Finite Group Databasebeta

← #241318

Subgroups — #241318

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37623C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#21987C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#10582C212yesmaximal, complemented
6#10581C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10575C212yesmaximal, complemented
9#8040C211yesmaximal, characteristic
10#7898C211yesmaximal, characteristic
11#7898C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2843—22no—
16#2660—21no—
17#2660—21no—
18#2647—21no—
19#2647—21no—
20#2627—22no—
21#2571C2211yescharacteristic
22#2552C2211yescharacteristic
23#2552C2211yescharacteristic
24#2551C2211yescharacteristic
25#2442—21no—
26#2442—21no—
27#2440—21no—
28#2440—21no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#2283C2211yescharacteristic
32#2283C2211yescharacteristic
33#1948C2212yes—
34#1948C2212yes—
35#1947C2212yes—
36#1947C2212yes—
37#1853—21no—
38#1853—21no—
39#1852—21no—
40#1852—21no—
41#1847—21no—
42#1847—21no—
43#1836—21no—
44#1836—21no—
45#1834—22no—
46#1834C2212yes—
47#1834C2212yes—
48C2×#340—22no—
49#1296C2212yes—
50#1296C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C23×C8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×#125—21no—
59C2×#125—21no—
60C2×C22⋊Q8—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C42C2311yescharacteristic
68#432—22no—
69#432—22no—
70#432Q814yescomplemented
71#432Q814yescomplemented
72C2×#107—22no—
73C2×#107—22no—
74C2×#106—22no—
75C2×#106—22no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×#97—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—22no—
84C2×C4×C8Q814yescomplemented
85C2×C4×C8C2312yes—
86#400—22no—
87#400—22no—
88#400—22no—
89#400—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#398—22no—
97#398—22no—
98#395—22no—
99#395—22no—
100#395—22no—
101#395—22no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#385—22no—
111#385—22no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384D412yes—
120#384C2312yes—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#383—21no—
124#383—21no—
125#382—21no—
126#382—21no—
127#382D411yescharacteristic
128#382D411yescharacteristic
129#382D411yescharacteristic
130#382D411yescharacteristic
131#381—21no—
132#381—21no—
133#380—22no—
134#380D411yescharacteristic
135#380D411yescharacteristic
136#380D411yescharacteristic
137#380D411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144#376—21no—
145#376—21no—
146#340—24no—
147#340—24no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152C22×Q8—22no—
153C23×C4—22no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C22×C8—24no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—42no—
163C22×C8—42no—
164C4⋊Q8—82no—
165C4⋊Q8—82no—
166#125—82no—
167#125—82no—
168C22⋊Q8—84no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42C2×Q812yes—
220C2×C42C2×Q812yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223#107—82no—
224#107—82no—
225#106—82no—
226#106—82no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8C2×Q814yes—
230C4⋊C8C2×Q814yes—
231C22⋊C8—24no—
232C22⋊C8—24no—
233#97—24no—
234#97—24no—
235C4×C8—24no—
236C4×C8—22no—
237C4×C8C2×Q814yes—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24#12011yescharacteristic
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—82no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q812yes—
292C22×C4C22⋊Q812yes—
293C22×C4C4⋊Q812yes—
294C22×C4C4⋊Q812yes—
295C22×C4C4⋊Q812yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—48no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42C22⋊Q812yes—
352C42C22⋊Q812yes—
353C42C22⋊Q812yes—
354C42C4⋊Q812yes—
355C42C4⋊Q812yes—
356C42C4⋊Q812yes—
357C42C22×Q811yescharacteristic
358C42C22×Q811yescharacteristic
359C42C22×Q811yescharacteristic
360C42C2×#4111yescharacteristic
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23#39512yes—
366C23#45412yes—
367C23#45512yes—
368C23C2×#12011yescharacteristic
369Q8—164no—
370C2×C4—24no—
371C2×C4—24no—
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—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
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—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—164no—
418C2×C4—164no—
419C2×C4#39512yes—
420C2×C4#39512yes—
421C2×C4#39512yes—
422C2×C4C2×C22⋊Q811yescharacteristic
423C2×C4C2×C22⋊Q811yescharacteristic
424C2×C4C2×C22⋊Q811yescharacteristic
425C2×C4C2×C4⋊Q811yescharacteristic
426C2×C4C2×C4⋊Q811yescharacteristic
427C2×C4C2×C4⋊Q811yescharacteristic
428C2×C4#57612yes—
429C2×C4#57712yes—
430C2×C4#57712yes—
431C2×C4#57812yes—
432C8—44no—
433C8—44no—
434C8—44no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22#206114yes—
442C22C2×#39511yescharacteristic
443C22#300811yescharacteristic
444C22#300911yescharacteristic
445C22#301011yescharacteristic
446C22#301111yescharacteristic
447C22C2×#45511yescharacteristic
448C22C2×#45411yescharacteristic
449C4—22no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—22no—
454C4—82no—
455C4—82no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—162no—
465C4#308312yes—
466C4#308412yes—
467C4#316312yes—
468C4#316412yes—
469C2—24no—
470C2#1868911yescharacteristic
471C2#1869011yescharacteristic
472C2#1869411yescharacteristic
473C2#1869511yescharacteristic
474C2#2355511yescharacteristic
475C2#2355611yescharacteristic
476C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #10582C2semidirect
6. #10581C2semidirect
7. #10578C2semidirect
8. #10575C2semidirect
70. #432Q8semidirect
71. #432Q8semidirect
84. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#376232
#219872
C2×#18342
#105822
#105812
#105782
#105752
#80402
#78982