Finite Group Databasebeta

← #243257

Subgroups — #243257

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21984C211yesmaximal, characteristic
2#21975C211yesmaximal, characteristic
3#21963C211yesmaximal, characteristic
4#21962C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10655C211yesmaximal, characteristic, complemented
8#10655C211yesmaximal, characteristic, complemented
9#10654C211yesmaximal, characteristic, complemented
10#10653C211yesmaximal, characteristic, complemented
11#10652C211yesmaximal, characteristic, complemented
12#10650C211yesmaximal, characteristic, complemented
13#8045C211yesmaximal, characteristic
14#8041C211yesmaximal, characteristic
15#7877C211yesmaximal, characteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2283C2211yescharacteristic
23#1949—21no—
24#1949—21no—
25#1949—21no—
26#1949—21no—
27#1949—21no—
28#1949—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1854—21no—
42#1854—21no—
43#1851—21no—
44#1851—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1843—21no—
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1832—21no—
51#1809—21no—
52#1809—21no—
53#1808—21no—
54#1807—21no—
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69C23×C8—21no—
70C2×#117—21no—
71C2×#117—21no—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87#423—22no—
88#423—22no—
89#423—22no—
90#423Q814yes—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8D411yescharacteristic
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8Q814yes—
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#383—21no—
117#382—21no—
118#382—21no—
119#382D411yescharacteristic
120#382D411yescharacteristic
121#381—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380D411yescharacteristic
127#380D411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135#376—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C2×M16—24no—
156C2×M16—42no—
157C2×M16—81no—
158C2×M16—81no—
159C2×M16—81no—
160C2×M16—81no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—81no—
169C22×C8—81no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174#117—81no—
175#117—81no—
176#117—82no—
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⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#107—81no—
210#107—81no—
211#107—81no—
212#107—81no—
213#107—81no—
214#107—81no—
215#106—81no—
216#106—81no—
217#106—81no—
218#106—81no—
219#106—81no—
220#106—81no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237C4⋊C8—81no—
238C4⋊C8C2×Q812yes—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241C4⋊C8C2×Q812yes—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24#12111yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4C22⋊Q812yes—
283C22×C4C22⋊Q812yes—
284C22×C4#12312yes—
285C22×C4#12512yes—
286C22×C4#12512yes—
287C22×C4C4⋊Q812yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C2×#4111yescharacteristic
291M16—84no—
292M16—84no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—84no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4—81no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—82no—
361C42—82no—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42#12312yes—
365C42#12512yes—
366C42#12512yes—
367C42C4⋊Q812yes—
368C42C22×Q811yescharacteristic
369C42C22×Q811yescharacteristic
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23#39812yes—
379C23C2×#12111yescharacteristic
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4#39812yes—
423C2×C4#39812yes—
424C2×C4#39812yes—
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C2×C4C2×#12511yescharacteristic
429C2×C4C2×#12511yescharacteristic
430C2×C4C2×C4⋊Q811yescharacteristic
431C2×C4#57611yescharacteristic
432C2×C4#57611yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57811yescharacteristic
438C2×C4#57811yescharacteristic
439C8—42no—
440C8—42no—
441C8—42no—
442C8—42no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—41no—
447C8—81no—
448C8—81no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22C2×#39811yescharacteristic
457C22#304511yescharacteristic
458C22#304611yescharacteristic
459C22#306011yescharacteristic
460C22#306111yescharacteristic
461C22#306811yescharacteristic
462C22#306911yescharacteristic
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4#308111yescharacteristic
477C4#308111yescharacteristic
478C4#308211yescharacteristic
479C4#308211yescharacteristic
480C4#309611yescharacteristic
481C4#309711yescharacteristic
482C4#316311yescharacteristic
483C4#316411yescharacteristic
484C2—42no—
485C2#1954011yescharacteristic
486C2#1954311yescharacteristic
487C2#1954811yescharacteristic
488C2#1955111yescharacteristic
489C2#2366211yescharacteristic
490C2#2366911yescharacteristic
491C2#2367511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
5. #10658C2semidirect
6. #10657C2semidirect
7. #10655C2semidirect
8. #10655C2semidirect
9. #10654C2semidirect
10. #10653C2semidirect
11. #10652C2semidirect
12. #10650C2semidirect

Maximal subgroups

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

subgroupindex
#219842
#219752
#219632
#219622
#106582
#106572
#106552
#106542
#106532
#106522
#106502
#80452
#80412
#78772