Finite Group Databasebeta

← #512502

Subgroups — #512502

Order 512.

505 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33261C211yesmaximal, characteristic, complemented
2#33247C211yesmaximal, characteristic
3#13283C211yesmaximal, characteristic, complemented
4#13282C211yesmaximal, characteristic, complemented
5#13205C211yesmaximal, characteristic
6#13204C211yesmaximal, characteristic
7#13150C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic
10#13149C211yesmaximal, characteristic
11#13148C211yesmaximal, characteristic, complemented
12#13148C211yesmaximal, characteristic, complemented
13#13147C211yesmaximal, characteristic
14#13147C211yesmaximal, characteristic
15#12539C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#489—21no—
19#2647—21no—
20#2640—21no—
21C2×#358C2211yescharacteristic
22C2×#358C2211yescharacteristic
23C2×#358C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#346C2211yescharacteristic
30C2×#346C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1973—21no—
36#1973—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1944—21no—
42#1943—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1846C2211yescharacteristic
48#1845C2211yescharacteristic
49#1836—21no—
50#1726C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#124—21no—
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68#500—22no—
69#500—22no—
70#495—22no—
71#495—22no—
72#489—44no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#97D411yescharacteristic
87C2×#97D411yescharacteristic
88C2×#97D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#400—41no—
93#400—41no—
94#398—41no—
95#398—41no—
96#397—41no—
97#397—41no—
98#393—41no—
99#393—41no—
100#391—21no—
101#391—21no—
102#390—21no—
103#390—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#384—21no—
109#382—21no—
110#376D411yescharacteristic
111#358—22no—
112#358—22no—
113#358—22no—
114#358—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#346—22no—
120#346—22no—
121#346—22no—
122C4×C16—22no—
123C4×C16—22no—
124C4×C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129C22×Q8—21no—
130C22×Q8—41no—
131C22×D4—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×SD16—42no—
139C2×SD16—42no—
140C2×SD16—42no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2411yescharacteristic
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153#124—42no—
154#124—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16#4112yes—
186C2×C16#4112yes—
187C2×C16#4112yes—
188C2×C16#4112yes—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—42no—
202#107—42no—
203#106—42no—
204#103—42no—
205#103—42no—
206#102—42no—
207#102—42no—
208#97—22no—
209#97—22no—
210#97SD1614yes—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8#4112yes—
216C4×C8#4112yes—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95D811yescharacteristic
228#95D811yescharacteristic
229#95D811yescharacteristic
230#95D811yescharacteristic
231C24—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—82no—
239C2×Q8—161no—
240C2×Q8—161no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C22×D411yescharacteristic
261Q16—41no—
262Q16—41no—
263Q16—41no—
264Q16—41no—
265SD16—84no—
266D8—41no—
267D8—41no—
268D8—41no—
269D8—41no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8#12412yes—
275C2×C8C2×SD1612yes—
276C2×C8C2×SD1612yes—
277C2×C8#13512yes—
278C2×C8#13512yes—
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C4⋊C4—161no—
299C4⋊C4—161no—
300C4⋊C4—161no—
301C4⋊C4—161no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—161no—
307C22⋊C4—161no—
308C22⋊C4—161no—
309C22⋊C4—161no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42#12412yes—
315C42C2×SD1612yes—
316C16—22no—
317C16—22no—
318C16—22no—
319C16—22no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23C22×D811yescharacteristic
328Q8—81no—
329Q8—81no—
330Q8—81no—
331Q8—81no—
332Q8—164no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4#48612yes—
375C2×C4#48612yes—
376C2×C4C2×#12411yescharacteristic
377C2×C4#53211yescharacteristic
378C2×C4#53211yescharacteristic
379C2×C4C22×SD1611yescharacteristic
380C2×C4C2×#13511yescharacteristic
381C2×C4C2×#13511yescharacteristic
382C8—22no—
383C8—22no—
384C8#48814yes—
385C8#52911yescharacteristic
386C8#52911yescharacteristic
387C8#52911yescharacteristic
388C8#52911yescharacteristic
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22C2×#48611yescharacteristic
397C22#316911yescharacteristic
398C22#316911yescharacteristic
399C22#341511yescharacteristic
400C22#341611yescharacteristic
401C22#341611yescharacteristic
402C22#341711yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—161no—
410C4—161no—
411C4—161no—
412C4—161no—
413C4—161no—
414C4—161no—
415C4—161no—
416C4#286111yescharacteristic
417C4C2×#48811yescharacteristic
418C4#316711yescharacteristic
419C4#316711yescharacteristic
420C2—321no—
421C2#3327711yescharacteristic
422C2#3370211yescharacteristic
423C2#3370311yescharacteristic
424C2#3373911yescharacteristic
425C2#3374211yescharacteristic
426C2#3374411yescharacteristic
427C2#3374511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33261C2semidirect
3. #13283C2semidirect
4. #13282C2semidirect
7. #13150C2semidirect
8. #13150C2semidirect
11. #13148C2semidirect
12. #13148C2semidirect
15. #12539C2semidirect

Maximal subgroups

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

subgroupindex
#332612
#332472
#132832
#132822
#132052
#132042
#131502
#131492
#131482
#131472
#125392