Finite Group Databasebeta

← #497075

Subgroups — #497075

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2243C211yesmaximal, characteristic
2#21725C211yesmaximal, characteristic, complemented
3#21696C211yesmaximal, characteristic
4#13282C211yesmaximal, characteristic, complemented
5#13204C211yesmaximal, characteristic
6#13133C211yesmaximal, characteristic, complemented
7#13132C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic
9#13130C211yesmaximal, characteristic
10#12938C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic
12#12936C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic
14#12908C211yesmaximal, characteristic, complemented
15#12592C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#440—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2331—21no—
22#2326—21no—
23#2243C418yescomplemented
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1993—21no—
39#1973—21no—
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1959—21no—
43#1943—21no—
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1846C2211yescharacteristic
47#1845C2211yescharacteristic
48#1776—21no—
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51#1726C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×SD16—21no—
62C22×D8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500C2×C414yescomplemented
73#495—22no—
74#495C2×C414yes—
75#440—44no—
76C4×Q16—41no—
77C4×Q16—41no—
78C4×D8—41no—
79C4×D8—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#97D411yescharacteristic
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—21no—
100#391—41no—
101#391—41no—
102#390—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#388—41no—
107#387—41no—
108#387—41no—
109#385—41no—
110#385—41no—
111#384—21no—
112#382—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115#376D411yescharacteristic
116#367—41no—
117#367—41no—
118#366—41no—
119#366—41no—
120#358—41no—
121#358—41no—
122#358C2×C414yescomplemented
123#358C2×C414yescomplemented
124#357—41no—
125#357—41no—
126#357C2×C414yes—
127#357C2×C414yes—
128#346C2×C414yescomplemented
129C4×C16—22no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×D4—21no—
137C23×C4—41no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16C22×C412yes—
144C2×SD16—24no—
145C2×SD16—42no—
146C2×SD16—42no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8C22×C412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2411yescharacteristic
157C4⋊Q8—24no—
158C4⋊Q8—42no—
159#124—24no—
160#124—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
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×C16C22×C412yes—
199C2×C16C22×C412yes—
200#107—24no—
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#107C22×C412yes—
213#107C22×C412yes—
214#106—42no—
215#103—42no—
216#103—42no—
217#102—42no—
218#102—42no—
219#97—22no—
220#97—22no—
221#97—22no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8C22×C412yes—
227C4×C8#4112yes—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95D811yescharacteristic
237#95D811yescharacteristic
238#95D811yescharacteristic
239#95D811yescharacteristic
240C24—41no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—44no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—161no—
249C2×Q8—161no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—82no—
264C22×C4—161no—
265C22×C4—161no—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C22×D411yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—41no—
276Q16—41no—
277SD16—48no—
278D8—22no—
279D8—22no—
280D8—41no—
281D8—41no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C4×D412yes—
289C2×C8AGL(1,ℤ/8)12yes—
290C2×C8#13712yes—
291C2×C8C23×C411yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C2×C82−1+411yescharacteristic
296C2×C82−1+411yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—82no—
318C22⋊C4—161no—
319C22⋊C4—161no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—82no—
325C42—161no—
326C42—161no—
327C42C4×D412yes—
328C42#13512yes—
329C16—22no—
330C16—22no—
331C16—41no—
332C16—41no—
333C23—42no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23C22×D811yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345Q8—164no—
346D4—42no—
347D4—42no—
348D4—81no—
349D4—81no—
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—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
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—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—164no—
388C2×C4C4×D812yes—
389C2×C4C4×D812yes—
390C2×C4C2×C4×D411yescharacteristic
391C2×C4#53511yescharacteristic
392C2×C4#53611yescharacteristic
393C2×C4C2×#13511yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C8—41no—
397C8—41no—
398C8#44014yes—
399C8#52011yescharacteristic
400C8#52011yescharacteristic
401C8#54211yescharacteristic
402C8#54211yescharacteristic
403C22—82no—
404C22—82no—
405C22—82no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22C2×C4×D811yescharacteristic
411C22#320511yescharacteristic
412C22#322611yescharacteristic
413C22#341511yescharacteristic
414C22#341611yescharacteristic
415C22#341611yescharacteristic
416C22#341711yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#232611yescharacteristic
431C4C2×#44011yescharacteristic
432C4#321011yescharacteristic
433C4#322811yescharacteristic
434C2—162no—
435C2#2222011yescharacteristic
436C2#3345411yescharacteristic
437C2#3345511yescharacteristic
438C2#3379011yescharacteristic
439C2#3379111yescharacteristic
440C2#3380911yescharacteristic
441C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×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
2. #21725C2semidirect
4. #13282C2semidirect
6. #13133C2semidirect
7. #13132C2semidirect
10. #12938C2semidirect
12. #12936C2semidirect
14. #12908C2semidirect
15. #12592C2semidirect
23. #2243C4semidirect
72. #500C2×C4semidirect
122. #358C2×C4semidirect
123. #358C2×C4semidirect
128. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
#217252
#216962
#132822
#132042
#131332
#131322
#131312
#131302
#129382
#129372
#129362
#129352
#129082
#125922