Finite Group Databasebeta

← #496596

Subgroups — #496596

Order 512.

820 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3175C211yesmaximal, characteristic
2C2×#2151C211yesmaximal, characteristic, complemented
3#20821C211yesmaximal, characteristic, complemented
4#13276C211yesmaximal, characteristic, complemented
5#13275C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic, complemented
7#13254C211yesmaximal, characteristic, complemented
8#13062C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic
10#13060C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic
12#12814C211yesmaximal, characteristic, complemented
13#12813C211yesmaximal, characteristic, complemented
14#12812C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C2×#542—21no—
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19#3175—24no—
20#3175C418yescomplemented
21C2×#500C2211yescharacteristic
22C2×#495C2211yescharacteristic
23C2×#489—21no—
24C2×#468—21no—
25C2×#467C2211yescharacteristic
26C2×#466C2211yescharacteristic
27C2×#440—21no—
28C2×C4×Q16C2211yescharacteristic
29C2×C4×D8C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2314—21no—
32#2151—24no—
33C2×C4⋊C16C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35C2×C4⋊C16C2211yescharacteristic
36C2×C4⋊C16C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#357C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1836—21no—
48#1809—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1776—21no—
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8C2311yescharacteristic
65C22×M16D411yescharacteristic
66C23×C8C2311yescharacteristic
67#542—28no—
68#542—44no—
69C2×C4⋊Q8—21no—
70C2×#126—41no—
71C2×#124—21no—
72C2×#124—41no—
73C2×#123—41no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—41no—
76C2×#121—21no—
77C2×C4×Q8—21no—
78C2×C4×D4—21no—
79C2×#117D411yescharacteristic
80C2×M32—22no—
81C2×M32—41no—
82C2×M32—41no—
83C22×C16—22no—
84C22×C16—41no—
85C22×C16—41no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#500—22no—
91#500C2×C414yes—
92#495—22no—
93#495C2×C414yes—
94#489—28no—
95#468—28no—
96#467—22no—
97#467C2×C414yes—
98#466—22no—
99#466C2×C414yes—
100#440—28no—
101C4×Q16—22no—
102C4×Q16C2×C414yes—
103C4×D8—22no—
104C4×D8C2×C414yes—
105C2×#107—21no—
106C2×#107—21no—
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#107C2311yescharacteristic
113C2×#106—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#102—21no—
117C2×#102—21no—
118#405C2×C414yes—
119#405D414yes—
120C2×#97D411yescharacteristic
121C2×#97D411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#388—41no—
125#388—41no—
126#387—41no—
127#387—41no—
128#384—21no—
129#382—21no—
130#380—21no—
131#380—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4⋊C16—22no—
135C4⋊C16—22no—
136C4⋊C16—22no—
137C4⋊C16—22no—
138#358—41no—
139#358—41no—
140#358—41no—
141#358—41no—
142#357—41no—
143#357—41no—
144#357—41no—
145#357—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150C22×Q8—21no—
151C22×Q8—41no—
152C22×D4—21no—
153C22×D4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16C22×C412yes—
162C2×SD16—24no—
163C2×SD16—42no—
164C2×SD16—42no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8C22×C412yes—
171C2×M16—41no—
172C2×M16—41no—
173C2×M16C22⋊C414yes—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C22×C412yes—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D412yes—
183C22×C8C2411yescharacteristic
184C4⋊Q8—24no—
185C4⋊Q8—42no—
186#126—48no—
187#124—24no—
188#124—48no—
189#124—42no—
190#123—48no—
191C22⋊Q8—24no—
192C22⋊Q8—42no—
193C22⋊Q8—48no—
194#121—24no—
195#121—42no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×D4—24no—
199C4×D4—42no—
200#117C22⋊C414yes—
201#117SD1614yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4D811yescharacteristic
214C2×C4⋊C4D811yescharacteristic
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4D811yescharacteristic
223C2×C22⋊C4D811yescharacteristic
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229M32—44no—
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—21no—
241C2×C16—21no—
242C2×C16—44no—
243#107—24no—
244#107—21no—
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—42no—
255#107C22×C412yes—
256#107C22×C412yes—
257#106—42no—
258#103—24no—
259#103—24no—
260#102—24no—
261#102—24no—
262#97C22⋊C414yes—
263#97C22⋊C414yes—
264C4×C8C22×C412yes—
265C4×C8C22×C412yes—
266C4×C8C2×D412yes—
267C4×C8C2×D412yes—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—41no—
275C24C2×D811yescharacteristic
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—44no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—161no—
284C2×Q8—161no—
285C2×D4—22no—
286C2×D4—44no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—161no—
293C2×D4—161no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—161no—
307C22×C4—161no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×D811yescharacteristic
310C22×C4C2×D811yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×D811yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×SD1612yes—
315C22×C4C22×D411yescharacteristic
316Q16—22no—
317Q16—22no—
318Q16—41no—
319Q16—41no—
320SD16—48no—
321D8—22no—
322D8—22no—
323D8—41no—
324D8—41no—
325M16—44no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8C2×C22⋊C412yes—
337C2×C8C2×C22⋊C412yes—
338C2×C8C2×C22⋊C412yes—
339C2×C8C23×C411yescharacteristic
340C2×C8C22×D411yescharacteristic
341C2×C8H(5,𝔽2)11yescharacteristic
342C2×C8H(5,𝔽2)11yescharacteristic
343C2×C82−1+411yescharacteristic
344C2×C82−1+411yescharacteristic
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
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—161no—
361C4⋊C4—161no—
362C4⋊C4#10214yes—
363C4⋊C4#10214yes—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—161no—
370C22⋊C4—161no—
371C22⋊C4#10214yes—
372C22⋊C4#10214yes—
373C42—44no—
374C42—161no—
375C42—161no—
376C42C2×C22⋊C412yes—
377C42C2×C22⋊C412yes—
378C42C2×SD1612yes—
379C42C2×SD1612yes—
380C16—41no—
381C16—41no—
382C16—41no—
383C16—41no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—42no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23C2×#10212yes—
398C23C22×D811yescharacteristic
399Q8—42no—
400Q8—42no—
401Q8—81no—
402Q8—81no—
403Q8—164no—
404D4—42no—
405D4—42no—
406D4—81no—
407D4—81no—
408D4—164no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
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—81no—
444C2×C4—164no—
445C2×C4C2×#10212yes—
446C2×C4C2×#10212yes—
447C2×C4C2×#10212yes—
448C2×C4C2×#10212yes—
449C2×C4C2×#10212yes—
450C2×C4C22×C22⋊C411yescharacteristic
451C2×C4#53711yescharacteristic
452C2×C4#53711yescharacteristic
453C2×C4C22×SD1611yescharacteristic
454C2×C4#57511yescharacteristic
455C2×C4#57511yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8#52011yescharacteristic
460C8#52011yescharacteristic
461C8#53511yescharacteristic
462C8#53611yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22C22×#10211yescharacteristic
478C22#324511yescharacteristic
479C22#324511yescharacteristic
480C22#341511yescharacteristic
481C22#341611yescharacteristic
482C22#341611yescharacteristic
483C22#341711yescharacteristic
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—161no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4#231111yescharacteristic
497C4#289611yescharacteristic
498C4#324611yescharacteristic
499C4#324611yescharacteristic
500C2—42no—
501C2—162no—
502C2#2188411yescharacteristic
503C2#3338111yescharacteristic
504C2#3338211yescharacteristic
505C2#3382311yescharacteristic
506C2#3382411yescharacteristic
507C2#3382811yescharacteristic
508C2#3382911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2151C2semidirect
3. #20821C2semidirect
4. #13276C2semidirect
5. #13275C2semidirect
6. #13254C2semidirect
7. #13254C2semidirect
8. #13062C2semidirect
10. #13060C2semidirect
12. #12814C2semidirect
13. #12813C2semidirect
14. #12812C2semidirect
15. #12811C2semidirect
20. #3175C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31752
C2×#21512
#208212
#132762
#132752
#132542
#130622
#130612
#130602
#130592
#128142
#128132
#128122
#128112