Finite Group Databasebeta

← #511402

Subgroups — #511402

Order 512.

465 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28014C211yesmaximal, characteristic, complemented
2#28013C211yesmaximal, characteristic, complemented
3#13528C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13448C211yesmaximal, characteristic
8#13448C211yesmaximal, characteristic
9#13239C211yesmaximal, characteristic
10#13178C211yesmaximal, characteristic, complemented
11#13177C211yesmaximal, characteristic, complemented
12#13168C211yesmaximal, characteristic, complemented
13#13160C211yesmaximal, characteristic, complemented
14#13131C211yesmaximal, characteristic, complemented
15#13131C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17#2513—21no—
18#2512—21no—
19C2×#367C2211yescharacteristic
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#2073—21no—
35#2071—21no—
36#2067—21no—
37#2065—21no—
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1926—21no—
47#1925—21no—
48#1846C2211yescharacteristic
49#1845C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×#126—41no—
62C2×D16—22no—
63C2×D16—41no—
64C2×D16—41no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C2×#107—21no—
70C2×#107C2311yescharacteristic
71C2×#107C2311yescharacteristic
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#106—21no—
77C2×C4⋊C8D411yescharacteristic
78C2×#102—21no—
79C2×#102—21no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×C4×C8C2311yescharacteristic
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2311yescharacteristic
87#397—41no—
88#397—41no—
89#397—41no—
90#396—41no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#388—41no—
100#388—41no—
101#388—41no—
102#384—21no—
103#382—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D411yescharacteristic
107#367—41no—
108#367—41no—
109#367—41no—
110#366—22no—
111#366—41no—
112C4⋊C16—22no—
113C4⋊C16—22no—
114C4⋊C16—41no—
115#357—22no—
116#357—22no—
117#357—41no—
118#357—41no—
119#357—41no—
120#357—41no—
121#346—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×D4—21no—
129C22×D4—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
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—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2411yescharacteristic
146#126—84no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4D811yescharacteristic
158C2×C4⋊C4D811yescharacteristic
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176D16—44no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16#4112yes—
188C2×C16#4112yes—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—81no—
202#106—81no—
203C4⋊C8—41no—
204#102—42no—
205#102—42no—
206#102—81no—
207#102—81no—
208#102—81no—
209#102—81no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95D811yescharacteristic
223#95D811yescharacteristic
224C24—41no—
225C24—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C2×D811yescharacteristic
249C22×C4C2×D811yescharacteristic
250C22×C4C22×D411yescharacteristic
251D8—41no—
252D8—41no—
253D8—41no—
254D8—41no—
255D8—41no—
256D8—41no—
257D8—41no—
258D8—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8C2×#4111yescharacteristic
269C2×C8C2×#4111yescharacteristic
270C2×C8H(5,𝔽2)11yescharacteristic
271C2×C8H(5,𝔽2)11yescharacteristic
272C2×C82−1+411yescharacteristic
273C2×C82−1+411yescharacteristic
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—161no—
291C4⋊C4—161no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—161no—
299C22⋊C4—161no—
300C22⋊C4—161no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—82no—
311C42—161no—
312C16—22no—
313C16—22no—
314C16—41no—
315C16—41no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23C22×D811yescharacteristic
331D4—81no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4#50912yes—
375C2×C4#53711yescharacteristic
376C2×C4#54711yescharacteristic
377C2×C4#55011yescharacteristic
378C2×C4#57511yescharacteristic
379C2×C4#57611yescharacteristic
380C2×C4#57811yescharacteristic
381C8—41no—
382C8—41no—
383C8—41no—
384C8#53811yescharacteristic
385C8#53911yescharacteristic
386C8#55611yescharacteristic
387C8#56311yescharacteristic
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22—161no—
401C22—161no—
402C22#324511yescharacteristic
403C22#333411yescharacteristic
404C22#339111yescharacteristic
405C22C2×#50911yescharacteristic
406C22#341411yescharacteristic
407C22#341511yescharacteristic
408C22#341611yescharacteristic
409C4—22no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4#272711yescharacteristic
422C4#326211yescharacteristic
423C4#335511yescharacteristic
424C4#340211yescharacteristic
425C2—321no—
426C2—321no—
427C2#3252411yescharacteristic
428C2#3381711yescharacteristic
429C2#3382311yescharacteristic
430C2#3391211yescharacteristic
431C2#3391411yescharacteristic
432C2#3396611yescharacteristic
433C2#3396811yescharacteristic
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. #28014C2semidirect
2. #28013C2semidirect
3. #13528C2semidirect
4. #13526C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
10. #13178C2semidirect
11. #13177C2semidirect
12. #13168C2semidirect
13. #13160C2semidirect
14. #13131C2semidirect
15. #13131C2semidirect

Maximal subgroups

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

subgroupindex
#280142
#280132
#135282
#135262
#134762
#134742
#134482
#132392
#131782
#131772
#131682
#131602
#131312