Finite Group Databasebeta

← #504078

Subgroups — #504078

Order 512.

521 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28736C211yesmaximal, characteristic
2#22727C211yesmaximal, characteristic, complemented
3#13528C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13482C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13264C211yesmaximal, characteristic, complemented
8#13252C211yesmaximal, characteristic
9#13206C211yesmaximal, characteristic
10#13178C211yesmaximal, characteristic, complemented
11#13131C211yesmaximal, characteristic, complemented
12#13131C211yesmaximal, characteristic, complemented
13#13131C211yesmaximal, characteristic, complemented
14#13130C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18#2548—21no—
19#2355—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#366C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#2054—21no—
35#2054—21no—
36#1982—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1949—21no—
42#1949—21no—
43#1948—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
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×#124—41no—
64C2×C4×Q8D411yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#493—22no—
70#493—22no—
71C8×Q8—22no—
72C8×Q8—22no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#394—41no—
93#394—41no—
94#391—41no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—41no—
100#387—41no—
101#387—41no—
102#384—21no—
103#382—21no—
104#382—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—41no—
110#367—22no—
111#367—41no—
112#366—22no—
113#366—41no—
114C4⋊C16—22no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—41no—
120#357—41no—
121#357—41no—
122#357—41no—
123#346—22no—
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#124—84no—
155C4×Q8—22no—
156C4×Q8—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4D811yescharacteristic
166C2×C4⋊C4D811yescharacteristic
167C2×C4⋊C4D811yescharacteristic
168C2×C22⋊C4—41no—
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×C42—41no—
177C2×C42—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C16—21no—
184C2×C16—21no—
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#4112yes—
195#107—21no—
196#107—21no—
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—81no—
207#107#4112yes—
208#106—81no—
209#106—81no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213#102—81no—
214#102—81no—
215#102—81no—
216#102—81no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8#4112yes—
222C4×C8#4112yes—
223#95—41no—
224#95—41no—
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—22no—
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#12312yes—
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8H(5,𝔽2)11yescharacteristic
290C2×C82−1+411yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
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—161no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—161no—
313C22⋊C4—161no—
314C22⋊C4—161no—
315C22⋊C4—161no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—82no—
321C42—82no—
322C42—161no—
323C42—161no—
324C42#12312yes—
325C42#13712yes—
326C16—22no—
327C16—41no—
328C16—41no—
329C16—41no—
330C23—81no—
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—44no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—164no—
355D4—164no—
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—21no—
371C2×C4—42no—
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—81no—
389C2×C4#48012yes—
390C2×C4#48012yes—
391C2×C4#50912yes—
392C2×C4C2×#12311yescharacteristic
393C2×C4#54811yescharacteristic
394C2×C4#55011yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57611yescharacteristic
397C2×C4#57711yescharacteristic
398C8—22no—
399C8—41no—
400C8—41no—
401C8#52911yescharacteristic
402C8#54211yescharacteristic
403C8#55611yescharacteristic
404C8#55911yescharacteristic
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22C2×#48011yescharacteristic
420C22#335811yescharacteristic
421C22#339111yescharacteristic
422C22C2×#50911yescharacteristic
423C22#341411yescharacteristic
424C22#341511yescharacteristic
425C22#341611yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4#244811yescharacteristic
439C4#309611yescharacteristic
440C4#337511yescharacteristic
441C4#340011yescharacteristic
442C2—321no—
443C2—321no—
444C2#2631611yescharacteristic
445C2#3364811yescharacteristic
446C2#3365011yescharacteristic
447C2#3393511yescharacteristic
448C2#3393711yescharacteristic
449C2#3396611yescharacteristic
450C2#3396811yescharacteristic
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. #22727C2semidirect
3. #13528C2semidirect
4. #13526C2semidirect
5. #13482C2semidirect
6. #13480C2semidirect
7. #13264C2semidirect
10. #13178C2semidirect
11. #13131C2semidirect
12. #13131C2semidirect
13. #13131C2semidirect
14. #13130C2semidirect
15. #12674C2semidirect

Maximal subgroups

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

subgroupindex
#287362
#227272
#135282
#135262
#134822
#134802
#132642
#132522
#132062
#131782
#131312
#131302
#126742