Finite Group Databasebeta

← #498062

Subgroups — #498062

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2177C211yesmaximal, characteristic
2#22810C211yesmaximal, characteristic, complemented
3#22807C211yesmaximal, characteristic, complemented
4#13239C211yesmaximal, characteristic
5#13235C211yesmaximal, characteristic
6#13131C211yesmaximal, characteristic, complemented
7#13131C211yesmaximal, characteristic, complemented
8#13130C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#12908C211yesmaximal, characteristic, complemented
11#12813C211yesmaximal, characteristic, complemented
12#12811C211yesmaximal, characteristic, complemented
13#12674C211yesmaximal, characteristic, complemented
14C4×#367C211yesmaximal, characteristic, complemented
15#12663C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2357—21no—
20#2354—21no—
21#2177C418yescomplemented
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1926—21no—
42#1926—21no—
43#1925—21no—
44#1925—21no—
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1804—21no—
48#1802—21no—
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×#117—41no—
65C2×D16—22no—
66C2×D16—22no—
67C2×D16C2×C414yes—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C4×D8—22no—
73C4×D8—22no—
74C4×D8C2×C414yes—
75C4×D8C2×C414yes—
76C2×#107—21no—
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#106—21no—
84C2×C4⋊C8D411yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#388—41no—
99#388—41no—
100#388—41no—
101#388—41no—
102#384—21no—
103#382—21no—
104#381—41no—
105#380—41no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111C4×C22⋊C4—41no—
112#367—22no—
113#367—41no—
114#367C2×C414yes—
115#366—22no—
116#366—41no—
117C4⋊C16—22no—
118#357—41no—
119#357—41no—
120#357—41no—
121#357—41no—
122#357C2×C414yes—
123#357C2×C414yes—
124#346—22no—
125#346C2×C414yes—
126C4×C16—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×D4—21no—
134C22×D4—21no—
135C23×C4—41no—
136C23×C4—41no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8C22×C412yes—
148C2×D8C22×C412yes—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2411yescharacteristic
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—42no—
156C4×D4—42no—
157#117—84no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4D811yescharacteristic
167C2×C4⋊C4D811yescharacteristic
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×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185D16—28no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16C22×C412yes—
195C2×C16C22×C412yes—
196C2×C16#4112yes—
197C2×C16#4112yes—
198#107—21no—
199#107—21no—
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—81no—
209#107C22×C412yes—
210#107C22×C412yes—
211#106—81no—
212C4⋊C8—22no—
213#102—24no—
214#102—24no—
215#102—81no—
216#102—81no—
217#102—81no—
218#102—81no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8C22×C412yes—
224C4×C8#4112yes—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95D811yescharacteristic
232#95D811yescharacteristic
233C24—41no—
234C24—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
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—44no—
250C22×C4—44no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—161no—
258C22×C4—161no—
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C22×D411yescharacteristic
266D8—22no—
267D8—22no—
268D8—22no—
269D8—22no—
270D8—41no—
271D8—41no—
272D8—41no—
273D8—41no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8C4×D412yes—
283C2×C8C23×C411yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
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—82no—
304C4⋊C4—161no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—161no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C42—82no—
321C42—161no—
322C42—161no—
323C42C4×D412yes—
324C42#13512yes—
325C16—22no—
326C16—41no—
327C16—41no—
328C16#11714yescomplemented
329C23—42no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23C22×D811yescharacteristic
344D4—42no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
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—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—88no—
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—164no—
387C2×C4C4×D812yes—
388C2×C4C4×D812yes—
389C2×C4#50912yes—
390C2×C4C2×C4×D411yescharacteristic
391C2×C4#54711yescharacteristic
392C2×C4#55011yescharacteristic
393C2×C4C2×#13511yescharacteristic
394C2×C4#57611yescharacteristic
395C2×C4#57811yescharacteristic
396C8—22no—
397C8—41no—
398C8—41no—
399C8C2×#11711yescharacteristic
400C8#52011yescharacteristic
401C8#55511yescharacteristic
402C8#55611yescharacteristic
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22C2×C4×D811yescharacteristic
418C22#333411yescharacteristic
419C22#339111yescharacteristic
420C22C2×#50911yescharacteristic
421C22#341411yescharacteristic
422C22#341511yescharacteristic
423C22#341611yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#235411yescharacteristic
437C4#295111yescharacteristic
438C4#334111yescharacteristic
439C4#339311yescharacteristic
440C2—162no—
441C2—162no—
442C2#2280711yescharacteristic
443C2C4×#50911yescharacteristic
444C2#3345411yescharacteristic
445C2#3391211yescharacteristic
446C2#3391411yescharacteristic
447C2#3396611yescharacteristic
448C2#3396811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C2×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
2. #22810C2semidirect
3. #22807C2semidirect
6. #13131C2semidirect
7. #13131C2semidirect
8. #13130C2semidirect
9. #13130C2semidirect
10. #12908C2semidirect
11. #12813C2semidirect
12. #12811C2semidirect
13. #12674C2semidirect
14. C4×#367C2semidirect
15. #12663C2semidirect
21. #2177C4semidirect
328. C16#117semidirect

Maximal subgroups

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

subgroupindex
C2×#21772
#228102
#228072
#132392
#132352
#131312
#131302
#129082
#128132
#128112
#126742
C4×#3672
#126632