Finite Group Databasebeta

← #507567

Subgroups — #507567

Order 512.

553 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21725C211yesmaximal, characteristic, complemented
2#21696C211yesmaximal, characteristic
3#13529C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic
5#13364C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic
7#13254C211yesmaximal, characteristic
8#13254C211yesmaximal, characteristic
9#13177C211yesmaximal, characteristic, complemented
10#13176C211yesmaximal, characteristic, complemented
11#12938C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic
15#12880C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#440—21no—
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2331—21no—
22#2326—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37#1993—21no—
38#1973—21no—
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1959—21no—
42#1943—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1776—21no—
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1726C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#124—21no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#500—22no—
71#500—22no—
72#495—22no—
73#495Q814yescomplemented
74#440—44no—
75C4×Q16—41no—
76C4×Q16—41no—
77C4×D8—41no—
78C4×D8—41no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×#103—21no—
89C2×#103—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×#97D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#391—21no—
99#391—41no—
100#391—41no—
101#390—21no—
102#389—21no—
103#389—21no—
104#388—41no—
105#388—41no—
106#387—41no—
107#387—41no—
108#385—41no—
109#385—41no—
110#384—21no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114#376D411yescharacteristic
115#367—41no—
116#367—41no—
117#366—41no—
118#366—41no—
119C4⋊C16—22no—
120C4⋊C16—22no—
121#358—22no—
122#358—22no—
123#358—41no—
124#358—41no—
125#357—22no—
126#357—22no—
127#357—41no—
128#357—41no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133C22×Q8—21no—
134C22×Q8—41no—
135C22×D4—21no—
136C23×C4—41no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16C2×Q812yes—
143C2×SD16—24no—
144C2×SD16—42no—
145C2×SD16—42no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8C2×Q812yes—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2411yescharacteristic
156C4⋊Q8—24no—
157C4⋊Q8—42no—
158#124—24no—
159#124—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—42no—
163C4×D4—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
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
187C2×C16—21no—
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—
199#107—24no—
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—42no—
211#107#4112yes—
212#107#4112yes—
213#106—42no—
214#103—42no—
215#103—42no—
216#102—42no—
217#102—42no—
218#97—22no—
219#97—22no—
220#97—41no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8C2×Q812yes—
226C4×C8#4112yes—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95D811yescharacteristic
236#95D811yescharacteristic
237#95D811yescharacteristic
238#95D811yescharacteristic
239C24—41no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—161no—
248C2×Q8—161no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—82no—
263C22×C4—161no—
264C22×C4—161no—
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C22×D411yescharacteristic
272Q16—22no—
273Q16—22no—
274Q16—41no—
275Q16—41no—
276SD16—48no—
277D8—22no—
278D8—22no—
279D8—41no—
280D8—41no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C22⋊Q812yes—
289C2×C8AGL(1,ℤ/8)12yes—
290C2×C8C22×Q811yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—44no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—82no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—82no—
324C42—161no—
325C42—161no—
326C42C22⋊Q812yes—
327C42#13712yes—
328C16—41no—
329C16—41no—
330C16—41no—
331C16—41no—
332C23—42no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23C22×D811yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344Q8—164no—
345D4—42no—
346D4—42no—
347D4—81no—
348D4—81no—
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—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
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—81no—
385C2×C4—81no—
386C2×C4—164no—
387C2×C4#47412yes—
388C2×C4#47412yes—
389C2×C4C2×C22⋊Q811yescharacteristic
390C2×C4#52511yescharacteristic
391C2×C4#53511yescharacteristic
392C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
393C2×C4C2×#13711yescharacteristic
394C2×C4#57511yescharacteristic
395C8—22no—
396C8—22no—
397C8—41no—
398C8#55411yescharacteristic
399C8#55411yescharacteristic
400C8#55911yescharacteristic
401C8#55911yescharacteristic
402C22—82no—
403C22—82no—
404C22—82no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22C2×#47411yescharacteristic
410C22#310011yescharacteristic
411C22#320511yescharacteristic
412C22#341511yescharacteristic
413C22#341611yescharacteristic
414C22#341611yescharacteristic
415C22#341711yescharacteristic
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#258011yescharacteristic
430C4#307711yescharacteristic
431C4#311211yescharacteristic
432C4#321611yescharacteristic
433C2—162no—
434C2#2925411yescharacteristic
435C2#3363311yescharacteristic
436C2#3363411yescharacteristic
437C2#3367811yescharacteristic
438C2#3368111yescharacteristic
439C2#3379011yescharacteristic
440C2#3379111yescharacteristic
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. #21725C2semidirect
3. #13529C2semidirect
5. #13364C2semidirect
9. #13177C2semidirect
10. #13176C2semidirect
11. #12938C2semidirect
13. #12936C2semidirect
15. #12880C2semidirect
73. #495Q8semidirect

Maximal subgroups

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

subgroupindex
#217252
#216962
#135292
#135282
#133642
#133632
#132542
#131772
#131762
#129382
#129372
#129362
#129352
#128802