Finite Group Databasebeta

← #360686

Subgroups — #360686

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#20800C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4C2×C4×#101C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#11136C214yesmaximal
7#10899C214yesmaximal
8#10851C211yesmaximal, characteristic
9#10851C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#428C2214yes—
12C22×#101C2212yes—
13C2×C4×M16C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2356—22no—
16#2356—22no—
17#2355—21no—
18#2355—21no—
19#2355—21no—
20#2355—21no—
21#2313—22no—
22#2310C2212yes—
23C2×#391C2212yes—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#2283—22no—
28#2274C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1847—22no—
31#1845—21no—
32#1844—21no—
33#1836—22no—
34#1834—21no—
35#1834—21no—
36C4×#101C418yescomplemented
37C2×#343C2212yes—
38#1736C2214yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1441—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C22×M16C2312yes—
46C22×M16C2312yes—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8Q814yes—
55C2×#117C2312yes—
56C2×#117C2312yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C42D412yes—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#428—28no—
67#428—44no—
68C2×#107—24no—
69C2×#106—24no—
70C2×#101—24no—
71C2×#101—24no—
72C2×#101C2×C418yes—
73C4×M16—22no—
74C4×M16—22no—
75C4×M16C2×C414yes—
76C4×M16C2×C414yes—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#391—44no—
86#387—24no—
87#387—24no—
88#387—24no—
89#387—24no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384D411yescharacteristic
97#384D411yescharacteristic
98#383—24no—
99#382—22no—
100#382—22no—
101#382D412yes—
102#382D412yes—
103#381—24no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C22⋊C4—24no—
121#376—24no—
122C2×#95—24no—
123C2×#95—21no—
124C2×#95—24no—
125#343C2×C418yes—
126#340—24no—
127#340—24no—
128#339—28no—
129#339—44no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C22×Q8C2×Q812yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16C22×C414yes—
156C2×M16C22×C414yes—
157C2×M16#4114yes—
158C2×M16#4114yes—
159C22×C8—22no—
160C22×C8—22no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169#117—24no—
170#117—24no—
171#117—24no—
172#117—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C4⋊C418yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×Q812yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42C4⋊C418yes—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—48no—
231#106—48no—
232#101—216no—
233#97—24no—
234#97—24no—
235C4×C8—24no—
236C4×C8—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—48no—
248#95—42no—
249#95—84no—
250C24C22×D411yescharacteristic
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4C2×C4⋊C414yes—
298C22×C4C2×C4⋊C412yes—
299C22×C4C2×C4⋊C414yes—
300C22×C4C2×C4⋊C412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q811yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4112yes—
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4H(5,𝔽2)12yes—
316C22×C42−1+412yes—
317M16—24no—
318M16—24no—
319M16—24no—
320M16—24no—
321M16—24no—
322M16—42no—
323M16—42no—
324M16#11718yes—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—28no—
336C4⋊C4—28no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—28no—
361C42—28no—
362C42—44no—
363C42—44no—
364C42—44no—
365C42—44no—
366C42—84no—
367C42—84no—
368C42C4×D412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C4×D412yes—
372C42C22⋊Q812yes—
373C42C22⋊Q812yes—
374C42C22⋊Q812yes—
375C42C22⋊Q812yes—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23C22×C4⋊C411yescharacteristic
381C23C2×#12011yescharacteristic
382C23C2×#12311yescharacteristic
383C23#53211yescharacteristic
384C23#53211yescharacteristic
385C23#53511yescharacteristic
386C23#53611yescharacteristic
387Q8—44no—
388Q8—44no—
389Q8—44no—
390Q8—44no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4#38512yes—
438C2×C4#38512yes—
439C2×C4#38512yes—
440C2×C4#38512yes—
441C2×C4#45412yes—
442C2×C4#45612yes—
443C2×C4C2×#11712yes—
444C2×C4C2×C4×D411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4#52012yes—
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4#53312yes—
450C2×C4#54412yes—
451C2×C4#54612yes—
452C2×C4#54712yes—
453C2×C4#54812yes—
454C2×C4#55012yes—
455C2×C4#55212yes—
456C2×C4#55312yes—
457C2×C4#55512yes—
458C2×C4#55612yes—
459C8—28no—
460C8—44no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22#228511yescharacteristic
474C22C2×#38511yescharacteristic
475C22#231711yescharacteristic
476C22#243111yescharacteristic
477C22#243211yescharacteristic
478C22#249911yescharacteristic
479C22#250011yescharacteristic
480C22C2×#45411yescharacteristic
481C22C2×#45611yescharacteristic
482C22#302111yescharacteristic
483C22#302211yescharacteristic
484C4—22no—
485C4—22no—
486C4—22no—
487C4—44no—
488C4—44no—
489C4—84no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—162no—
495C4#185714yes—
496C4#235212yes—
497C4#235412yes—
498C4#243912yes—
499C4#244112yes—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2#1387911yescharacteristic
505C2C2×#185711yescharacteristic
506C2#2085811yescharacteristic
507C2#2562311yescharacteristic
508C2#2562411yescharacteristic
509C2#2569511yescharacteristic
510C2#2569611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. C4×#101C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#208002
#207642
C2×C4×#1012
#139262
#111362
#108992
#108512