Finite Group Databasebeta

← #315763

Subgroups — #315763

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33174C211yesmaximal, characteristic
2#27974C211yesmaximal, characteristic
3#27907C211yesmaximal, characteristic
4#27878C211yesmaximal, characteristic, complemented
5#27875C211yesmaximal, characteristic, complemented
6#20824C211yesmaximal, characteristic, complemented
7#10681C211yesmaximal, characteristic, complemented
8#10679C211yesmaximal, characteristic, complemented
9#10678C211yesmaximal, characteristic, complemented
10#10672C211yesmaximal, characteristic, complemented
11#9842C211yesmaximal, characteristic, complemented
12#9820C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#482—22no—
18C2×#468—22no—
19C2×C4×M16C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#2092—22no—
25#2077—22no—
26#2076—22no—
27#2073—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#1921—22no—
40#1920—22no—
41#1875—22no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1844C2211yescharacteristic
48#1844C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1809—22no—
56#1776—22no—
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—
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#492—22no—
75#492—22no—
76#482—48no—
77#468—48no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#103—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×C22⋊C8D412yes—
92C2×C22⋊C8D412yes—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97D412yes—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#391—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#382—21no—
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#340—22no—
127#340—22no—
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×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×SD16—44no—
149C2×SD16—44no—
150C2×SD16—44no—
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—24no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164#127—21no—
165#127—21no—
166#127—21no—
167#127—21no—
168C22⋊Q8—44no—
169#121—44no—
170#121—44no—
171#121—44no—
172#117—44no—
173#117—82no—
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—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2411yescharacteristic
211#107—44no—
212#107—44no—
213#107—44no—
214#107—44no—
215#106—44no—
216#106—44no—
217#106—44no—
218#103—44no—
219#102—44no—
220#102—44no—
221#102—44no—
222#102—82no—
223#102—82no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226#97—24no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8#4112yes—
238C4×C8#4112yes—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245C24—41no—
246C24#12711yescharacteristic
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4#12711yescharacteristic
286C22×C4#12711yescharacteristic
287C22×C4#12711yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291SD16—88no—
292M16—48no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8#12314yes—
308C2×C8#13514yes—
309C2×C8AGL(1,ℤ/8)14yes—
310C2×C8AGL(1,ℤ/8)14yes—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
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—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—82no—
347C42—82no—
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C422−1+411yescharacteristic
351C422−1+411yescharacteristic
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23C2×#12711yescharacteristic
367Q8—82no—
368Q8—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373D4—82no—
374D4—82no—
375D4—164no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
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×C4C2×#12311yescharacteristic
413C2×C4#53611yescharacteristic
414C2×C4#54711yescharacteristic
415C2×C4#54711yescharacteristic
416C2×C4#54711yescharacteristic
417C2×C4#54711yescharacteristic
418C2×C4C2×#13512yes—
419C2×C4C2×AGL(1,ℤ/8)12yes—
420C2×C4C2×AGL(1,ℤ/8)12yes—
421C2×C4#57512yes—
422C2×C4#57712yes—
423C2×C4#57812yes—
424C8—24no—
425C8—42no—
426C8—42no—
427C8—42no—
428C8—42no—
429C8#48218yescomplemented
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22#250111yescharacteristic
445C22#314711yescharacteristic
446C22#315111yescharacteristic
447C22#315111yescharacteristic
448C22#315311yescharacteristic
449C22#315411yescharacteristic
450C22#315611yescharacteristic
451C4—41no—
452C4—41no—
453C4—41no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4C2×#48212yes—
462C4#322812yes—
463C4#334312yes—
464C4#335012yes—
465C4#335112yes—
466C4#335312yes—
467C2—81no—
468C2—321no—
469C2#1964111yescharacteristic
470C2#1964311yescharacteristic
471C2#1972511yescharacteristic
472C2#1974711yescharacteristic
473C2#2790711yescharacteristic
474C2#2794411yescharacteristic
475C2#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
4. #27878C2semidirect
5. #27875C2semidirect
6. #20824C2semidirect
7. #10681C2semidirect
8. #10679C2semidirect
9. #10678C2semidirect
10. #10672C2semidirect
11. #9842C2semidirect
12. #9820C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect
15. #8146C2semidirect
429. C8#482semidirect

Maximal subgroups

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

subgroupindex
#331742
#279742
#279072
#278782
#278752
#208242
#106812
#106792
#106782
#106722
#98422
#98202
#88202
#88142
#81462