Finite Group Databasebeta

← #502367

Subgroups — #502367

Order 512.

651 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2243C211yesmaximal, characteristic, complemented
2C2×#2210C211yesmaximal, characteristic
3#28527C211yesmaximal, characteristic, complemented
4#23186C211yesmaximal, characteristic, complemented
5#13481C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13449C211yesmaximal, characteristic, complemented
8#13448C211yesmaximal, characteristic
9#13276C211yesmaximal, characteristic, complemented
10#13160C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#13012C211yesmaximal, characteristic
13#12812C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#474—21no—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24#2542—21no—
25#2367—21no—
26#2243—24no—
27#2210—24no—
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#346C2211yescharacteristic
40#2093—21no—
41#2093—21no—
42#1983—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1880—21no—
48#1877—21no—
49#1845C2211yescharacteristic
50#1845C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66C2×#124—21no—
67C2×C22⋊Q8—41no—
68C2×C22⋊Q8—41no—
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32—41no—
75C2×D16—22no—
76C2×D16—22no—
77C2×D16—41no—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#500—22no—
83#500—22no—
84#495—22no—
85#495D414yes—
86#474—44no—
87C4×Q16—22no—
88C4×Q16—41no—
89C4×D8—22no—
90C4×D8—22no—
91C4×D8—22no—
92C4×D8—41no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#399—41no—
113#399—41no—
114#391—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#388—41no—
119#388—41no—
120#385—41no—
121#384—21no—
122#384—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—41no—
126C4⋊C16—22no—
127C4⋊C16—41no—
128C4⋊C16—41no—
129#358—22no—
130#358—22no—
131#358—22no—
132#357—22no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357—22no—
137#346—22no—
138#340—21no—
139#336D411yescharacteristic
140#336D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—41no—
143C22×D4—21no—
144C22×D4—21no—
145C23×C4—41no—
146C23×C4—41no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16C2×D412yes—
153C2×SD16—24no—
154C2×SD16—81no—
155C2×SD16—81no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8C2×D412yes—
167C2×D8#4112yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2411yescharacteristic
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174#124—24no—
175#124—42no—
176C22⋊Q8—84no—
177C22⋊Q8—84no—
178C4×Q8—42no—
179C4×Q8—81no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—81no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4D811yescharacteristic
196C2×C4⋊C4D811yescharacteristic
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212SD32—44no—
213D16—44no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—42no—
236#107—42no—
237#107—42no—
238#107#4112yes—
239C4⋊C8—22no—
240#103—42no—
241#103—81no—
242#102—42no—
243#102—42no—
244#102—42no—
245#102—42no—
246#102—81no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8C2×D412yes—
252C4×C8#4112yes—
253#95—41no—
254#95—41no—
255#95—41no—
256#95D811yescharacteristic
257#95D811yescharacteristic
258C24—41no—
259C24—41no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—161no—
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C22×D411yescharacteristic
300Q16—22no—
301Q16—22no—
302Q16—41no—
303Q16—41no—
304SD16—84no—
305D8—22no—
306D8—22no—
307D8—22no—
308D8—22no—
309D8—41no—
310D8—41no—
311D8—41no—
312D8—41no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8#12112yes—
322C2×C8C22×D411yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8H(5,𝔽2)11yescharacteristic
327C2×C82−1+411yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
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—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—161no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—82no—
361C42—161no—
362C42#12112yes—
363C42AGL(1,ℤ/8)12yes—
364C16—41no—
365C16—41no—
366C16—41no—
367C16—41no—
368C23—42no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23C22×D811yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—81no—
386Q8—81no—
387Q8—164no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
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—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
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—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—164no—
429C2×C4—164no—
430C2×C4#45912yes—
431C2×C4#45912yes—
432C2×C4#50912yes—
433C2×C4#50912yes—
434C2×C4C2×#12111yescharacteristic
435C2×C4#54611yescharacteristic
436C2×C4#54811yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C8—22no—
441C8—41no—
442C8—41no—
443C8#52911yescharacteristic
444C8#54211yescharacteristic
445C8#54611yescharacteristic
446C8#54711yescharacteristic
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22C2×#45911yescharacteristic
462C22#329411yescharacteristic
463C22#335811yescharacteristic
464C22C2×#50911yescharacteristic
465C22C2×#50911yescharacteristic
466C22#341611yescharacteristic
467C22#341711yescharacteristic
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—82no—
475C4—82no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4#242711yescharacteristic
481C4#304211yescharacteristic
482C4#333211yescharacteristic
483C4#337311yescharacteristic
484C2—162no—
485C2—162no—
486C2#2531811yescharacteristic
487C2#3358211yescharacteristic
488C2#3359011yescharacteristic
489C2#3387311yescharacteristic
490C2#3387511yescharacteristic
491C2#3393411yescharacteristic
492C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2243C2semidirect
3. #28527C2semidirect
4. #23186C2semidirect
5. #13481C2semidirect
6. #13480C2semidirect
7. #13449C2semidirect
9. #13276C2semidirect
10. #13160C2semidirect
11. #13013C2semidirect
13. #12812C2semidirect
14. #12811C2semidirect
15. #12637C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
C2×#22102
#285272
#231862
#134812
#134802
#134492
#134482
#132762
#131602
#130132
#130122
#128122
#128112
#126372