Finite Group Databasebeta

← #511511

Subgroups — #511511

Order 512.

505 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28736C211yesmaximal, characteristic
2#28679C211yesmaximal, characteristic, complemented
3#13476C211yesmaximal, characteristic, complemented
4#13474C211yesmaximal, characteristic, complemented
5#13407C211yesmaximal, characteristic
6#13407C211yesmaximal, characteristic
7#13365C211yesmaximal, characteristic, complemented
8#13365C211yesmaximal, characteristic, complemented
9#13264C211yesmaximal, characteristic, complemented
10#13149C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic, complemented
12#13131C211yesmaximal, characteristic, complemented
13#13130C211yesmaximal, characteristic, complemented
14#13111C211yesmaximal, characteristic, complemented
15#13109C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19#2584—21no—
20#2548—21no—
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33#2088—21no—
34#2088—21no—
35#2054—21no—
36#2054—21no—
37#1989—21no—
38#1982—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8—21no—
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C2×#127—21no—
63C2×#125—21no—
64C2×#124—41no—
65C2×C4×Q8D411yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#499—41no—
71#499—41no—
72#493—41no—
73#493—41no—
74C8×Q8—41no—
75C8×Q8—41no—
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#106—21no—
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×C4⋊C8D411yescharacteristic
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#398—41no—
94#395—41no—
95#394—41no—
96#394—41no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—41no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—41no—
107#389—41no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110#367—22no—
111#366—22no—
112C4⋊C16—41no—
113#357—22no—
114#357—22no—
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#340—21no—
127#340—21no—
128C22×Q8D811yescharacteristic
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—41no—
132C22×D4—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—21no—
145C2×D8—42no—
146C2×D8—81no—
147C2×D8—81no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152#127—42no—
153#127—42no—
154#125—42no—
155#125—42no—
156#124—84no—
157C4×Q8—41no—
158C4×Q8—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4D811yescharacteristic
172C2×C4⋊C4D811yescharacteristic
173C2×C4⋊C4D811yescharacteristic
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×C22⋊C4—41no—
181C2×C22⋊C4—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—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#4112yes—
210#107#4112yes—
211#106—42no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
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×C8—21no—
224C4×C8—21no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231C24—41no—
232C24—41no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—41no—
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—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—82no—
252C2×D4—161no—
253C2×D4—161no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C22×D411yescharacteristic
267D8—41no—
268D8—41no—
269D8—41no—
270D8—41no—
271D8—41no—
272D8—41no—
273D8—41no—
274D8—41no—
275D8—84no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C8H(5,𝔽2)11yescharacteristic
289C2×C82−1+411yescharacteristic
290C2×C82−1+411yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
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—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C22⋊C4—161no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—82no—
327C42—161no—
328C16—41no—
329C16—41no—
330C16—41no—
331C16—41no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23C22×D811yescharacteristic
347Q8—44no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—164no—
357D4—164no—
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—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
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—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4#50912yes—
394C2×C4#53411yescharacteristic
395C2×C4#54811yescharacteristic
396C2×C4#54811yescharacteristic
397C2×C4#57511yescharacteristic
398C2×C4#57711yescharacteristic
399C2×C4#57711yescharacteristic
400C8—41no—
401C8—41no—
402C8—41no—
403C8#54211yescharacteristic
404C8#54211yescharacteristic
405C8#55511yescharacteristic
406C8#56311yescharacteristic
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22#318611yescharacteristic
422C22#335811yescharacteristic
423C22#335811yescharacteristic
424C22C2×#50911yescharacteristic
425C22#341411yescharacteristic
426C22#341511yescharacteristic
427C22#341611yescharacteristic
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—82no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4#273111yescharacteristic
441C4#320011yescharacteristic
442C4#337111yescharacteristic
443C4#337111yescharacteristic
444C2—321no—
445C2—321no—
446C2#3262811yescharacteristic
447C2#3376011yescharacteristic
448C2#3376411yescharacteristic
449C2#3393311yescharacteristic
450C2#3393511yescharacteristic
451C2#3393711yescharacteristic
452C2#3393811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28679C2semidirect
3. #13476C2semidirect
4. #13474C2semidirect
7. #13365C2semidirect
8. #13365C2semidirect
9. #13264C2semidirect
10. #13149C2semidirect
11. #13149C2semidirect
12. #13131C2semidirect
13. #13130C2semidirect
14. #13111C2semidirect
15. #13109C2semidirect

Maximal subgroups

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

subgroupindex
#287362
#286792
#134762
#134742
#134072
#133652
#132642
#131492
#131312
#131302
#131112
#131092