Finite Group Databasebeta

← #308478

Subgroups — #308478

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26526C211yesmaximal, characteristic
2#26047C211yesmaximal, characteristic, complemented
3#23003C211yesmaximal, characteristic
4#22740C211yesmaximal, characteristic, complemented
5#10671C211yesmaximal, characteristic
6#10655C211yesmaximal, characteristic, complemented
7#10587C211yesmaximal, characteristic
8#10581C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9537C211yesmaximal, characteristic, complemented
11#9222C211yesmaximal, characteristic
12#8979C211yesmaximal, characteristic
13#8921C211yesmaximal, characteristic
14#8892C211yesmaximal, characteristic, complemented
15#8846C211yesmaximal, characteristic, complemented
16C2×#479—21no—
17C2×#460C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×#440—21no—
20#2552C2211yescharacteristic
21#2366C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24#2093—21no—
25#2086—21no—
26#2067—21no—
27#2059—21no—
28#1982C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947—21no—
34#1947—21no—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1936—21no—
42#1880—21no—
43#1863—21no—
44#1862—21no—
45#1862—21no—
46#1845C2211yescharacteristic
47#1834—21no—
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1803C2211yescharacteristic
51#1802—21no—
52C4×#107—21no—
53C4×#107C2211yescharacteristic
54#1770C2211yescharacteristic
55C4×#103C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16—21no—
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#479—44no—
76#460—22no—
77#460Q814yes—
78C8×Q8—22no—
79C8×Q8—22no—
80#440—44no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107D411yescharacteristic
88C2×#107D411yescharacteristic
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103D411yescharacteristic
99C2×#103D411yescharacteristic
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—41no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#398—41no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—41no—
121#389—41no—
122#386—41no—
123#384—21no—
124#384—21no—
125#384—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—41no—
143#340D411yescharacteristic
144#340D411yescharacteristic
145#340D411yescharacteristic
146#340D411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#334—22no—
151#331—22no—
152Q8⋊C8—22no—
153C22×Q8C2×D411yescharacteristic
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C23×C4—41no—
158C2×SD16—24no—
159C2×SD16—42no—
160C2×SD16—42no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170#127—21no—
171#127—21no—
172#127—21no—
173#127C2×Q812yes—
174#125—42no—
175#125—81no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—81no—
181C4×Q8—81no—
182C4×Q8C2×Q812yes—
183C4×Q8#4112yes—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—81no—
187C4×D4—81no—
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—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C2411yescharacteristic
222#107—41no—
223#107—41no—
224#107—42no—
225#107—42no—
226#107—42no—
227#107—42no—
228#107—81no—
229#107—81no—
230#106—42no—
231C4⋊C8—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—81no—
240C4⋊C8C2×Q812yes—
241C4⋊C8#4112yes—
242#103—22no—
243#103—42no—
244#103D814yes—
245#102—24no—
246#102—42no—
247#102—42no—
248#102—81no—
249#102—81no—
250#102—81no—
251#102—81no—
252#102—82no—
253#97—42no—
254C4×C8—24no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95—41no—
266#95—41no—
267C24—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8C22⋊Q812yes—
272C2×Q8C2×D812yes—
273C2×Q8#13712yes—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—81no—
289C2×D4—81no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—161no—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4H(5,𝔽2)11yescharacteristic
307SD16—48no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8C2×D812yes—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—161no—
369C4⋊C4—161no—
370C4⋊C4—161no—
371C4⋊C4C22⋊Q812yes—
372C4⋊C4C2×D812yes—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C22⋊C4—161no—
376C42—22no—
377C42—22no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—82no—
385C42—82no—
386C42C22×Q811yescharacteristic
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C422−1+411yescharacteristic
390C23—42no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23D4211yescharacteristic
398Q8—22no—
399Q8—22no—
400Q8—22no—
401Q8#47414yes—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—82no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4—81no—
411D4—161no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—164no—
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4#52511yescharacteristic
450C2×C4#53911yescharacteristic
451C2×C4#54711yescharacteristic
452C2×C4D4×Q811yescharacteristic
453C2×C4#55411yescharacteristic
454C2×C4C22×D811yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4#57511yescharacteristic
457C2×C4#57511yescharacteristic
458C2×C4#57611yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57811yescharacteristic
462C8—22no—
463C8—41no—
464C8—41no—
465C8—42no—
466C8—41no—
467C8—42no—
468C8—81no—
469C8—81no—
470C8—81no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22#244911yescharacteristic
479C22#327011yescharacteristic
480C22#327111yescharacteristic
481C22#327511yescharacteristic
482C22D4×D811yescharacteristic
483C22#328811yescharacteristic
484C22#329011yescharacteristic
485C4—21no—
486C4—21no—
487C4—24no—
488C4—42no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—161no—
496C4—161no—
497C4—161no—
498C4C2×#47411yescharacteristic
499C4#308311yescharacteristic
500C4#311111yescharacteristic
501C4#312611yescharacteristic
502C4#333411yescharacteristic
503C4#334611yescharacteristic
504C4#338811yescharacteristic
505C4#338811yescharacteristic
506C2—162no—
507C2#1976211yescharacteristic
508C2#1988611yescharacteristic
509C2#2012111yescharacteristic
510C2#2021911yescharacteristic
511C2#2633611yescharacteristic
512C2#2639911yescharacteristic
513C2#2640211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26047C2semidirect
4. #22740C2semidirect
6. #10655C2semidirect
8. #10581C2semidirect
9. #9620C2semidirect
10. #9537C2semidirect
14. #8892C2semidirect
15. #8846C2semidirect

Maximal subgroups

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

subgroupindex
#265262
#260472
#230032
#227402
#106712
#106552
#105872
#105812
#96202
#95372
#92222
#89792
#89212
#88922
#88462