Finite Group Databasebeta

← #330827

Subgroups — #330827

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10653C211yesmaximal, characteristic, complemented
7#10650C211yesmaximal, characteristic, complemented
8#10649C211yesmaximal, characteristic, complemented
9#10619C211yesmaximal, characteristic
10#10619C211yesmaximal, characteristic
11#10616C211yesmaximal, characteristic
12#10615C211yesmaximal, characteristic
13#9899C211yesmaximal, characteristic, complemented
14#9898C211yesmaximal, characteristic, complemented
15#8786C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#501—21no—
18C2×#433—21no—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21#2293C2211yescharacteristic
22#2099—21no—
23#2093—21no—
24#2086—21no—
25#2086—21no—
26#2067—21no—
27#2067—21no—
28#2057—21no—
29#2023—21no—
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1924—21no—
39#1924—21no—
40#1847—21no—
41#1847—21no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1750—21no—
54C4×#97C2211yescharacteristic
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×#125—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#501—44no—
74#501—44no—
75#433—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390D411yescharacteristic
114#390D411yescharacteristic
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389D411yescharacteristic
120#389D411yescharacteristic
121#388—21no—
122#388—21no—
123#384—21no—
124#384—21no—
125#384—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#382—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147C22×Q8—41no—
148C22×D4C2×D411yescharacteristic
149C23×C4—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—81no—
163C4⋊Q8—81no—
164C4⋊Q8—82no—
165#125—42no—
166#125—42no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42—21no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#107—42no—
209#107—42no—
210#107—42no—
211#107—42no—
212#107—81no—
213#107—81no—
214#107—81no—
215#107—81no—
216#106—42no—
217#106—42no—
218#106—42no—
219#106—42no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236#102—81no—
237#102—81no—
238#102—81no—
239#102—81no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244#97—22no—
245#97—22no—
246#97—22no—
247#97D814yes—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—42no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258C24—21no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8C2×D812yes—
315C2×C8C2×D812yes—
316C2×C8#13712yes—
317C2×C8#13712yes—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42C2×D812yes—
368C42AGL(1,ℤ/8)12yes—
369C422−1+411yescharacteristic
370C422−1+411yescharacteristic
371C422−1+411yescharacteristic
372C422−1+411yescharacteristic
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23C2×#12111yescharacteristic
381Q8—164no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—41no—
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—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
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—81no—
422C2×C4—81no—
423C2×C4#45912yes—
424C2×C4#45912yes—
425C2×C4#53611yescharacteristic
426C2×C4#53611yescharacteristic
427C2×C4#53611yescharacteristic
428C2×C4#53611yescharacteristic
429C2×C4#54111yescharacteristic
430C2×C4#54111yescharacteristic
431C2×C4C22×D811yescharacteristic
432C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57711yescharacteristic
438C2×C4#57811yescharacteristic
439C8—22no—
440C8—22no—
441C8—22no—
442C8—22no—
443C8—81no—
444C8—81no—
445C8—81no—
446C8—81no—
447C8—81no—
448C8—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#262111yescharacteristic
457C22C2×#45911yescharacteristic
458C22#303911yescharacteristic
459C22#306211yescharacteristic
460C22#306311yescharacteristic
461C22#306811yescharacteristic
462C22#306911yescharacteristic
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—41no—
468C4—41no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4#322611yescharacteristic
477C4#322611yescharacteristic
478C4#323911yescharacteristic
479C4#323911yescharacteristic
480C4#324111yescharacteristic
481C4#324211yescharacteristic
482C4#324311yescharacteristic
483C4#324411yescharacteristic
484C2—161no—
485C2#1923011yescharacteristic
486C2#1923111yescharacteristic
487C2#1937811yescharacteristic
488C2#1937911yescharacteristic
489C2#2986411yescharacteristic
490C2#2988511yescharacteristic
491C2#2988911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33316C2semidirect
5. #10653C2semidirect
6. #10653C2semidirect
7. #10650C2semidirect
8. #10649C2semidirect
13. #9899C2semidirect
14. #9898C2semidirect
15. #8786C2semidirect

Maximal subgroups

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

subgroupindex
#333162
#299942
#299802
#212682
#106532
#106502
#106492
#106192
#106162
#106152
#98992
#98982
#87862