Finite Group Databasebeta

← #499459

Subgroups — #499459

Order 512.

632 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2176C211yesmaximal, characteristic, complemented
2C2×#2175C211yesmaximal, characteristic
3C4×#495C211yesmaximal, characteristic
4#21276C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13483C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic
8#13475C211yesmaximal, characteristic, complemented
9#13450C211yesmaximal, characteristic, complemented
10#13449C211yesmaximal, characteristic
11#13408C211yesmaximal, characteristic
12#13329C211yesmaximal, characteristic
13#12653C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×SD16—21no—
22C2×C4×D8C2211yescharacteristic
23C4×#124—21no—
24#2295—21no—
25#2176—24no—
26#2175—24no—
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1968—21no—
46#1958—21no—
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1933—21no—
50#1933—21no—
51#1846C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#103—21no—
54C42×C8C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8C2311yescharacteristic
65C2×#124—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×Q32—22no—
70C2×Q32—22no—
71C2×Q32—22no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32—22no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#495—22no—
80#495—22no—
81C4×Q16—22no—
82C4×Q16—22no—
83C4×Q16—22no—
84C4×Q16D414yes—
85C4×SD16—44no—
86C4×D8—22no—
87C4×D8—22no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#391—21no—
109#391—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#388—41no—
114#388—41no—
115#387—41no—
116#387—41no—
117#382—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—41no—
120C4×C22⋊C4—41no—
121C4×C22⋊C4—41no—
122#376—41no—
123C43D411yescharacteristic
124#367—22no—
125#366—22no—
126C4⋊C16—22no—
127C4⋊C16—22no—
128#358—22no—
129#358—22no—
130#358—22no—
131#358—22no—
132#358—22no—
133#357—22no—
134#357—22no—
135#357—22no—
136#346—22no—
137#346—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×D4—21no—
144C23×C4—41no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16C2×D412yes—
156C2×Q16#4112yes—
157C2×SD16—24no—
158C2×SD16—42no—
159C2×SD16—42no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8#4112yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170#124—24no—
171#124—42no—
172C4×Q8—24no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×D4—42no—
177C4×D4—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42D811yescharacteristic
196C2×C42D811yescharacteristic
197C2×C42D811yescharacteristic
198C2×C42D811yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202Q32—28no—
203SD32—28no—
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#4112yes—
215C2×C16#4112yes—
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#107C2×D412yes—
226#106—42no—
227#103—24no—
228#103—42no—
229#103—42no—
230#103—42no—
231#103—42no—
232#102—42no—
233#102—42no—
234C4×C8—22no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—22no—
240C4×C8—41no—
241C4×C8C2×D412yes—
242C4×C8#4112yes—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254C24—41no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C22×D411yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—22no—
293Q16—22no—
294Q16—41no—
295Q16—41no—
296Q16—41no—
297Q16—41no—
298SD16—48no—
299D8—22no—
300D8—22no—
301D8—41no—
302D8—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8#12112yes—
311C2×C8#13512yes—
312C2×C8C22×D411yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—44no—
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—82no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—22no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—44no—
346C42—82no—
347C42—82no—
348C42#12112yes—
349C42AGL(1,ℤ/8)12yes—
350C16—22no—
351C16—22no—
352C16—41no—
353C16—41no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23C22×D811yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—42no—
365Q8—42no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369Q8—81no—
370D4—42no—
371D4—42no—
372D4—81no—
373D4—81no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
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—164no—
411C2×C4#45912yes—
412C2×C4#45912yes—
413C2×C4#51012yes—
414C2×C4C2×#12111yescharacteristic
415C2×C4#52511yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
419C2×C4#57511yescharacteristic
420C8—22no—
421C8—22no—
422C8—41no—
423C8#52511yescharacteristic
424C8#52511yescharacteristic
425C8#52911yescharacteristic
426C8#52911yescharacteristic
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22C2×#45911yescharacteristic
435C22#310011yescharacteristic
436C22#316911yescharacteristic
437C22C2×#51011yescharacteristic
438C22#341411yescharacteristic
439C22#341411yescharacteristic
440C22#341411yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4#239211yescharacteristic
455C4#303511yescharacteristic
456C4#310511yescharacteristic
457C4#317111yescharacteristic
458C2—162no—
459C2#2367611yescharacteristic
460C2#3358111yescharacteristic
461C2#3358411yescharacteristic
462C2#3367111yescharacteristic
463C2#3367611yescharacteristic
464C2#3374711yescharacteristic
465C2#3375011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2176C2semidirect
4. #21276C2semidirect
5. #13507C2semidirect
6. #13483C2semidirect
8. #13475C2semidirect
9. #13450C2semidirect
13. #12653C2semidirect
14. #12638C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21762
C2×#21752
C4×#4952
#212762
#135072
#134832
#134762
#134752
#134502
#134492
#134082
#133292
#126532
#126382
#126372