Finite Group Databasebeta

← #315770

Subgroups — #315770

Order 512.

866 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33175C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic, complemented
3#27909C211yesmaximal, characteristic, complemented
4#27891C211yesmaximal, characteristic, complemented
5#27891C211yesmaximal, characteristic, complemented
6#20800C211yesmaximal, characteristic, complemented
7#10673C211yesmaximal, characteristic, complemented
8#10672C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic, complemented
11#9837C211yesmaximal, characteristic, complemented
12#9824C211yesmaximal, characteristic, complemented
13#8820C211yesmaximal, characteristic, complemented
14#8814C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#480—22no—
18C2×#469—22no—
19C2×#428—22no—
20C2×C4×M16C2211yescharacteristic
21#2856C2211yescharacteristic
22#2856C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#2093—22no—
26#2071—22no—
27#2033—22no—
28#2033—22no—
29#1972—22no—
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1925—22no—
41#1925—22no—
42#1874—22no—
43#1847—22no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
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×D8—22no—
64C22×M16D412yes—
65C2×#127C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C42C2311yescharacteristic
78#493—22no—
79#493—22no—
80#480—48no—
81#469—48no—
82#428—48no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×C22⋊C8D412yes—
103C2×C22⋊C8D412yes—
104C4×M16—22no—
105C4×M16—22no—
106C2×#97D412yes—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#336D412yes—
138#336D412yes—
139#322—22no—
140C82—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—41no—
148C22×D4—41no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×D8—44no—
154C2×D8—44no—
155C2×D8—44no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16D818yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165#127—21no—
166#127—21no—
167#127—21no—
168#127—21no—
169#127—21no—
170#127—21no—
171#127—21no—
172#127—21no—
173#121—44no—
174#121—44no—
175#121—44no—
176#121—44no—
177#121—82no—
178#121—82no—
179#117—44no—
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—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—82no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—44no—
221#107—44no—
222#107—44no—
223#107—44no—
224#107—82no—
225#107—82no—
226#106—44no—
227#106—44no—
228#106—44no—
229#102—44no—
230#102—44no—
231#102—44no—
232#102—44no—
233#102—44no—
234#102—44no—
235#102—82no—
236#102—82no—
237#102—82no—
238#102—82no—
239C22⋊C8—24no—
240C22⋊C8—42no—
241#97—24no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8#4112yes—
253C4×C8#4112yes—
254#95—41no—
255#95—41no—
256C24—41no—
257C24—41no—
258C24#12711yescharacteristic
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4#12711yescharacteristic
310C22×C4#12711yescharacteristic
311C22×C4#12711yescharacteristic
312C22×C4C2×D814yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316D8—88no—
317M16—216no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8#12314yes—
332C2×C8C2×D814yes—
333C2×C8C2×D814yes—
334C2×C8AGL(1,ℤ/8)14yes—
335C2×C8#13714yes—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—82no—
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C422−1+411yescharacteristic
374C422−1+411yescharacteristic
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23#49312yes—
396C23C2×#12711yescharacteristic
397D4—82no—
398D4—82no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—82no—
409D4—164no—
410D4—164no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—164no—
444C2×C4#49312yes—
445C2×C4C2×#12311yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4#54711yescharacteristic
449C2×C4#54711yescharacteristic
450C2×C4#54711yescharacteristic
451C2×C4C22×D812yes—
452C2×C4C2×AGL(1,ℤ/8)12yes—
453C2×C4C2×#13712yes—
454C2×C4#57512yes—
455C2×C4#57612yes—
456C2×C4#57712yes—
457C8—24no—
458C8—24no—
459C8—42no—
460C8—42no—
461C8—42no—
462C8#48018yescomplemented
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22—161no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22#250111yescharacteristic
485C22C2×#49311yescharacteristic
486C22#315111yescharacteristic
487C22#315211yescharacteristic
488C22#315311yescharacteristic
489C22#315411yescharacteristic
490C22#315511yescharacteristic
491C4—22no—
492C4—41no—
493C4—41no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4—161no—
498C4—161no—
499C4—161no—
500C4C2×#48012yes—
501C4#322612yes—
502C4#334512yes—
503C4#334712yes—
504C4#335412yes—
505C4#335512yes—
506C2—42no—
507C2—321no—
508C2—321no—
509C2#1962911yescharacteristic
510C2#1963111yescharacteristic
511C2#1972611yescharacteristic
512C2#1974811yescharacteristic
513C2#2790911yescharacteristic
514C2#2794111yescharacteristic
515C2#2794211yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33175C2
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. #27973C2semidirect
3. #27909C2semidirect
4. #27891C2semidirect
5. #27891C2semidirect
6. #20800C2semidirect
7. #10673C2semidirect
8. #10672C2semidirect
9. #10672C2semidirect
10. #10671C2semidirect
11. #9837C2semidirect
12. #9824C2semidirect
13. #8820C2semidirect
14. #8814C2semidirect
15. #8146C2semidirect
462. C8#480semidirect

Maximal subgroups

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

subgroupindex
#331752
#279732
#279092
#278912
#208002
#106732
#106722
#106712
#98372
#98242
#88202
#88142
#81462