Finite Group Databasebeta

← #267249

Subgroups — #267249

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2#32957C211yesmaximal, characteristic
3#25813C211yesmaximal, characteristic
4#22510C211yesmaximal, characteristic
5C2×#1696C212yesmaximal, complemented
6#10482C212yesmaximal, complemented
7#10279C212yesmaximal, complemented
8#9224C212yesmaximal, complemented
9#9127C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8342C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#500C2212yes—
15C2×#477—28no—
16C2×#477—21no—
17C2×#477—21no—
18C2×#477C2212yes—
19C2×#477C2211yescharacteristic
20C2×#477C2212yes—
21C2×#477C2211yescharacteristic
22C2×#446C2212yes—
23C22×#107—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103—21no—
26#2857C2211yescharacteristic
27#2057—22no—
28#1990C2212yes—
29#1985—22no—
30#1984C2212yes—
31#1863—21no—
32#1863—21no—
33#1852—21no—
34#1852—21no—
35#1852—21no—
36#1803C2211yescharacteristic
37#1803C2211yescharacteristic
38#1797—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1787—21no—
42#1749—21no—
43C4×C22⋊C8C2211yescharacteristic
44#1696—28no—
45C2×#334C2212yes—
46C2×#333C2212yes—
47C2×#333C2212yes—
48C2×#333C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51C23×Q8—21no—
52C23×Q8D411yescharacteristic
53C23×C8—21no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#500—24no—
77#500—42no—
78#477—28no—
79#477—28no—
80#477—24no—
81#477—24no—
82#477—24no—
83#477—24no—
84#477—42no—
85#477D418yes—
86#446—24no—
87#446—42no—
88C2×#107—28no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—21no—
94C2×#107—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×#103—28no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8D411yescharacteristic
112C2×C22⋊C8D411yescharacteristic
113C2×#97—22no—
114C2×C4×C8C2312yes—
115#391—21no—
116#389—21no—
117#389—21no—
118#384C2311yescharacteristic
119#380—21no—
120#380—21no—
121#340—22no—
122#340—22no—
123#340—21no—
124#340—21no—
125#336—22no—
126#336—22no—
127#336—21no—
128#336—21no—
129#336D412yes—
130#334—24no—
131#333—24no—
132#333—24no—
133#333—42no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—41no—
142C22×Q8—41no—
143C22×Q8—44no—
144C22×Q8—41no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×Q8C2×D411yescharacteristic
148C22×Q8C2×D411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C4⋊Q8—24no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4⋊Q8—42no—
173C4⋊Q8C2×D414yes—
174C22⋊Q8—82no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—44no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—44no—
184C4×Q8C2×D414yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—44no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—28no—
221#107—28no—
222#107—44no—
223#107—44no—
224#107—42no—
225#107—42no—
226C4⋊C8—24no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—44no—
233C4⋊C8C2×D414yes—
234#103—28no—
235#103—28no—
236#103—42no—
237#103—42no—
238#103—42no—
239#103—42no—
240C22⋊C8—22no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243C22⋊C8Q1614yes—
244#97—44no—
245C4×C8—22no—
246C4×C8—22no—
247#95—41no—
248#95—41no—
249C24#12111yescharacteristic
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—28no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—24no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—82no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8—816no—
278C2×Q8#12012yes—
279C2×Q8#13712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—81no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4C2×Q1612yes—
309C22×C4#13712yes—
310C22×C4#13712yes—
311C22×C4#13712yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—48no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8C2×Q1614yes—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4#12012yes—
357C22⋊C4—82no—
358C42—21no—
359C42—21no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42C22×D412yes—
367C42H(5,𝔽2)12yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23#46212yes—
373C23#47012yes—
374C23#47212yes—
375C23C2×#12111yescharacteristic
376Q8—22no—
377Q8—22no—
378Q8—22no—
379Q8—22no—
380Q8—48no—
381Q8—41no—
382Q8—41no—
383Q8—41no—
384Q8—41no—
385Q8—84no—
386Q8—84no—
387Q8—88no—
388Q8—162no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#46212yes—
430C2×C4#47012yes—
431C2×C4C2×#12011yescharacteristic
432C2×C4#53411yescharacteristic
433C2×C4D4212yes—
434C2×C4#54612yes—
435C2×C4C22×Q1611yescharacteristic
436C2×C4C2×#13711yescharacteristic
437C2×C4C2×#13711yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57812yes—
442C8—24no—
443C8—44no—
444C8—42no—
445C8—81no—
446C8—81no—
447C8—82no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22#169614yescomplemented
455C22#241011yescharacteristic
456C22C2×#46211yescharacteristic
457C22#304411yescharacteristic
458C22C2×#47011yescharacteristic
459C22#305611yescharacteristic
460C22C2×#47211yescharacteristic
461C22#306811yescharacteristic
462C4—22no—
463C4—22no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—42no—
469C4—81no—
470C4—84no—
471C4—161no—
472C4—161no—
473C4#300411yescharacteristic
474C4#300711yescharacteristic
475C4#319511yescharacteristic
476C4#319611yescharacteristic
477C4#327012yes—
478C4D4×Q1612yes—
479C2—24no—
480C2C2×#169611yescharacteristic
481C2#1917311yescharacteristic
482C2#1944111yescharacteristic
483C2#1947711yescharacteristic
484C2#2436011yescharacteristic
485C2#2438611yescharacteristic
486C2#2439811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C2×#1696C2semidirect
6. #10482C2semidirect
7. #10279C2semidirect
8. #9224C2semidirect
454. C22#1696semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
#329572
#258132
#225102
C2×#16962
#104822
#102792
#92242
#91272
#88302
#83422