Finite Group Databasebeta

← #511494

Subgroups — #511494

Order 512.

541 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28736C211yesmaximal, characteristic
2#28265C211yesmaximal, characteristic, complemented
3#13526C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13474C211yesmaximal, characteristic, complemented
6#13449C211yesmaximal, characteristic, complemented
7#13407C211yesmaximal, characteristic
8#13328C211yesmaximal, characteristic
9#13264C211yesmaximal, characteristic, complemented
10#13238C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic, complemented
12#13133C211yesmaximal, characteristic, complemented
13#13112C211yesmaximal, characteristic, complemented
14#13013C211yesmaximal, characteristic, complemented
15#13013C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21#2582—21no—
22#2548—21no—
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2068—21no—
36#2068—21no—
37#2054—21no—
38#2054—21no—
39#1982—21no—
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1845C2211yescharacteristic
49#1803—21no—
50C4×#107C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×D8—21no—
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#127—21no—
64C2×#124—41no—
65C2×C4×Q8—21no—
66C2×C4×Q8D411yescharacteristic
67C2×SD32—22no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32—41no—
71C2×SD32—41no—
72C2×SD32—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#493—41no—
78#493—41no—
79C8×Q8—41no—
80C8×Q8—41no—
81C4×Q16—41no—
82C4×Q16—41no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#398—41no—
103#398—41no—
104#394—41no—
105#394—41no—
106#391—21no—
107#391—21no—
108#391—41no—
109#390—21no—
110#390—21no—
111#390—21no—
112#390—21no—
113#390—41no—
114#389—21no—
115#389—21no—
116#389—41no—
117#384—21no—
118C4×C4⋊C4—21no—
119#366—22no—
120#366—41no—
121C4⋊C16—41no—
122#358—22no—
123#358—41no—
124#357—22no—
125#357—41no—
126#357—41no—
127#357—41no—
128#357—41no—
129#357—41no—
130#346—41no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C22×Q8D811yescharacteristic
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—41no—
139C22×D4—41no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16#4112yes—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—42no—
159C2×D8—81no—
160C2×D8—81no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2411yescharacteristic
165#127—42no—
166#127—42no—
167#124—84no—
168C4×Q8—41no—
169C4×Q8—41no—
170C4×Q8—42no—
171C4×Q8—42no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D811yescharacteristic
183C2×C4⋊C4D811yescharacteristic
184C2×C4⋊C4D811yescharacteristic
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198SD32—44no—
199SD32—44no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16#4112yes—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—81no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226#103—42no—
227#103—81no—
228#103—81no—
229#102—81no—
230#102—81no—
231#102—81no—
232#102—81no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246C24—41no—
247C24—41no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—161no—
274C2×D4—161no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C22×D411yescharacteristic
288Q16—22no—
289Q16—22no—
290Q16—41no—
291Q16—41no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—41no—
297D8—41no—
298D8—41no—
299D8—41no—
300D8—84no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C82−1+411yescharacteristic
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—161no—
332C4⋊C4—161no—
333C4⋊C4—161no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—161no—
339C22⋊C4—161no—
340C22⋊C4—161no—
341C22⋊C4—161no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—82no—
349C42—82no—
350C16—22no—
351C16—41no—
352C16—41no—
353C16—41no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23C22×D811yescharacteristic
369Q8—44no—
370Q8—42no—
371Q8—42no—
372Q8—81no—
373Q8—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—164no—
383D4—164no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#50912yes—
419C2×C4#51012yes—
420C2×C4#53411yescharacteristic
421C2×C4#54811yescharacteristic
422C2×C4#55011yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57611yescharacteristic
425C2×C4#57711yescharacteristic
426C8—41no—
427C8—41no—
428C8—41no—
429C8#53811yescharacteristic
430C8#54211yescharacteristic
431C8#55111yescharacteristic
432C8#56211yescharacteristic
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22#318611yescharacteristic
448C22#335811yescharacteristic
449C22#339111yescharacteristic
450C22C2×#50911yescharacteristic
451C22C2×#51011yescharacteristic
452C22#341511yescharacteristic
453C22#341611yescharacteristic
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—82no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4#273011yescharacteristic
467C4#319811yescharacteristic
468C4#337011yescharacteristic
469C4#340111yescharacteristic
470C2—321no—
471C2—321no—
472C2#3259611yescharacteristic
473C2#3375911yescharacteristic
474C2#3376411yescharacteristic
475C2#3393511yescharacteristic
476C2#3394211yescharacteristic
477C2#3396611yescharacteristic
478C2#3396711yescharacteristic
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. #28265C2semidirect
3. #13526C2semidirect
4. #13480C2semidirect
5. #13474C2semidirect
6. #13449C2semidirect
9. #13264C2semidirect
10. #13238C2semidirect
11. #13149C2semidirect
12. #13133C2semidirect
13. #13112C2semidirect
14. #13013C2semidirect
15. #13013C2semidirect

Maximal subgroups

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

subgroupindex
#287362
#282652
#135262
#134802
#134742
#134492
#134072
#133282
#132642
#132382
#131492
#131332
#131122
#130132