Finite Group Databasebeta

← #330847

Subgroups — #330847

Order 512.

521 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33072C211yesmaximal, characteristic
2#33066C211yesmaximal, characteristic
3#30307C211yesmaximal, characteristic, complemented
4#22008C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10653C211yesmaximal, characteristic, complemented
8#10649C211yesmaximal, characteristic, complemented
9#10625C211yesmaximal, characteristic
10#10622C211yesmaximal, characteristic
11#10619C211yesmaximal, characteristic
12#10615C211yesmaximal, characteristic
13#9901C211yesmaximal, characteristic, complemented
14#9897C211yesmaximal, characteristic, complemented
15#8788C211yesmaximal, characteristic, complemented
16#2859C2211yescharacteristic
17#2859C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#2085—21no—
20#2085—21no—
21#2059—21no—
22#2059—21no—
23#1954C2211yescharacteristic
24#1954C2211yescharacteristic
25#1948C2211yescharacteristic
26#1947C2211yescharacteristic
27#1947C2211yescharacteristic
28#1947C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1923—21no—
32#1923—21no—
33#1921—21no—
34#1919—21no—
35#1918—21no—
36#1918—21no—
37#1864—21no—
38#1862—21no—
39#1862—21no—
40#1862—21no—
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1817—21no—
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1797—21no—
53C4×C4⋊C8C2211yescharacteristic
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—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—21no—
113#382—21no—
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C43C2311yescharacteristic
121#340—21no—
122#340—21no—
123#340—21no—
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#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336D411yescharacteristic
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144C22×D4C2×D411yescharacteristic
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158#125—81no—
159#125—81no—
160#125—82no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—21no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—41no—
168C4×D4—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
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—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#107—42no—
204#107—42no—
205#107—42no—
206#107—81no—
207#107—81no—
208#106—42no—
209#106—81no—
210#106—81no—
211C4⋊C8—21no—
212C4⋊C8—21no—
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—81no—
224C4⋊C8—81no—
225#102—42no—
226#102—42no—
227#102—42no—
228#102—42no—
229C22⋊C8—81no—
230C22⋊C8—81no—
231C22⋊C8—81no—
232C22⋊C8—81no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—81no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245C24—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4#13512yes—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
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—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
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—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
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—81no—
330C4⋊C4—81no—
331C4⋊C4—81no—
332C4⋊C4—81no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4#13512yes—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—81no—
355C42—81no—
356C422−1+411yescharacteristic
357C422−1+411yescharacteristic
358C422−1+411yescharacteristic
359C422−1+411yescharacteristic
360C23—22no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23C2×#12111yescharacteristic
368D4—22no—
369D4—22no—
370D4—41no—
371D4—41no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—84no—
396C2×C4—84no—
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—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4#53611yescharacteristic
414C2×C4#53611yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#54111yescharacteristic
418C2×C4#54111yescharacteristic
419C2×C4C2×#13511yescharacteristic
420C2×C4C2×#13511yescharacteristic
421C2×C4#57511yescharacteristic
422C2×C4#57511yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57611yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C8—41no—
428C8—41no—
429C8—41no—
430C8—41no—
431C8—81no—
432C8—81no—
433C8—81no—
434C8—81no—
435C8—81no—
436C8—81no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22#262111yescharacteristic
445C22#304111yescharacteristic
446C22#304611yescharacteristic
447C22#305011yescharacteristic
448C22#305411yescharacteristic
449C22#306811yescharacteristic
450C22#306911yescharacteristic
451C4—21no—
452C4—21no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—81no—
459C4—81no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4#323211yescharacteristic
465C4#323211yescharacteristic
466C4#323311yescharacteristic
467C4#323311yescharacteristic
468C4#324111yescharacteristic
469C4#324111yescharacteristic
470C4#324311yescharacteristic
471C4#324311yescharacteristic
472C2—82no—
473C2#1938911yescharacteristic
474C2#1941811yescharacteristic
475C2#1942411yescharacteristic
476C2#1945511yescharacteristic
477C2#2987211yescharacteristic
478C2#2987911yescharacteristic
479C2#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
3. #30307C2semidirect
5. #10657C2semidirect
6. #10656C2semidirect
7. #10653C2semidirect
8. #10649C2semidirect
13. #9901C2semidirect
14. #9897C2semidirect
15. #8788C2semidirect

Maximal subgroups

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

subgroupindex
#330722
#330662
#303072
#220082
#106572
#106562
#106532
#106492
#106252
#106222
#106192
#106152
#99012
#98972
#87882