Finite Group Databasebeta

← #351516

Subgroups — #351516

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33153C211yesmaximal, characteristic
2#33153C211yesmaximal, characteristic
3#29876C211yesmaximal, characteristic
4#20550C211yesmaximal, characteristic
5#9739C211yesmaximal, characteristic, complemented
6#9730C211yesmaximal, characteristic, complemented
7#9619C211yesmaximal, characteristic, complemented
8#9613C211yesmaximal, characteristic, complemented
9#9576C211yesmaximal, characteristic, complemented
10#9575C211yesmaximal, characteristic, complemented
11#9568C211yesmaximal, characteristic, complemented
12#9567C211yesmaximal, characteristic, complemented
13#9135C211yesmaximal, characteristic
14#9135C211yesmaximal, characteristic
15#8600C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#2086—21no—
23#2086—21no—
24#2075—21no—
25#2075—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1987C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1919—21no—
37#1919—21no—
38#1918—21no—
39#1918—21no—
40#1854—21no—
41#1854—21no—
42#1834C2211yescharacteristic
43#1825—21no—
44#1825—21no—
45#1797C2211yescharacteristic
46#1797C2211yescharacteristic
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1767—21no—
52C4×C4⋊C8—21no—
53#1753—21no—
54#1753—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C42C2311yescharacteristic
75#500—22no—
76#500—22no—
77#431—22no—
78#431—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85#423—22no—
86#423—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D411yescharacteristic
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#391—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125C43—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
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—21no—
143#340—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#335—22no—
151#334—22no—
152#333—22no—
153#333—22no—
154#333—22no—
155#333—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—24no—
162C2×M16—81no—
163C2×M16—81no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C22⋊Q8—82no—
183C22⋊Q8—82no—
184#117—22no—
185#117—22no—
186#117—41no—
187#117—41no—
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⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—42no—
223#107—42no—
224#107—42no—
225#107—42no—
226#106—42no—
227#106—42no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
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—21no—
240C4⋊C8—21no—
241C4⋊C8#4112yes—
242C4⋊C8#4112yes—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—81no—
248#97—81no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—81no—
252C4×C8—81no—
253C4×C8#4112yes—
254C4×C8#4112yes—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24#12111yescharacteristic
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8—81no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302M16—84no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
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—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8#12312yes—
332C2×C8#12312yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
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—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4#13712yes—
360C4⋊C4#13712yes—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—42no—
379C42C2×#4111yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23C2×#12111yescharacteristic
391Q8—161no—
392Q8—161no—
393Q8—161no—
394Q8—161no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
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—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#47212yes—
440C2×C4C2×#12311yescharacteristic
441C2×C4#52511yescharacteristic
442C2×C4#52911yescharacteristic
443C2×C4#52911yescharacteristic
444C2×C4#53211yescharacteristic
445C2×C4#53211yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4#57511yescharacteristic
449C2×C4#57511yescharacteristic
450C2×C4#57711yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57811yescharacteristic
453C2×C4#57811yescharacteristic
454C8—22no—
455C8—22no—
456C8—41no—
457C8—41no—
458C8—42no—
459C8—42no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#285411yescharacteristic
472C22#304611yescharacteristic
473C22#304611yescharacteristic
474C22#305611yescharacteristic
475C22#305611yescharacteristic
476C22C2×#47211yescharacteristic
477C22#306711yescharacteristic
478C4—21no—
479C4—21no—
480C4—22no—
481C4—42no—
482C4—42no—
483C4—41no—
484C4—41no—
485C4—81no—
486C4—81no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4—161no—
491C4#309711yescharacteristic
492C4#309711yescharacteristic
493C4#312311yescharacteristic
494C4#312711yescharacteristic
495C4#317211yescharacteristic
496C4#317211yescharacteristic
497C4#318311yescharacteristic
498C4#318511yescharacteristic
499C2—81no—
500C2#1948711yescharacteristic
501C2#1949111yescharacteristic
502C2#1949911yescharacteristic
503C2#1950311yescharacteristic
504C2#3311711yescharacteristic
505C2#3312311yescharacteristic
506C2#3314411yescharacteristic
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. #9739C2semidirect
6. #9730C2semidirect
7. #9619C2semidirect
8. #9613C2semidirect
9. #9576C2semidirect
10. #9575C2semidirect
11. #9568C2semidirect
12. #9567C2semidirect

Maximal subgroups

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

subgroupindex
#331532
#298762
#205502
#97392
#97302
#96192
#96132
#95762
#95752
#95682
#95672
#91352
#86002