Finite Group Databasebeta

← #501547

Subgroups — #501547

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic
3#28535C211yesmaximal, characteristic, complemented
4#22807C211yesmaximal, characteristic
5#13482C211yesmaximal, characteristic
6#13480C211yesmaximal, characteristic
7#13449C211yesmaximal, characteristic
8#13449C211yesmaximal, characteristic
9#13284C211yesmaximal, characteristic, complemented
10#13178C211yesmaximal, characteristic, complemented
11#13176C211yesmaximal, characteristic, complemented
12#13160C211yesmaximal, characteristic, complemented
13#13133C211yesmaximal, characteristic, complemented
14#13132C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#479—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2565—21no—
22#2354—21no—
23#2228—24no—
24#2175—24no—
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2097—21no—
39#2096—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1925—21no—
45#1925—21no—
46#1846C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1802—21no—
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C2×#117—41no—
67C2×SD32—22no—
68C2×SD32—22no—
69C2×SD32—22no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#499—22no—
75#499—22no—
76#479—44no—
77C4×Q16—22no—
78C4×Q16—22no—
79C4×D8—22no—
80C4×D8D414yes—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×C4⋊C8D411yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#398—41no—
102#398—41no—
103#391—21no—
104#391—21no—
105#390—21no—
106#390—21no—
107#388—41no—
108#388—41no—
109#387—41no—
110#387—41no—
111#384—21no—
112#384—21no—
113#382—21no—
114#380—41no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D411yescharacteristic
118#367—22no—
119#366—22no—
120#366—22no—
121C4⋊C16—22no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130#346—22no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C22×D4—21no—
137C23×C4—41no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16#4112yes—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8C2×D412yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2411yescharacteristic
154#125—42no—
155#125—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×D4—24no—
159C4×D4—42no—
160#117—84no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C4⋊C4D811yescharacteristic
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187SD32—28no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16#4112yes—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—42no—
212#107—81no—
213#107C2×D412yes—
214#106—42no—
215#106—42no—
216C4⋊C8—22no—
217#103—42no—
218#103—42no—
219#103—42no—
220#102—24no—
221#102—81no—
222#102—81no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8C2×D412yes—
228C4×C8#4112yes—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95D811yescharacteristic
238#95D811yescharacteristic
239C24—41no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C22×D411yescharacteristic
269Q16—22no—
270Q16—22no—
271Q16—41no—
272Q16—41no—
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8#12112yes—
286C2×C8C22×D411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C82−1+411yescharacteristic
291C2×C82−1+411yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—161no—
313C22⋊C4—44no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—82no—
323C42—82no—
324C42—82no—
325C42#12112yes—
326C42AGL(1,ℤ/8)12yes—
327C16—22no—
328C16—41no—
329C16—41no—
330C16—41no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23C22×D811yescharacteristic
339Q8—42no—
340Q8—42no—
341Q8—81no—
342Q8—81no—
343D4—42no—
344D4—42no—
345D4—81no—
346D4—81no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—88no—
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—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4#45912yes—
385C2×C4#45912yes—
386C2×C4#50912yes—
387C2×C4C2×#12111yescharacteristic
388C2×C4#54711yescharacteristic
389C2×C4#54811yescharacteristic
390C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
391C2×C4#57711yescharacteristic
392C2×C4#57811yescharacteristic
393C8—22no—
394C8—41no—
395C8—41no—
396C8#52511yescharacteristic
397C8#53611yescharacteristic
398C8#55211yescharacteristic
399C8#55611yescharacteristic
400C22—82no—
401C22—82no—
402C22—82no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22C2×#45911yescharacteristic
408C22#333411yescharacteristic
409C22#335811yescharacteristic
410C22C2×#50911yescharacteristic
411C22#341411yescharacteristic
412C22#341611yescharacteristic
413C22#341711yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4#241511yescharacteristic
428C4#304611yescharacteristic
429C4#335411yescharacteristic
430C4#336711yescharacteristic
431C2—162no—
432C2#2467611yescharacteristic
433C2#3358011yescharacteristic
434C2#3359011yescharacteristic
435C2#3391011yescharacteristic
436C2#3391511yescharacteristic
437C2#3393511yescharacteristic
438C2#3394411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28535C2semidirect
9. #13284C2semidirect
10. #13178C2semidirect
11. #13176C2semidirect
12. #13160C2semidirect
13. #13133C2semidirect
14. #13132C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#21752
#285352
#228072
#134822
#134802
#134492
#132842
#131782
#131762
#131602
#131332
#131322
#128122