Finite Group Databasebeta

← #331113

Subgroups — #331113

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic
2#33209C211yesmaximal, characteristic
3#33207C211yesmaximal, characteristic
4#33172C211yesmaximal, characteristic
5#33043C211yesmaximal, characteristic
6#33043C211yesmaximal, characteristic
7#10684C211yesmaximal, characteristic, complemented
8#10680C211yesmaximal, characteristic, complemented
9#10679C211yesmaximal, characteristic, complemented
10#10674C211yesmaximal, characteristic, complemented
11#10318C211yesmaximal, characteristic, complemented
12#10314C211yesmaximal, characteristic, complemented
13#10313C211yesmaximal, characteristic, complemented
14#10309C211yesmaximal, characteristic, complemented
15C2×#1273C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#498C2211yescharacteristic
19C2×#494C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#487C2211yescharacteristic
22C2×#467—22no—
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1969—22no—
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1956C2211yescharacteristic
43#1942—22no—
44#1942—22no—
45#1941—22no—
46#1933—22no—
47#1933—22no—
48#1892—22no—
49#1892—22no—
50#1845C2211yescharacteristic
51#1845C2211yescharacteristic
52#1845C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56#1808—22no—
57#1808—22no—
58#1807—22no—
59#1273C2211yescharacteristic
60#1273C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×Q16—22no—
64C23×C8D412yes—
65C23×C8D412yes—
66C23×C8D412yes—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C22×C42C2311yescharacteristic
76#500—41no—
77#500—41no—
78#500—41no—
79#500—41no—
80#498—41no—
81#498—41no—
82#494—41no—
83#494—41no—
84#487—41no—
85#487—41no—
86#487—41no—
87#487—41no—
88#467—48no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×C4×C8—41no—
103C2×C4×C8—41no—
104C2×C4×C8—41no—
105C2×C4×C8—41no—
106C2×C4×C8—41no—
107C2×C4×C8—41no—
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#391—21no—
120#391—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×Q16—44no—
159C2×Q16—44no—
160C2×Q16—44no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8Q1618yescomplemented
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
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—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232#107—44no—
233#107—44no—
234#107—44no—
235#107—44no—
236#107—44no—
237#106—44no—
238#106—44no—
239#103—44no—
240#103—44no—
241#103—44no—
242#103—44no—
243#103—44no—
244#103—44no—
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—21no—
257C4×C8—42no—
258C4×C8—42no—
259C4×C8—42no—
260#95—41no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268C24#12711yescharacteristic
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4#12711yescharacteristic
303C22×C4#12711yescharacteristic
304C22×C4#12711yescharacteristic
305C22×C4C2×Q1614yes—
306C22×C4AGL(1,ℤ/8)14yes—
307C22×C4#13714yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311Q16—88no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
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—48no—
327C2×C8—48no—
328C2×C8—48no—
329C2×C8C2×Q1614yes—
330C2×C8C2×Q1614yes—
331C2×C8#13514yes—
332C2×C8#13514yes—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
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—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42H(5,𝔽2)11yescharacteristic
376C42H(5,𝔽2)11yescharacteristic
377C42H(5,𝔽2)11yescharacteristic
378C422−1+411yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23#49412yes—
384C23#49612yes—
385C23#49712yes—
386C23C2×#12711yescharacteristic
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394Q8—82no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
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—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#49412yes—
432C2×C4#49612yes—
433C2×C4#49712yes—
434C2×C4#53411yescharacteristic
435C2×C4#53411yescharacteristic
436C2×C4#53411yescharacteristic
437C2×C4#53711yescharacteristic
438C2×C4#53711yescharacteristic
439C2×C4#53711yescharacteristic
440C2×C4C22×Q1612yes—
441C2×C4C2×#13512yes—
442C2×C4C2×#13512yes—
443C2×C4C2×AGL(1,ℤ/8)12yes—
444C2×C4C2×#13712yes—
445C2×C4#57512yes—
446C8—24no—
447C8—24no—
448C8—24no—
449C8—24no—
450C8—42no—
451C8—42no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22#172314yes—
459C22#265611yescharacteristic
460C22C2×#49411yescharacteristic
461C22#314711yescharacteristic
462C22#314711yescharacteristic
463C22C2×#49611yescharacteristic
464C22C2×#49711yescharacteristic
465C22#315011yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#318812yes—
478C4#318912yes—
479C4#319012yes—
480C4#324712yes—
481C4#324812yes—
482C4#324912yes—
483C2—24no—
484C2#1960411yescharacteristic
485C2C2×#172311yescharacteristic
486C2#1961911yescharacteristic
487C2#1962011yescharacteristic
488C2#3024211yescharacteristic
489C2#3024311yescharacteristic
490C2#3024411yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
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
7. #10684C2semidirect
8. #10680C2semidirect
9. #10679C2semidirect
10. #10674C2semidirect
11. #10318C2semidirect
12. #10314C2semidirect
13. #10313C2semidirect
14. #10309C2semidirect
169. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332092
#332072
#331722
#330432
#106842
#106802
#106792
#106742
#103182
#103142
#103132
#103092
C2×#12732