Finite Group Databasebeta

← #244199

Subgroups — #244199

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#493C211yesmaximal, characteristic
2C4×#492C211yesmaximal, characteristic, complemented
3C4×#487C211yesmaximal, characteristic, complemented
4C4×#486C211yesmaximal, characteristic, complemented
5C4×#486C211yesmaximal, characteristic, complemented
6C4×#486C211yesmaximal, characteristic, complemented
7#10684C211yesmaximal, characteristic, complemented
8#10679C211yesmaximal, characteristic, complemented
9#10678C211yesmaximal, characteristic, complemented
10#10677C211yesmaximal, characteristic, complemented
11#10311C211yesmaximal, characteristic, complemented
12#10310C211yesmaximal, characteristic, complemented
13#10307C211yesmaximal, characteristic
14#10305C211yesmaximal, characteristic
15#8131C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#486C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×C4×SD16—22no—
23C2×C4×D8—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1970—22no—
33#1968—22no—
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1934—22no—
43#1934—22no—
44#1934—22no—
45#1933—22no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52C4×#103—22no—
53C4×#102—22no—
54C4×#102—22no—
55C4×#102—22no—
56C42×C8C2211yescharacteristic
57C42×C8C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C22×D8—22no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#493—22no—
78#493—22no—
79#492—22no—
80#492—22no—
81#487—22no—
82#487—22no—
83#486—22no—
84#486—22no—
85#486—22no—
86#486—22no—
87#486—22no—
88#486—22no—
89C4×SD16—48no—
90C4×D8—48no—
91C2×#107—22no—
92C2×#106—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×C4×C8D412yes—
106C2×C4×C8D412yes—
107C2×C4×C8D412yes—
108C2×C4×C8D412yes—
109C2×C4×C8D412yes—
110C2×C4×C8D412yes—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117#391—21no—
118#391—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C43C2311yescharacteristic
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C23×C4—41no—
155C23×C4—41no—
156C2×SD16—28no—
157C2×SD16—44no—
158C2×SD16—44no—
159C2×D8—28no—
160C2×D8—44no—
161C2×D8—44no—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8#4112yes—
175C4⋊Q8#4112yes—
176#127—21no—
177#127—21no—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127#4112yes—
183#127#4112yes—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
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—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
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×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225#107—44no—
226#106—44no—
227#103—28no—
228#103—44no—
229#103—44no—
230#102—28no—
231#102—28no—
232#102—28no—
233#102—44no—
234#102—44no—
235#102—44no—
236#102—44no—
237#102—44no—
238#102—44no—
239C4×C8—24no—
240C4×C8—24no—
241C4×C8—24no—
242C4×C8—24no—
243C4×C8—24no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—42no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261C24—41no—
262C24—41no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4#12711yescharacteristic
308C22×C4#12711yescharacteristic
309C22×C4#12711yescharacteristic
310C22×C4#12711yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314SD16—416no—
315D8—416no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8#13514yes—
333C2×C8#13514yes—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—82no—
366C42—82no—
367C42#13514yes—
368C42AGL(1,ℤ/8)14yes—
369C42AGL(1,ℤ/8)14yes—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—42no—
375C23—42no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23C2×#12711yescharacteristic
389Q8—44no—
390Q8—44no—
391Q8—82no—
392Q8—82no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—44no—
397D4—44no—
398D4—44no—
399D4—82no—
400D4—82no—
401D4—82no—
402D4—82no—
403D4—82no—
404D4—82no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
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—81no—
429C2×C4—164no—
430C2×C4—164no—
431C2×C4#49512yes—
432C2×C4#49512yes—
433C2×C4#49612yes—
434C2×C4#49612yes—
435C2×C4#49612yes—
436C2×C4#49612yes—
437C2×C4#52511yescharacteristic
438C2×C4#52511yescharacteristic
439C2×C4#52511yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4#52511yescharacteristic
442C2×C4#52511yescharacteristic
443C2×C4C2×#13512yes—
444C2×C4C2×#13512yes—
445C2×C4C2×#13512yes—
446C2×C4C2×AGL(1,ℤ/8)12yes—
447C2×C4C2×AGL(1,ℤ/8)12yes—
448C2×C4#57512yes—
449C8—24no—
450C8—24no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—42no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22#239711yescharacteristic
470C22C2×#49511yescharacteristic
471C22#314711yescharacteristic
472C22#314711yescharacteristic
473C22C2×#49611yescharacteristic
474C22C2×#49611yescharacteristic
475C22#315011yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—82no—
480C4—82no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4#171714yes—
486C4#310212yes—
487C4#310212yes—
488C4#310512yes—
489C4#310512yes—
490C4#310612yes—
491C4#310612yes—
492C2—162no—
493C2—162no—
494C2#1960011yescharacteristic
495C2C2×#171711yescharacteristic
496C2#1961911yescharacteristic
497C2#1961911yescharacteristic
498C2#2379611yescharacteristic
499C2#2379611yescharacteristic
500C2#2380011yescharacteristic
subgroupG/H
centerC22×C4#127
commutator, Frattini, cosocleC2×C42C24
socleC23C2×#127
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#492C2semidirect
3. C4×#487C2semidirect
4. C4×#486C2semidirect
5. C4×#486C2semidirect
6. C4×#486C2semidirect
7. #10684C2semidirect
8. #10679C2semidirect
9. #10678C2semidirect
10. #10677C2semidirect
11. #10311C2semidirect
12. #10310C2semidirect
15. #8131C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4932
C4×#4922
C4×#4872
C4×#4862
#106842
#106792
#106782
#106772
#103112
#103102
#103072
#103052
#81312