Finite Group Databasebeta

← #510462

Subgroups — #510462

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28531C211yesmaximal, characteristic, complemented
2#26254C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic, complemented
6#13363C211yesmaximal, characteristic, complemented
7#13363C211yesmaximal, characteristic, complemented
8#13328C211yesmaximal, characteristic
9#13202C211yesmaximal, characteristic
10#13148C211yesmaximal, characteristic, complemented
11#13104C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic, complemented
13#13013C211yesmaximal, characteristic, complemented
14#13013C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#493C2211yescharacteristic
20C2×#477—21no—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23#2567—21no—
24#2445—21no—
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2095—21no—
36#2095—21no—
37#2054—21no—
38#2041—21no—
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1942C2211yescharacteristic
45#1879—21no—
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
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×C4⋊Q8—21no—
64C2×#127—21no—
65C2×#123—41no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32—22no—
71C2×SD32—41no—
72C2×SD32—41no—
73C2×SD32—41no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#500—41no—
79#500—41no—
80#493—22no—
81#493—41no—
82#477—44no—
83C4×Q16—41no—
84C4×Q16—41no—
85C4×D8—22no—
86C4×D8—41no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#395—41no—
107#395—41no—
108#394—41no—
109#394—41no—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#389—21no—
116#389—41no—
117#389—41no—
118#386—41no—
119#384—21no—
120#384—21no—
121C4×C4⋊C4D411yescharacteristic
122C4⋊C16—41no—
123#358—22no—
124#358—41no—
125#357—41no—
126#357—41no—
127#357—41no—
128#357—41no—
129#357—41no—
130#357—41no—
131C4×C16—41no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—41no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—41no—
140C22×D4—41no—
141C23×C4—41no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16#4112yes—
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—21no—
159C2×D8—42no—
160C2×D8—42no—
161C2×D8—81no—
162C2×D8#4112yes—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2411yescharacteristic
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169#127—42no—
170#127—81no—
171#123—84no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×D4—42no—
175C4×D4—81no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4D811yescharacteristic
189C2×C4⋊C4D811yescharacteristic
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201SD32—44no—
202SD32—44no—
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—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—42no—
226#107—42no—
227#107—42no—
228C4⋊C8—22no—
229#103—42no—
230#103—42no—
231#103—42no—
232#102—42no—
233#102—81no—
234#102—81no—
235#102—81no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95D811yescharacteristic
248#95D811yescharacteristic
249C24—41no—
250C24—41no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—82no—
258C2×Q8—161no—
259C2×Q8—161no—
260C2×D4—22no—
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—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—161no—
277C2×D4—161no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—161no—
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C22×D411yescharacteristic
294Q16—22no—
295Q16—22no—
296Q16—41no—
297Q16—41no—
298D8—22no—
299D8—22no—
300D8—41no—
301D8—41no—
302D8—41no—
303D8—41no—
304D8—41no—
305D8—41no—
306D8—84no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8AGL(1,ℤ/8)12yes—
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8H(5,𝔽2)11yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C2×C82−1+411yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—161no—
337C4⋊C4—161no—
338C4⋊C4—161no—
339C4⋊C4—161no—
340C4⋊C4SD3214yescomplemented
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C22⋊C4—161no—
347C22⋊C4—161no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—82no—
355C16—41no—
356C16—41no—
357C16—41no—
358C16—41no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23C22×D811yescharacteristic
374Q8—42no—
375Q8—42no—
376Q8—81no—
377Q8—81no—
378Q8—164no—
379D4—42no—
380D4—42no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—164no—
388D4—164no—
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—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
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—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—164no—
422C2×C4C2×SD3212yes—
423C2×C4C2×SD3212yes—
424C2×C4C2×SD3212yes—
425C2×C4#51012yes—
426C2×C4#53511yescharacteristic
427C2×C4#54611yescharacteristic
428C2×C4#54811yescharacteristic
429C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C8—22no—
433C8—41no—
434C8—41no—
435C8#53811yescharacteristic
436C8#53911yescharacteristic
437C8#55111yescharacteristic
438C8#56211yescharacteristic
439C22—82no—
440C22—82no—
441C22—82no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22#320511yescharacteristic
454C22#329411yescharacteristic
455C22#335811yescharacteristic
456C22C22×SD3211yescharacteristic
457C22C2×#51011yescharacteristic
458C22#341511yescharacteristic
459C22#341511yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—41no—
465C4—41no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#269611yescharacteristic
473C4#322111yescharacteristic
474C4#330611yescharacteristic
475C4#337311yescharacteristic
476C2—162no—
477C2—321no—
478C2#3163411yescharacteristic
479C2#3377211yescharacteristic
480C2#3378611yescharacteristic
481C2#3386211yescharacteristic
482C2#3388111yescharacteristic
483C2#3392911yescharacteristic
484C2#3394211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
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. #28531C2semidirect
3. #13538C2semidirect
4. #13480C2semidirect
5. #13449C2semidirect
6. #13363C2semidirect
7. #13363C2semidirect
10. #13148C2semidirect
11. #13104C2semidirect
12. #13014C2semidirect
13. #13013C2semidirect
14. #13013C2semidirect
15. #12811C2semidirect
340. C4⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
#285312
#262542
#135382
#134802
#134492
#133632
#133282
#132022
#131482
#131042
#130142
#130132
#128112