Finite Group Databasebeta

← #511303

Subgroups — #511303

Order 512.

526 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28527C211yesmaximal, characteristic, complemented
2#28013C211yesmaximal, characteristic, complemented
3#13527C211yesmaximal, characteristic, complemented
4#13482C211yesmaximal, characteristic, complemented
5#13480C211yesmaximal, characteristic, complemented
6#13452C211yesmaximal, characteristic, complemented
7#13448C211yesmaximal, characteristic
8#13363C211yesmaximal, characteristic, complemented
9#13239C211yesmaximal, characteristic
10#13177C211yesmaximal, characteristic, complemented
11#13131C211yesmaximal, characteristic, complemented
12#13105C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic
14#12937C211yesmaximal, characteristic, complemented
15#12890C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#474—21no—
19C2×C4×D8C2211yescharacteristic
20#2542—21no—
21#2513—21no—
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2093—21no—
34#2093—21no—
35#2073—21no—
36#2065—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1942C2211yescharacteristic
43#1926—21no—
44#1846C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C22⋊Q8—41no—
63C2×C22⋊Q8—41no—
64C2×C4×D4—21no—
65C2×D16—22no—
66C2×D16—22no—
67C2×D16—41no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—41no—
73#500—41no—
74#474—44no—
75C4×D8—41no—
76C4×D8—41no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#399—41no—
98#399—41no—
99#397—41no—
100#397—41no—
101#390—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#389—21no—
107#388—41no—
108#388—41no—
109#384—21no—
110#384—21no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—41no—
115#367—41no—
116#367—41no—
117C4⋊C16—22no—
118#357—22no—
119#357—22no—
120#357—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124#346—22no—
125#340—21no—
126#340—21no—
127#336D411yescharacteristic
128#336D411yescharacteristic
129C22×Q8—41no—
130C22×D4—21no—
131C22×D4—21no—
132C23×C4—41no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8#4112yes—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2411yescharacteristic
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C22⋊Q8—84no—
152C22⋊Q8—84no—
153C4×D4—42no—
154C4×D4—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4D811yescharacteristic
168C2×C4⋊C4D811yescharacteristic
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×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184D16—44no—
185C2×C16—21no—
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×C16—21no—
195C2×C16—21no—
196C2×C16#4112yes—
197#107—21no—
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—21no—
209#107—42no—
210#107—42no—
211#107—42no—
212#106—81no—
213C4⋊C8—22no—
214#102—42no—
215#102—42no—
216#102—42no—
217#102—42no—
218#102—81no—
219#102—81no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95D811yescharacteristic
231#95D811yescharacteristic
232C24—41no—
233C24—41no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—82no—
237C2×D4—22no—
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—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C22×D411yescharacteristic
265D8—22no—
266D8—22no—
267D8—41no—
268D8—41no—
269D8—41no—
270D8—41no—
271D8—41no—
272D8—41no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8AGL(1,ℤ/8)12yes—
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8H(5,𝔽2)11yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C2×C82−1+411yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
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—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—161no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—161no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—161no—
326C16—22no—
327C16—41no—
328C16—41no—
329C16—41no—
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—
343C23—81no—
344C23C22×D811yescharacteristic
345Q8—164no—
346D4—42no—
347D4—42no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
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—21no—
368C2×C4—21no—
369C2×C4—21no—
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—81no—
388C2×C4—164no—
389C2×C4#50912yes—
390C2×C4#50912yes—
391C2×C4#53511yescharacteristic
392C2×C4#54711yescharacteristic
393C2×C4#54811yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4#57711yescharacteristic
396C2×C4#57811yescharacteristic
397C8—22no—
398C8—41no—
399C8—41no—
400C8#53911yescharacteristic
401C8#54211yescharacteristic
402C8#55111yescharacteristic
403C8#56311yescharacteristic
404C22—82no—
405C22—82no—
406C22—82no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22#320511yescharacteristic
419C22#333411yescharacteristic
420C22#335811yescharacteristic
421C22C2×#50911yescharacteristic
422C22C2×#50911yescharacteristic
423C22#341611yescharacteristic
424C22#341611yescharacteristic
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#272311yescharacteristic
438C4#322511yescharacteristic
439C4#334511yescharacteristic
440C4#336711yescharacteristic
441C2—162no—
442C2—321no—
443C2#3242611yescharacteristic
444C2#3378311yescharacteristic
445C2#3378711yescharacteristic
446C2#3391211yescharacteristic
447C2#3392111yescharacteristic
448C2#3394011yescharacteristic
449C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
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
1. #28527C2semidirect
2. #28013C2semidirect
3. #13527C2semidirect
4. #13482C2semidirect
5. #13480C2semidirect
6. #13452C2semidirect
8. #13363C2semidirect
10. #13177C2semidirect
11. #13131C2semidirect
12. #13105C2semidirect
14. #12937C2semidirect
15. #12890C2semidirect

Maximal subgroups

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

subgroupindex
#285272
#280132
#135272
#134822
#134802
#134522
#134482
#133632
#132392
#131772
#131312
#131052
#130122
#129372
#128902