Finite Group Databasebeta

← #315919

Subgroups — #315919

Order 512.

747 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29842C211yesmaximal, characteristic
2#27960C211yesmaximal, characteristic, complemented
3#27908C211yesmaximal, characteristic, complemented
4#27906C211yesmaximal, characteristic
5#27891C211yesmaximal, characteristic
6#21982C211yesmaximal, characteristic, complemented
7#10675C211yesmaximal, characteristic, complemented
8#10675C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic, complemented
11#9830C211yesmaximal, characteristic, complemented
12#9822C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#481—22no—
18C2×#485—22no—
19C2×#431C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2099—22no—
25#2097—22no—
26#2094—22no—
27#2067—22no—
28#2033—22no—
29#2033—22no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1956C2211yescharacteristic
39#1956C2211yescharacteristic
40#1947—22no—
41#1925—22no—
42#1852—22no—
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1844C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1823—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#127C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#486—22no—
75#486—22no—
76#485—48no—
77#481—48no—
78#431—22no—
79#431—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×C4⋊C8D412yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×C22⋊C8D412yes—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8D412yes—
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
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159#127—21no—
160#127—21no—
161#127—21no—
162#127—21no—
163C22⋊Q8—44no—
164#121—44no—
165#121—82no—
166#121—82no—
167#117—44no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
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—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—82no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
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×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#107—44no—
210#107—82no—
211#107—82no—
212#107—82no—
213#107—82no—
214#106—44no—
215#106—44no—
216#106—44no—
217C4⋊C8—24no—
218#103—44no—
219#103—44no—
220#102—44no—
221#102—44no—
222#102—44no—
223#102—44no—
224#102—82no—
225#102—82no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228C22⋊C8D818yes—
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—21no—
238C4×C8—21no—
239C4×C8#4112yes—
240C4×C8#4112yes—
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—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#12711yescharacteristic
288C22×C4#12711yescharacteristic
289C22×C4#12711yescharacteristic
290C22×C4C2×D814yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
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—22no—
308C2×C8—22no—
309C2×C8#12314yes—
310C2×C8C2×D814yes—
311C2×C8C2×D814yes—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
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—42no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—82no—
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C422−1+411yescharacteristic
353C422−1+411yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23#49312yes—
368C23C2×#12711yescharacteristic
369Q8—82no—
370Q8—82no—
371D4—82no—
372D4—82no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—82no—
377D4—164no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
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×C4—81no—
413C2×C4—164no—
414C2×C4#49312yes—
415C2×C4C2×#12311yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#54711yescharacteristic
418C2×C4#54711yescharacteristic
419C2×C4#54711yescharacteristic
420C2×C4#54711yescharacteristic
421C2×C4C22×D812yes—
422C2×C4#57512yes—
423C2×C4#57712yes—
424C2×C4#57712yes—
425C2×C4#57712yes—
426C2×C4#57812yes—
427C8—24no—
428C8—24no—
429C8—24no—
430C8—42no—
431C8—42no—
432C8—42no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22#250111yescharacteristic
448C22C2×#49311yescharacteristic
449C22#315311yescharacteristic
450C22#315411yescharacteristic
451C22#315411yescharacteristic
452C22#315411yescharacteristic
453C22#315611yescharacteristic
454C4—22no—
455C4—41no—
456C4—41no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4#309912yes—
465C4#324412yes—
466C4#333412yes—
467C4#333412yes—
468C4#334812yes—
469C4#335112yes—
470C2—42no—
471C2—321no—
472C2#1963111yescharacteristic
473C2#1966011yescharacteristic
474C2#1972311yescharacteristic
475C2#1973911yescharacteristic
476C2#2792011yescharacteristic
477C2#2792911yescharacteristic
478C2#2794611yescharacteristic
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
2. #27960C2semidirect
3. #27908C2semidirect
6. #21982C2semidirect
7. #10675C2semidirect
8. #10675C2semidirect
9. #10672C2semidirect
10. #10671C2semidirect
11. #9830C2semidirect
12. #9822C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect
15. #8148C2semidirect

Maximal subgroups

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

subgroupindex
#298422
#279602
#279082
#279062
#278912
#219822
#106752
#106722
#106712
#98302
#98222
#88202
#88142
#81482