Finite Group Databasebeta

← #507443

Subgroups — #507443

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33208C211yesmaximal, characteristic, complemented
2C4×#466C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13453C211yesmaximal, characteristic, complemented
6#13365C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic
10#13061C211yesmaximal, characteristic
11#12937C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic, complemented
15#12544C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2851—21no—
23C4×#121—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32#1991—21no—
33#1970C2211yescharacteristic
34#1942C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1941—21no—
38#1934—21no—
39#1893—21no—
40#1846C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1808C2211yescharacteristic
44#1807—21no—
45C4×#107C2211yescharacteristic
46C4×#102—21no—
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×C4×D4—41no—
66C22×C42D411yescharacteristic
67C2×M32—22no—
68C2×M32—22no—
69C2×M32—22no—
70C2×M32—22no—
71C2×M32—41no—
72C2×M32—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#499—41no—
78#499—41no—
79#466—22no—
80#466Q814yes—
81C4×D8—41no—
82C4×D8—41no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C4×C8—41no—
96C2×C4×C8—41no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#387—41no—
107#387—41no—
108#386—41no—
109#385—41no—
110#385—41no—
111#382—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—41no—
117#367—22no—
118#367—41no—
119#366—22no—
120#366—41no—
121#357—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C22×D4—21no—
133C22×D4—41no—
134C23×C4—41no—
135C23×C4D811yescharacteristic
136C23×C4D811yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8C2×Q812yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—41no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×Q812yes—
155C22×C8#4112yes—
156C22×C8C2411yescharacteristic
157#125—42no—
158#125—42no—
159#121—24no—
160#121—42no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—84no—
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—21no—
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×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42D811yescharacteristic
187C2×C42D811yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190M32—44no—
191M32—44no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16#4112yes—
203C2×C16#4112yes—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107C2×Q812yes—
218#106—42no—
219#106—42no—
220#102—24no—
221#102—42no—
222#102—42no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—41no—
228C4×C8—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24—41no—
239C24C2×D811yescharacteristic
240C2×D4—22no—
241C2×D4—44no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4C22⋊Q812yes—
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4#13712yes—
276C22×C4C22×D411yescharacteristic
277D8—22no—
278D8—22no—
279D8—41no—
280D8—41no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—44no—
293C2×C8C22⋊Q812yes—
294C2×C8#13512yes—
295C2×C8C22×Q811yescharacteristic
296C2×C8C2×#4111yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8H(5,𝔽2)11yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—44no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—41no—
333C42—82no—
334C16—22no—
335C16—22no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23#47412yes—
352C23C22×D811yescharacteristic
353D4—42no—
354D4—42no—
355D4—88no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
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—44no—
376C2×C4—44no—
377C2×C4—42no—
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—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—164no—
399C2×C4#47412yes—
400C2×C4#50912yes—
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4#53211yescharacteristic
403C2×C4#53411yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57511yescharacteristic
407C8—22no—
408C8—22no—
409C8—41no—
410C8#55111yescharacteristic
411C8#55111yescharacteristic
412C8#55411yescharacteristic
413C8#55411yescharacteristic
414C22—22no—
415C22—22no—
416C22—22no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22C2×#47411yescharacteristic
429C22#316911yescharacteristic
430C22#318611yescharacteristic
431C22C2×#50911yescharacteristic
432C22#341411yescharacteristic
433C22#341411yescharacteristic
434C22#341411yescharacteristic
435C4—22no—
436C4—22no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—82no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4#256411yescharacteristic
448C4#307611yescharacteristic
449C4#318111yescharacteristic
450C4#319011yescharacteristic
451C2—42no—
452C2—162no—
453C2#2915711yescharacteristic
454C2#3362711yescharacteristic
455C2#3363011yescharacteristic
456C2#3373711yescharacteristic
457C2#3375011yescharacteristic
458C2#3376011yescharacteristic
459C2#3376311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33208C2semidirect
3. #13539C2semidirect
4. #13528C2semidirect
5. #13453C2semidirect
6. #13365C2semidirect
7. #13111C2semidirect
8. #13109C2semidirect
11. #12937C2semidirect
12. #12935C2semidirect
13. #12836C2semidirect
14. #12836C2semidirect
15. #12544C2semidirect

Maximal subgroups

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

subgroupindex
#332082
C4×#4662
#135392
#135282
#134532
#133652
#131112
#131092
#130612
#129372
#129352
#128362
#125442