Finite Group Databasebeta

← #496833

Subgroups — #496833

Order 512.

714 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
1#33521C211yesmaximal, characteristic
2#21527C211yesmaximal, characteristic, complemented
3C2×#1975C211yesmaximal, characteristic
4#13250C212yesmaximal, complemented
5#13153C212yesmaximal, complemented
6#13143C212yesmaximal, complemented
7#13038C212yesmaximal, complemented
8#13003C211yesmaximal, characteristic, complemented
9#12928C211yesmaximal, characteristic, complemented
10#12735C211yesmaximal, characteristic
11#12605C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C2×#510C2212yes—
14C2×#467C2211yescharacteristic
15C2×#441C2211yescharacteristic, complemented
16C22×#108C2211yescharacteristic
17C22×#104—21no—
18C2×#408—21no—
19C22×#97C2211yescharacteristic
20C4×C22⋊Q8—21no—
21#2189C414yescomplemented
22#2188C412yescomplemented
23#2187C412yescomplemented
24C2×#365C2212yescomplemented
25C2×#362C2212yescomplemented
26C2×#360C2212yes—
27C2×#360C2212yes—
28#2139C2212yes—
29#2112C2211yescharacteristic
30#2112C2211yescharacteristic
31C2×#347C2212yescomplemented
32#1975—21no—
33#1975—21no—
34#1975C2212yes—
35#1975C2212yes—
36#1971C2211yescharacteristic, complemented
37#1943C2211yescharacteristic, complemented
38#1935—21no—
39#1892—21no—
40#1807C2211yescharacteristic
41#1776C2211yescharacteristic, complemented
42#1763—21no—
43#1387C2212yescomplemented
44#1383C2212yescomplemented
45#1375C2212yescomplemented
46C22×#41—21no—
47C2×#135—21no—
48C2×#135—21no—
49C2×#135—42no—
50C2×#135—42no—
51C2×#135C2×C412yes—
52C2×#135C2312yes—
53C22×Q16C2×C411yescharacteristic
54C22×Q16C2311yescharacteristic
55C22×SD16—21no—
56C22×D8C2×C411yescharacteristic
57C22×M16—21no—
58C23×C8D411yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C22⋊Q8—21no—
61C2×C4×Q8—21no—
62C22×C42D411yescharacteristic
63#510—42no—
64#510—42no—
65C2×Q32C2×C414yescomplemented
66C2×SD32C2×C414yescomplemented
67C2×M32C2312yes—
68C2×M32C2312yes—
69#467—21no—
70#467—21no—
71#467C2×C412yes—
72#467C2×C412yes—
73#441—42no—
74C2×#108—21no—
75C2×#108—42no—
76C2×#108C2312yes—
77C2×#107C2311yescharacteristic
78C2×#106C2311yescharacteristic
79C2×#106C2311yescharacteristic
80C2×#104—22no—
81C2×#104—22no—
82C2×#104—44no—
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86#408—22no—
87#408—22no—
88#408—42no—
89C2×C22⋊C8—42no—
90C2×#97—42no—
91C2×#97D411yescharacteristic
92C2×#97D411yescharacteristic
93C2×#97C2311yescharacteristic
94C2×#97C2311yescharacteristic
95#391—21no—
96#389—21no—
97#387—41no—
98#385—41no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—41no—
102C4×C22⋊C4—41no—
103#365—42no—
104#362—42no—
105#360—42no—
106#360—42no—
107#358C2×C414yescomplemented
108#357C2×C414yescomplemented
109C22⋊C16—21no—
110C22⋊C16—21no—
111C22⋊C16C2×C412yes—
112C22⋊C16C2×C412yes—
113#347—42no—
114#340—21no—
115#340—21no—
116#340—21no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—42no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—42no—
123C2×#41—41no—
124C2×#41—41no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×D4—21no—
128C22×D4—41no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132#135—22no—
133#135—22no—
134#135—22no—
135#135—22no—
136#135—42no—
137#135—42no—
138#135—42no—
139#135—42no—
140#135—44no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—22no—
144C2×Q16—22no—
145C2×Q16—42no—
146C2×Q16C22×C412yes—
147C2×Q16C22×C412yes—
148C2×SD16—22no—
149C2×SD16—22no—
150C2×SD16—44no—
151C2×D8—21no—
152C2×D8—42no—
153C2×D8C22×C412yes—
154C2×M16—22no—
155C2×M16—41no—
156C2×M16—42no—
157C2×M16—42no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—41no—
163C22×C8—41no—
164C22×C8—41no—
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8#4111yescharacteristic
171C22×C8#4111yescharacteristic
172C22×C8C2411yescharacteristic
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C4×Q8—42no—
178C4×Q8—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194Q32—44no—
195SD32—44no—
196M32—22no—
197M32—22no—
198M32—22no—
199M32—22no—
200C2×C16—22no—
201C2×C16C22×C414yes—
202#108—22no—
203#108—22no—
204#108—42no—
205#107—21no—
206#107C22×C412yes—
207#106—22no—
208#106—22no—
209#104—44no—
210#104—44no—
211#103—22no—
212#103—22no—
213#103—42no—
214#103—42no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217#97—22no—
218#97—22no—
219#97—41no—
220#97—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226C24—41no—
227C24AGL(1,ℤ/8)11yescharacteristic
228#41—44no—
229#41—44no—
230#41—42no—
231#41—42no—
232#41—42no—
233#41—42no—
234#41—82no—
235#41—82no—
236#41—82no—
237#41—82no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—82no—
255C2×D4—82no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—81no—
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4AGL(1,ℤ/8)11yescharacteristic
277C22×C4AGL(1,ℤ/8)11yescharacteristic
278C22×C4AGL(1,ℤ/8)11yescharacteristic
279C22×C4C22×D411yescharacteristic
280Q16—24no—
281Q16—22no—
282Q16—22no—
283Q16—42no—
284Q16—42no—
285SD16—44no—
286SD16—42no—
287SD16—42no—
288D8—22no—
289D8—22no—
290D8—42no—
291M16—42no—
292M16—42no—
293M16—82no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—82no—
310C2×C8C4×D411yescharacteristic
311C2×C8C4×D411yescharacteristic
312C2×C8C23×C411yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8H(5,𝔽2)12yes—
315C2×C8H(5,𝔽2)12yes—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—82no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—42no—
333C42—82no—
334C42—82no—
335C42AGL(1,ℤ/8)12yes—
336C16—24no—
337C16—42no—
338C23—21no—
339C23—22no—
340C23—21no—
341C23—41no—
342C23—42no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23#44211yescharacteristic
347C23#44211yescharacteristic
348C23C2×AGL(1,ℤ/8)11yescharacteristic
349Q8—42no—
350Q8—42no—
351Q8—42no—
352Q8—42no—
353Q8—82no—
354Q8—81no—
355Q8—81no—
356D4—42no—
357D4—42no—
358D4—82no—
359D4—84no—
360D4—82no—
361D4—82no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4#44211yescharacteristic
397C2×C4#44211yescharacteristic
398C2×C4C2×C4×D411yescharacteristic
399C2×C4#53411yescharacteristic
400C2×C4#53511yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
403C2×C4#57511yescharacteristic
404C8—21no—
405C8—21no—
406C8—41no—
407C8—41no—
408C8—82no—
409C8#51812yes—
410C8#53812yes—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#218012yes—
425C22C2×#44211yescharacteristic
426C22#319211yescharacteristic
427C22#321311yescharacteristic
428C4—21no—
429C4—21no—
430C4—42no—
431C4—81no—
432C4—82no—
433C4—82no—
434C4—82no—
435C4—161no—
436C4—161no—
437C4#232111yescharacteristic
438C4C4×AGL(1,ℤ/8)11yescharacteristic
439C4#319411yescharacteristic
440C4#321711yescharacteristic
441C2—21no—
442C2—21no—
443C2—22no—
444C2—41no—
445C2—162no—
446C2#2204611yescharacteristic
447C2C2×#218011yescharacteristic
448C2#3346811yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21527C2semidirect
4. #13250C2semidirect
5. #13153C2semidirect
6. #13143C2semidirect
7. #13038C2semidirect
8. #13003C2semidirect
9. #12928C2semidirect
11. #12605C2semidirect
15. C2×#441C22semidirect
21. #2189C4semidirect
22. #2188C4semidirect
23. #2187C4semidirect
24. C2×#365C22semidirect
25. C2×#362C22semidirect
31. C2×#347C22semidirect
36. #1971C22semidirect
37. #1943C22semidirect
41. #1776C22semidirect
43. #1387C22semidirect
44. #1383C22semidirect
45. #1375C22semidirect
65. C2×Q32C2×C4semidirect
66. C2×SD32C2×C4semidirect
107. #358C2×C4semidirect
108. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
#335212
#215272
C2×#19752
#132502
#131532
#131432
#130382
#130032
#129282
#127352
#126052