Finite Group Databasebeta

← #330887

Subgroups — #330887

Order 512.

529 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33199C211yesmaximal, characteristic
2#33066C211yesmaximal, characteristic
3#30307C211yesmaximal, characteristic, complemented
4#22012C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10653C211yesmaximal, characteristic, complemented
8#10650C211yesmaximal, characteristic, complemented
9#10625C211yesmaximal, characteristic
10#10621C211yesmaximal, characteristic
11#10619C211yesmaximal, characteristic
12#10615C211yesmaximal, characteristic
13#9907C211yesmaximal, characteristic, complemented
14#9902C211yesmaximal, characteristic, complemented
15#8789C211yesmaximal, characteristic, complemented
16C2×#478—21no—
17#2859C2211yescharacteristic
18#2859C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#2085—21no—
21#2085—21no—
22#2059—21no—
23#2059—21no—
24#1989—21no—
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1949—21no—
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1924—21no—
35#1924—21no—
36#1918—21no—
37#1918—21no—
38#1862—21no—
39#1862—21no—
40#1845C2211yescharacteristic
41#1844C2211yescharacteristic
42#1842C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1831C2211yescharacteristic
46#1817—21no—
47#1804—21no—
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1797—21no—
53#1767C2211yescharacteristic
54#1744C2211yescharacteristic
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×#125—21no—
68C2×#125—21no—
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C22×C42—21no—
73#478—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#382—21no—
116#382—21no—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C43C2311yescharacteristic
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×D4C2×D411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159#125—42no—
160#125—42no—
161#125—81no—
162#125—81no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—41no—
170C4×D4—41no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—81no—
197C2×C42—81no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42C2411yescharacteristic
205#107—42no—
206#107—42no—
207#107—42no—
208#107—81no—
209#107—81no—
210#106—42no—
211#106—81no—
212#106—81no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229#102—42no—
230#102—42no—
231#102—42no—
232#102—42no—
233C22⋊C8—81no—
234C22⋊C8—81no—
235C22⋊C8—81no—
236C22⋊C8—81no—
237#97—81no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249C24—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4AGL(1,ℤ/8)12yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
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—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
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—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4#13712yes—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—81no—
359C42—81no—
360C422−1+411yescharacteristic
361C422−1+411yescharacteristic
362C422−1+411yescharacteristic
363C422−1+411yescharacteristic
364C23—22no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23C2×#12111yescharacteristic
372D4—22no—
373D4—22no—
374D4—41no—
375D4—41no—
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—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—84no—
400C2×C4—84no—
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#53611yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#53611yescharacteristic
420C2×C4#53611yescharacteristic
421C2×C4#54111yescharacteristic
422C2×C4#54111yescharacteristic
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4#57511yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57811yescharacteristic
431C8—41no—
432C8—41no—
433C8—41no—
434C8—41no—
435C8—81no—
436C8—81no—
437C8—81no—
438C8—81no—
439C8—81no—
440C8—81no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#262111yescharacteristic
449C22#304111yescharacteristic
450C22#304611yescharacteristic
451C22#305511yescharacteristic
452C22#305611yescharacteristic
453C22#306711yescharacteristic
454C22#306711yescharacteristic
455C4—21no—
456C4—21no—
457C4—41no—
458C4—42no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—81no—
463C4—81no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4#323411yescharacteristic
469C4#323711yescharacteristic
470C4#323811yescharacteristic
471C4#324011yescharacteristic
472C4#324111yescharacteristic
473C4#324111yescharacteristic
474C4#324311yescharacteristic
475C4#324311yescharacteristic
476C2—82no—
477C2#1946011yescharacteristic
478C2#1946111yescharacteristic
479C2#1948611yescharacteristic
480C2#1949911yescharacteristic
481C2#2987211yescharacteristic
482C2#2988311yescharacteristic
483C2#2988811yescharacteristic
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
3. #30307C2semidirect
5. #10657C2semidirect
6. #10656C2semidirect
7. #10653C2semidirect
8. #10650C2semidirect
13. #9907C2semidirect
14. #9902C2semidirect
15. #8789C2semidirect

Maximal subgroups

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

subgroupindex
#331992
#330662
#303072
#220122
#106572
#106562
#106532
#106502
#106252
#106212
#106192
#106152
#99072
#99022
#87892