Finite Group Databasebeta

← #338929

Subgroups — #338929

Order 512.

513 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28792C211yesmaximal, characteristic, complemented
2#27748C211yesmaximal, characteristic
3#26301C211yesmaximal, characteristic
4#26090C211yesmaximal, characteristic
5#10686C211yesmaximal, characteristic, complemented
6#10625C211yesmaximal, characteristic
7#10588C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#10416C211yesmaximal, characteristic
10#10357C211yesmaximal, characteristic, complemented
11#9611C211yesmaximal, characteristic, complemented
12#9565C211yesmaximal, characteristic, complemented
13#9542C211yesmaximal, characteristic
14#9534C211yesmaximal, characteristic, complemented
15#8897C211yesmaximal, characteristic
16#2571C2211yescharacteristic
17#2552C2211yescharacteristic
18#2547C2211yescharacteristic
19#2354C2211yescharacteristic
20#2094—21no—
21#2085—21no—
22#2077—21no—
23#2076—21no—
24#2074—21no—
25#2065—21no—
26#2059—21no—
27#2042—21no—
28#1991—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1988C2211yescharacteristic
32#1981C2211yescharacteristic
33#1958—21no—
34#1956C2211yescharacteristic
35#1954C2211yescharacteristic
36#1948—21no—
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1942—21no—
42#1939C2211yescharacteristic
43#1933—21no—
44#1920—21no—
45#1854—21no—
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1825—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×#124—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106D411yescharacteristic
79C2×#106D411yescharacteristic
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102D411yescharacteristic
92C2×#102D411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—41no—
101#398—41no—
102#397—41no—
103#391—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#388—41no—
110#384—21no—
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#382—21no—
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
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#340—21no—
134#340—21no—
135#340—41no—
136#340—41no—
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#334—22no—
145#332—22no—
146D4⋊C8—22no—
147C22×Q8—21no—
148C22×D4C2×D411yescharacteristic
149C23×C4—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
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163#125—42no—
164#125—81no—
165#124—42no—
166#124—81no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171#117—42no—
172#117—81no—
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—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—41no—
204C2×C42C2411yescharacteristic
205#107—42no—
206#107—81no—
207#107—81no—
208#106—22no—
209#106—41no—
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—42no—
221C4⋊C8#4112yes—
222#103—42no—
223#103—81no—
224#103—81no—
225#103—81no—
226#102—22no—
227#102—41no—
228#102—81no—
229#102—81no—
230C22⋊C8—42no—
231C22⋊C8—81no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—81no—
239C4×C8—81no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24—21no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
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×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
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#13512yes—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
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—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—161no—
345C4⋊C4—161no—
346C4⋊C4#13512yes—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—161no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—82no—
360C42C2×#4111yescharacteristic
361C422−1+411yescharacteristic
362C422−1+411yescharacteristic
363C422−1+411yescharacteristic
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23D4211yescharacteristic
372Q8—81no—
373Q8—81no—
374Q8—81no—
375Q8—81no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
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—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
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—81no—
418C2×C4—81no—
419C2×C4#53611yescharacteristic
420C2×C4#53611yescharacteristic
421C2×C4#54211yescharacteristic
422C2×C4#54711yescharacteristic
423C2×C4#54811yescharacteristic
424C2×C4#56311yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4#57511yescharacteristic
428C2×C4#57511yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57711yescharacteristic
433C8—22no—
434C8—41no—
435C8—42no—
436C8—41no—
437C8—41no—
438C8—81no—
439C8—81no—
440C8—81no—
441C8—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22#267411yescharacteristic
450C22#326711yescharacteristic
451C22#326811yescharacteristic
452C22#328211yescharacteristic
453C22#328411yescharacteristic
454C22#328911yescharacteristic
455C22#329011yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4#323311yescharacteristic
470C4#323311yescharacteristic
471C4#324111yescharacteristic
472C4#324411yescharacteristic
473C4#333911yescharacteristic
474C4#335111yescharacteristic
475C4#336011yescharacteristic
476C4#337111yescharacteristic
477C2—161no—
478C2#1983111yescharacteristic
479C2#1987811yescharacteristic
480C2#1999911yescharacteristic
481C2#2013711yescharacteristic
482C2#3078811yescharacteristic
483C2#3081711yescharacteristic
484C2#3084011yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28792C2semidirect
5. #10686C2semidirect
7. #10588C2semidirect
8. #10578C2semidirect
10. #10357C2semidirect
11. #9611C2semidirect
12. #9565C2semidirect
14. #9534C2semidirect

Maximal subgroups

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

subgroupindex
#287922
#277482
#263012
#260902
#106862
#106252
#105882
#105782
#104162
#103572
#96112
#95652
#95422
#95342
#88972