Finite Group Databasebeta

← #315758

Subgroups — #315758

Order 512.

795 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#33174C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#27948C211yesmaximal, characteristic, complemented
4#27890C211yesmaximal, characteristic
5#27876C211yesmaximal, characteristic, complemented
6#20800C211yesmaximal, characteristic, complemented
7#10686C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10683C211yesmaximal, characteristic, complemented
10#10672C211yesmaximal, characteristic, complemented
11#9842C211yesmaximal, characteristic, complemented
12#9820C211yesmaximal, characteristic, complemented
13#8819C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#468—22no—
18C2×#428—22no—
19C2×C4×M16C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2084—22no—
25#2077—22no—
26#2071—22no—
27#2034—22no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1973—22no—
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1956C2211yescharacteristic
39#1926—22no—
40#1919—22no—
41#1874—22no—
42#1853—22no—
43#1847—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1834C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C22×M16D412yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×C22⋊Q8—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#492—22no—
77#492—22no—
78#468—48no—
79#428—48no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#103—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×C22⋊C8D412yes—
96C2×C22⋊C8D412yes—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97D412yes—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336D412yes—
137#336D412yes—
138#322—22no—
139C82—22no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×SD16—44no—
150C2×SD16—44no—
151C2×SD16—44no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16SD1618yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165#127—21no—
166#127—21no—
167#127—21no—
168#127—21no—
169C22⋊Q8—44no—
170#121—44no—
171#121—44no—
172#121—82no—
173#117—44no—
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—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—41no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2411yescharacteristic
215#107—44no—
216#107—44no—
217#107—44no—
218#106—44no—
219#106—44no—
220#106—44no—
221#106—82no—
222#106—82no—
223#106—82no—
224#103—44no—
225#102—44no—
226#102—44no—
227#102—82no—
228#102—82no—
229#102—82no—
230C22⋊C8—24no—
231C22⋊C8—42no—
232#97—24no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8#4112yes—
244C4×C8#4112yes—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249C24—41no—
250C24#12711yescharacteristic
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4#12711yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4#12711yescharacteristic
294C22×C4C2×SD1614yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298SD16—88no—
299M16—216no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8#12314yes—
314C2×C8C2×SD1614yes—
315C2×C8C2×SD1614yes—
316C2×C8AGL(1,ℤ/8)14yes—
317C2×C8#13714yes—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—82no—
356C42C2×#4111yescharacteristic
357C42C2×#4111yescharacteristic
358C422−1+411yescharacteristic
359C422−1+411yescharacteristic
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23#49212yes—
374C23C2×#12711yescharacteristic
375Q8—82no—
376Q8—82no—
377D4—82no—
378D4—82no—
379D4—82no—
380D4—82no—
381D4—82no—
382D4—82no—
383D4—164no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
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—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—164no—
420C2×C4#49212yes—
421C2×C4C2×#12311yescharacteristic
422C2×C4#53611yescharacteristic
423C2×C4#54711yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4#54711yescharacteristic
426C2×C4#54711yescharacteristic
427C2×C4C22×SD1612yes—
428C2×C4C2×AGL(1,ℤ/8)12yes—
429C2×C4C2×#13712yes—
430C2×C4#57512yes—
431C2×C4#57612yes—
432C2×C4#57712yes—
433C8—24no—
434C8—24no—
435C8—42no—
436C8—42no—
437C8—42no—
438C8#48118yes—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22#250111yescharacteristic
454C22C2×#49211yescharacteristic
455C22#315111yescharacteristic
456C22#315211yescharacteristic
457C22#315311yescharacteristic
458C22#315411yescharacteristic
459C22#315511yescharacteristic
460C4—22no—
461C4—41no—
462C4—41no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4C2×#48112yes—
471C4#322712yes—
472C4#334612yes—
473C4#335012yes—
474C4#335112yes—
475C4#335712yes—
476C2—42no—
477C2—321no—
478C2#1962211yescharacteristic
479C2#1962311yescharacteristic
480C2#1971911yescharacteristic
481C2#1974811yescharacteristic
482C2#2790611yescharacteristic
483C2#2794511yescharacteristic
484C2#2794711yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
3. #27948C2semidirect
5. #27876C2semidirect
6. #20800C2semidirect
7. #10686C2semidirect
8. #10684C2semidirect
9. #10683C2semidirect
10. #10672C2semidirect
11. #9842C2semidirect
12. #9820C2semidirect
13. #8819C2semidirect
14. #8814C2semidirect
15. #8146C2semidirect

Maximal subgroups

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

subgroupindex
#331742
#279732
#279482
#278902
#278762
#208002
#106862
#106842
#106832
#106722
#98422
#98202
#88192
#88142
#81462