Finite Group Databasebeta

← #330828

Subgroups — #330828

Order 512.

523 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10656C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10625C211yesmaximal, characteristic
10#10625C211yesmaximal, characteristic
11#10622C211yesmaximal, characteristic
12#10621C211yesmaximal, characteristic
13#9898C211yesmaximal, characteristic, complemented
14#9897C211yesmaximal, characteristic, complemented
15#8786C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2293C2211yescharacteristic
20#2099—21no—
21#2093—21no—
22#2088—21no—
23#2088—21no—
24#2065—21no—
25#2065—21no—
26#2057—21no—
27#2023—21no—
28#1993—21no—
29#1993—21no—
30#1992—21no—
31#1992—21no—
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1924—21no—
41#1924—21no—
42#1842C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
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#433—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97D411yescharacteristic
97C2×#97D411yescharacteristic
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#390D411yescharacteristic
104#390D411yescharacteristic
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389D411yescharacteristic
110#389D411yescharacteristic
111#388—21no—
112#388—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121C4×C4⋊C4C2311yescharacteristic
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#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145C22×Q8—41no—
146C22×D4C2×D411yescharacteristic
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
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—81no—
161C4⋊Q8—81no—
162C4⋊Q8—82no—
163#125—42no—
164#125—42no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169#117—42no—
170#117—42no—
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—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C42—21no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206#107—81no—
207#107—81no—
208#106—81no—
209#106—81no—
210C4⋊C8—21no—
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—42no—
223C4⋊C8—42no—
224C4⋊C8—81no—
225C4⋊C8—81no—
226#102—81no—
227#102—81no—
228#102—81no—
229#102—81no—
230C22⋊C8—42no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233C22⋊C8—42no—
234#97—22no—
235#97—22no—
236#97—41no—
237#97—41no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—42no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—41no—
247#95—41no—
248C24—21no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
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#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—21no—
314C4⋊C4—42no—
315C4⋊C4—42no—
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—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—81no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42#13512yes—
358C42AGL(1,ℤ/8)12yes—
359C422−1+411yescharacteristic
360C422−1+411yescharacteristic
361C422−1+411yescharacteristic
362C422−1+411yescharacteristic
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C2×#12111yescharacteristic
371Q8—164no—
372D4—41no—
373D4—41no—
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—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
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#46312yes—
414C2×C4#46312yes—
415C2×C4#53611yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#53611yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#54111yescharacteristic
420C2×C4#54111yescharacteristic
421C2×C4C2×#13511yescharacteristic
422C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57511yescharacteristic
425C2×C4#57611yescharacteristic
426C2×C4#57711yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57811yescharacteristic
429C8—41no—
430C8—41no—
431C8—41no—
432C8—41no—
433C8—81no—
434C8—81no—
435C8—81no—
436C8—81no—
437C8—81no—
438C8—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#262111yescharacteristic
447C22C2×#46311yescharacteristic
448C22#304011yescharacteristic
449C22#306211yescharacteristic
450C22#306311yescharacteristic
451C22#306811yescharacteristic
452C22#306911yescharacteristic
453C4—22no—
454C4—22no—
455C4—21no—
456C4—21no—
457C4—41no—
458C4—41no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4#323311yescharacteristic
467C4#323311yescharacteristic
468C4#323511yescharacteristic
469C4#323511yescharacteristic
470C4#324111yescharacteristic
471C4#324211yescharacteristic
472C4#324311yescharacteristic
473C4#324411yescharacteristic
474C2—161no—
475C2#1923411yescharacteristic
476C2#1923511yescharacteristic
477C2#1938211yescharacteristic
478C2#1938311yescharacteristic
479C2#2986511yescharacteristic
480C2#2988511yescharacteristic
481C2#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. #33297C2semidirect
5. #10658C2semidirect
6. #10657C2semidirect
7. #10656C2semidirect
8. #10655C2semidirect
13. #9898C2semidirect
14. #9897C2semidirect
15. #8786C2semidirect

Maximal subgroups

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

subgroupindex
#332972
#299982
#299802
#212682
#106582
#106572
#106562
#106552
#106252
#106222
#106212
#98982
#98972
#87862