Finite Group Databasebeta

← #474168

Subgroups — #474168

Order 512.

1050 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
1C2×#1984C211yesmaximal, characteristic
2C2×#1933C211yesmaximal, characteristic
3#15201C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic
5#11316C212yesmaximal
6#11227C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11122C212yesmaximal
9#11077C211yesmaximal, characteristic
10#11067C212yesmaximal
11#10861C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#395—21no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24#1984—24no—
25#1933—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1446—21no—
35#1442—21no—
36#1442—22no—
37#1441—22no—
38#1441—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8D411yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—42no—
51C2×C4×Q8—42no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—42no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103Q814yescomplemented
76C2×C4×C8—42no—
77C2×C4×C8D414yes—
78#395—44no—
79#391—22no—
80#391—22no—
81#391—42no—
82#391—42no—
83#389—22no—
84#389—22no—
85#389—42no—
86#389—42no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101#340—22no—
102#340—22no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#336—24no—
108#336—24no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8SD1614yes—
123C22×Q8C2×Q812yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—48no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4C2×Q812yes—
191C2×C4⋊C4#4114yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C2×D412yes—
201C2×C42#4112yes—
202#106—48no—
203C4⋊C8—44no—
204#103—28no—
205#103—48no—
206#103—44no—
207C4×C8—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95—44no—
216#95—42no—
217#95—42no—
218#95—81no—
219#95—81no—
220#95—82no—
221#95—82no—
222C24C22×D411yescharacteristic
223C2×Q8—28no—
224C2×Q8—22no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—48no—
229C2×Q8—44no—
230C2×Q8—48no—
231C2×Q8—44no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4C2×SD1614yes—
304C22×C4C2×SD1612yes—
305C22×C4#13512yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C42−1+412yes—
315C22×C42−1+411yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8C22⋊Q814yes—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C4⋊C4—84no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C23C2×#12111yescharacteristic
363C23#53412yes—
364C23#54812yes—
365C23D4×Q811yescharacteristic
366C23#55011yescharacteristic
367C23C22×SD1611yescharacteristic
368C23C2×#13511yescharacteristic
369C23#57512yes—
370C23#57711yescharacteristic
371C23#57711yescharacteristic
372C23#57812yes—
373Q8—416no—
374Q8—42no—
375Q8—42no—
376Q8—44no—
377Q8—81no—
378Q8—82no—
379Q8—81no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4#46514yes—
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4#53211yescharacteristic
442C2×C4#53411yescharacteristic
443C2×C4#53511yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4#53711yescharacteristic
446C2×C4#53812yes—
447C2×C4#53812yes—
448C2×C4#54212yes—
449C2×C4#54212yes—
450C2×C4#54412yes—
451C2×C4#55112yes—
452C2×C4#55412yes—
453C2×C4#55411yescharacteristic
454C2×C4#55611yescharacteristic
455C2×C4#55612yes—
456C2×C4#56212yes—
457C2×C4#56312yes—
458C8—24no—
459C8—42no—
460C8—82no—
461C22#253811yescharacteristic
462C22#254511yescharacteristic
463C22#261711yescharacteristic
464C22#272912yes—
465C22#273012yes—
466C22C2×#46511yescharacteristic
467C22#305411yescharacteristic
468C22#305911yescharacteristic
469C22#306011yescharacteristic
470C22#318712yes—
471C22#319112yes—
472C22#320112yes—
473C22#320112yes—
474C22#335912yes—
475C22#336412yes—
476C22#337112yes—
477C22#337612yes—
478C22Q8×SD1611yescharacteristic
479C22#338111yescharacteristic
480C22#339012yes—
481C22#339211yescharacteristic
482C22#339411yescharacteristic
483C22#340212yes—
484C4—24no—
485C4—44no—
486C4—42no—
487C4—82no—
488C4—82no—
489C4—84no—
490C4—82no—
491C4—161no—
492C4—162no—
493C4—161no—
494C4#256411yescharacteristic
495C4#262811yescharacteristic
496C4#266211yescharacteristic
497C4#275912yes—
498C4#278812yes—
499C4#281411yescharacteristic
500C2#1690111yescharacteristic
501C2#2826411yescharacteristic
502C2#2827111yescharacteristic
503C2#2863911yescharacteristic
504C2#2864411yescharacteristic
505C2#2973411yescharacteristic
506C2#2973711yescharacteristic
507C2#3256612yes—
508C2#3257112yes—
509C2#3259812yes—
510C2#3260312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
75. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
C2×#19332
#152012
#151582
#113162
#112272
#112202
#111222
#110772
#110672
#108612