Finite Group Databasebeta

← #500322

Subgroups — #500322

Order 512.

895 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3334C211yesmaximal, characteristic
2C2×#2234C211yesmaximal, characteristic, complemented
3C2×#2159C211yesmaximal, characteristic, complemented
4#27660C211yesmaximal, characteristic, complemented
5#13452C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic, complemented
10#12977C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic, complemented
15#12867C211yesmaximal, characteristic, complemented
16C2×#547—21no—
17C22×M32C2211yescharacteristic
18#3334—24no—
19#3334—24no—
20C2×#500C2211yescharacteristic
21C2×#480—21no—
22C2×#480—21no—
23C2×#474—21no—
24C2×#466C2211yescharacteristic
25C2×#466C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×C8×D4C2211yescharacteristic
28C2×#427C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31#2485—21no—
32#2234—24no—
33#2159—24no—
34C2×#367C2211yescharacteristic
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#366C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46C2×C22⋊C16C2211yescharacteristic
47#2088—21no—
48#2077—21no—
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1825—21no—
52#1808C2211yescharacteristic
53#1808C2211yescharacteristic
54#1808C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×#41—41no—
64C22×D8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#547—28no—
68#547—44no—
69C2×C4⋊Q8—21no—
70C2×#123—41no—
71C2×#123—41no—
72C2×C22⋊Q8—41no—
73C2×#121—21no—
74C2×#121—21no—
75C2×C4×D4—21no—
76C2×C4×D4D411yescharacteristic
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C2×M32—22no—
81C2×M32—22no—
82C2×M32—22no—
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#500—22no—
88#500—22no—
89#480—28no—
90#480—28no—
91#474—28no—
92#466—22no—
93#466—22no—
94#466—22no—
95#466D414yes—
96C4×D8—22no—
97C4×D8—22no—
98C8×D4—22no—
99C8×D4—22no—
100#427—22no—
101#427—22no—
102C2×#107—22no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#107—21no—
106C2×#107—22no—
107C2×#107—22no—
108C2×#107D414yes—
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#107C2311yescharacteristic
113C2×#107C2311yescharacteristic
114C2×#107C2311yescharacteristic
115C2×#107C2311yescharacteristic
116C2×#106—21no—
117C2×C4⋊C8D411yescharacteristic
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#397—41no—
125#394—41no—
126#389—21no—
127#389—21no—
128#388—41no—
129#387—41no—
130#380—21no—
131#380—21no—
132#380—21no—
133#367—22no—
134#367—22no—
135#367—22no—
136#366—22no—
137#366—22no—
138#366—22no—
139#357—22no—
140#357—22no—
141#357—22no—
142#357—22no—
143C22⋊C16—22no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151C2×#41—48no—
152C2×#41—84no—
153C2×#41—84no—
154C22×Q8—41no—
155C22×D4—21no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4D811yescharacteristic
159C23×C4—41no—
160C23×C4—41no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8C2×D412yes—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8C2411yescharacteristic
186C4⋊Q8—24no—
187C4⋊Q8—42no—
188#123—48no—
189#123—48no—
190C22⋊Q8—48no—
191#121—24no—
192#121—24no—
193#121—42no—
194#121—42no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—24no—
198C4×D4—42no—
199#117—42no—
200#117—42no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4D811yescharacteristic
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4D811yescharacteristic
226C2×C22⋊C4D811yescharacteristic
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229M32—28no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16#4112yescomplemented
241C2×C16#4112yescomplemented
242#107—24no—
243#107—24no—
244#107—24no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107C2×D412yes—
257#107C2×D412yes—
258#107C2×D412yes—
259#106—42no—
260C4⋊C8—22no—
261#102—24no—
262#102—24no—
263#102—24no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C4×C8—21no—
267C4×C8—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24—41no—
274C24C2×D811yescharacteristic
275C24C2×D811yescharacteristic
276#41—832no—
277C2×Q8—44no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C2×D4—82no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#12012yes—
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4C22×D411yescharacteristic
328D8—22no—
329D8—22no—
330D8—41no—
331D8—41no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8#12012yes—
349C2×C8C22×D411yescharacteristic
350C2×C8C22×D411yescharacteristic
351C2×C8C2×#4111yescharacteristic
352C2×C8C2×#4111yescharacteristic
353C2×C82−1+411yescharacteristic
354C2×C82−1+411yescharacteristic
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—82no—
374C4⋊C4—161no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—161no—
383C22⋊C4—161no—
384C42—21no—
385C42—21no—
386C42—82no—
387C16—22no—
388C16—22no—
389C16—41no—
390C16—41no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—42no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23#44712yes—
411C23#50912yes—
412C23C22×D811yescharacteristic
413Q8—88no—
414D4—28no—
415D4—42no—
416D4—42no—
417D4—88no—
418D4—88no—
419D4—81no—
420D4—81no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—88no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4#44712yes—
464C2×C4#50912yes—
465C2×C4C2×#12011yescharacteristic
466C2×C4#54711yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4#57811yescharacteristic
470C2×C4#57811yescharacteristic
471C8—22no—
472C8—41no—
473C8—41no—
474C8#52511yescharacteristic
475C8#53611yescharacteristic
476C8#54711yescharacteristic
477C8#54711yescharacteristic
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—161no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22C2×#44711yescharacteristic
500C22#333411yescharacteristic
501C22#333411yescharacteristic
502C22C2×#50911yescharacteristic
503C22C2×#50911yescharacteristic
504C22#341611yescharacteristic
505C22#341711yescharacteristic
506C4—22no—
507C4—22no—
508C4—41no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4#240611yescharacteristic
518C4#301111yescharacteristic
519C4#334311yescharacteristic
520C4#334311yescharacteristic
521C2—42no—
522C2—42no—
523C2—162no—
524C2#2408811yescharacteristic
525C2#3351711yescharacteristic
526C2#3352711yescharacteristic
527C2#3391011yescharacteristic
528C2#3391211yescharacteristic
529C2#3392011yescharacteristic
530C2#3392111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3334C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2234C2semidirect
3. C2×#2159C2semidirect
4. #27660C2semidirect
5. #13452C2semidirect
6. #13451C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13061C2semidirect
10. #12977C2semidirect
11. #12937C2semidirect
12. #12935C2semidirect
13. #12920C2semidirect
14. #12919C2semidirect
15. #12867C2semidirect
240. C2×C16#41semidirect
241. C2×C16#41semidirect

Maximal subgroups

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

subgroupindex
C2×#33342
C2×#22342
C2×#21592
#276602
#134522
#134512
#133812
#133802
#130612
#129772
#129372
#129352
#129202
#129192
#128672