Finite Group Databasebeta

← #341479

Subgroups — #341479

Order 512.

557 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28791C211yesmaximal, characteristic, complemented
2#28790C211yesmaximal, characteristic, complemented
3#26529C211yesmaximal, characteristic
4#26460C211yesmaximal, characteristic
5#10678C211yesmaximal, characteristic
6#10658C211yesmaximal, characteristic, complemented
7#10656C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#10575C211yesmaximal, characteristic, complemented
10#10354C211yesmaximal, characteristic
11#9564C211yesmaximal, characteristic
12#9564C211yesmaximal, characteristic
13#9535C211yesmaximal, characteristic, complemented
14#9533C211yesmaximal, characteristic, complemented
15#8924C211yesmaximal, characteristic
16C2×#474—21no—
17#2571C2211yescharacteristic
18#2552C2211yescharacteristic
19#2551C2211yescharacteristic
20#2366C2211yescharacteristic
21#2092—21no—
22#2085—21no—
23#2084—21no—
24#2067—21no—
25#2067—21no—
26#2059—21no—
27#2058—21no—
28#2023—21no—
29#1991—21no—
30#1990—21no—
31#1982C2211yescharacteristic
32#1982C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1947C2211yescharacteristic
42#1941—21no—
43#1941—21no—
44#1939C2211yescharacteristic
45#1939C2211yescharacteristic
46#1864—21no—
47#1862—21no—
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1804—21no—
52C4×#107C2211yescharacteristic
53C4×#106C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56C2×#331C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×#127C2311yescharacteristic
70C2×#125—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73#474—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107D411yescharacteristic
77C2×#107D411yescharacteristic
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106D411yescharacteristic
83C2×#106D411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—41no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#398—41no—
106#395—41no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#386—41no—
116#384—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#382—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
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—41no—
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#331Q814yescomplemented
146C22×Q8—41no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C23×C4—41no—
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—
164#127—21no—
165#127—21no—
166#127—21no—
167#127C2×Q812yes—
168#125—42no—
169#125—81no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—81no—
173C4×D4—81no—
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⋊C4—41no—
188C2×C4⋊C4—41no—
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×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C2411yescharacteristic
205#107—22no—
206#107—22no—
207#107—42no—
208#107—81no—
209#106—22no—
210#106—41no—
211#106—42no—
212#106—42no—
213#106—81no—
214#106—81no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—42no—
222C4⋊C8—81no—
223C4⋊C8C2×Q812yes—
224C4⋊C8C2×Q812yes—
225#102—42no—
226#102—42no—
227#102—42no—
228#102—42no—
229#102—81no—
230#102—81no—
231#102—81no—
232#102—81no—
233#102—82no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—81no—
241C4×C8—81no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—41no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×Q8—161no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—161no—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
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×C8—41no—
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—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
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—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—161no—
348C4⋊C4—161no—
349C4⋊C4#13512yes—
350C4⋊C4#13512yes—
351C4⋊C4AGL(1,ℤ/8)12yes—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—161no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42C22×Q811yescharacteristic
364C422−1+411yescharacteristic
365C422−1+411yescharacteristic
366C422−1+411yescharacteristic
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23D4211yescharacteristic
375Q8—164no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—42no—
380D4—82no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—161no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
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—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
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—164no—
422C2×C4#53611yescharacteristic
423C2×C4#53611yescharacteristic
424C2×C4#54411yescharacteristic
425C2×C4D4×Q811yescharacteristic
426C2×C4D4×Q811yescharacteristic
427C2×C4#56311yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
431C2×C4#57511yescharacteristic
432C2×C4#57611yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57711yescharacteristic
436C8—42no—
437C8—42no—
438C8—41no—
439C8—41no—
440C8—41no—
441C8—41no—
442C8—81no—
443C8—81no—
444C8—81no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22#267711yescharacteristic
453C22#326511yescharacteristic
454C22#327111yescharacteristic
455C22#328011yescharacteristic
456C22#328211yescharacteristic
457C22#328811yescharacteristic
458C22#329011yescharacteristic
459C4—21no—
460C4—21no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—81no—
468C4—81no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#322811yescharacteristic
473C4#323311yescharacteristic
474C4#324111yescharacteristic
475C4#324411yescharacteristic
476C4#338111yescharacteristic
477C4#338111yescharacteristic
478C4#338411yescharacteristic
479C4#338911yescharacteristic
480C2—162no—
481C2#1977311yescharacteristic
482C2#1980211yescharacteristic
483C2#1999111yescharacteristic
484C2#2009111yescharacteristic
485C2#3094711yescharacteristic
486C2#3095011yescharacteristic
487C2#3098111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #28791C2semidirect
2. #28790C2semidirect
6. #10658C2semidirect
7. #10656C2semidirect
8. #10578C2semidirect
9. #10575C2semidirect
13. #9535C2semidirect
14. #9533C2semidirect
145. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#287912
#287902
#265292
#264602
#106782
#106582
#106562
#105782
#105752
#103542
#95642
#95352
#95332
#89242