Finite Group Databasebeta

← #404657

Subgroups — #404657

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1981C211yesmaximal, characteristic
2C2×#1921C211yesmaximal, characteristic
3#15188C211yesmaximal, characteristic
4#13812C211yesmaximal, characteristic
5#11255C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11124C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10932C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1981—24no—
25#1921—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1442—21no—
35#1442—21no—
36#1441—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—21no—
40#1434—22no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2312yes—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—22no—
60C22×C42—21no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—42no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80#391—24no—
81#391—24no—
82#391—42no—
83#391—42no—
84#389—22no—
85#389—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D414yes—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95D411yescharacteristic
102C2×#95D412yes—
103C2×#95D411yescharacteristic
104#340—22no—
105#340—22no—
106#340—22no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—42no—
111#336—42no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8#4114yes—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8#4112yes—
154C4⋊Q8—28no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4#4114yes—
187C2×C4⋊C4#4112yes—
188C2×C4⋊C4#4114yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—28no—
198C2×C42—24no—
199C2×C42—82no—
200C2×C42—82no—
201C2×C42—82no—
202C2×C42C2×D412yes—
203C2×C42#4112yes—
204#107—28no—
205C4⋊C8—44no—
206#103—48no—
207#103—48no—
208#103—48no—
209#103—48no—
210#95—28no—
211#95—24no—
212#95—22no—
213#95—42no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—81no—
222#95—82no—
223#95Q1614yes—
224C24C22×D411yescharacteristic
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4C2×Q1614yes—
309C22×C4C2×Q1612yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4112yes—
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4112yes—
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)12yes—
320C22×C4H(5,𝔽2)11yescharacteristic
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—84no—
358C4⋊C4#12414yes—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—84no—
367C42—88no—
368C23C2×#12111yescharacteristic
369C23#52512yes—
370C23#54611yescharacteristic
371C23#54612yes—
372C23#55011yescharacteristic
373C23C22×Q1611yescharacteristic
374C23C2×#13711yescharacteristic
375C23#57512yes—
376C23#57712yes—
377C23#57711yescharacteristic
378C23#57811yescharacteristic
379Q8—44no—
380Q8—44no—
381Q8—44no—
382Q8—44no—
383Q8—82no—
384Q8—82no—
385Q8—82no—
386Q8—82no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
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—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#46214yes—
445C2×C4#52511yescharacteristic
446C2×C4#52511yescharacteristic
447C2×C4C2×#12411yescharacteristic
448C2×C4#52912yes—
449C2×C4#52912yes—
450C2×C4#52912yes—
451C2×C4#53211yescharacteristic
452C2×C4#53511yescharacteristic
453C2×C4#53511yescharacteristic
454C2×C4#54012yes—
455C2×C4#54012yes—
456C2×C4#55112yes—
457C2×C4#55112yes—
458C2×C4#55112yes—
459C2×C4#55312yes—
460C2×C4#55311yescharacteristic
461C2×C4#55512yes—
462C2×C4#55511yescharacteristic
463C8—42no—
464C8—81no—
465C8—82no—
466C22#241111yescharacteristic
467C22#241411yescharacteristic
468C22#247112yes—
469C22#247612yes—
470C22#285011yescharacteristic
471C22C2×#46211yescharacteristic
472C22#304011yescharacteristic
473C22#304211yescharacteristic
474C22#304611yescharacteristic
475C22#310412yes—
476C22#311612yes—
477C22#312012yes—
478C22#312312yes—
479C22#329911yescharacteristic
480C22#329912yes—
481C22#330712yes—
482C22#331312yes—
483C22#332012yes—
484C22#332612yes—
485C22#332811yescharacteristic
486C22#339511yescharacteristic
487C22#340011yescharacteristic
488C22#340112yes—
489C4—24no—
490C4—44no—
491C4—44no—
492C4—42no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4—82no—
497C4—84no—
498C4—161no—
499C4#239811yescharacteristic
500C4#252612yes—
501C4#253112yes—
502C4#260311yescharacteristic
503C4#261011yescharacteristic
504C4#264111yescharacteristic
505C2#1489811yescharacteristic
506C2#2448211yescharacteristic
507C2#2449711yescharacteristic
508C2#2463211yescharacteristic
509C2#2464311yescharacteristic
510C2#2719612yes—
511C2#2722012yes—
512C2#2739612yes—
513C2#2742012yes—
514C2#3302711yescharacteristic
515C2#3304011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19812
C2×#19212
#151882
#138122
#112552
#112282
#111902
#111242
#110102
#109322