2semidirect
10. #12935C2semidirect
11. #12783C2semidirect
72. #500D4semidirect
74. #493D4semidirect
149. C2×D8#41semidirect
214. #107#41semidirect
281. C22×C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#22242
C2×#22062
C4×#4662
#133802
#133042
#132752
#132522
#130122
#129352
#127832