Finite Group Databasebeta

← #414487

Subgroups — #414487

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic
3#15192C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic
5#11315C212yesmaximal
6#11227C211yesmaximal, characteristic
7#11077C211yesmaximal, characteristic
8#11067C212yesmaximal
9#10926C212yesmaximal
10#10880C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2212yes—
18C2×#398—21no—
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1933—24no—
25C8×C4⋊C4—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1445—21no—
35#1443—22no—
36#1442—22no—
37#1442—21no—
38#1441—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—24no—
59C2×C4⋊C8—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×C4×C8—24no—
69C2×C4×C8—24no—
70C2×C4×C8—42no—
71C2×C4×C8—42no—
72C2×C4×C8D414yes—
73#398—44no—
74#391—24no—
75#391—22no—
76#391—22no—
77#391—42no—
78#389—24no—
79#389—22no—
80#389—22no—
81#389—42no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95D412yes—
99#340—22no—
100#340—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#336—24no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8#4112yes—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C22×C8#4114yes—
144C22×C8#4114yes—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
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—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4SD1614yes—
180C2×C4⋊C4#4114yes—
181C2×C4⋊C4#4112yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C2×D412yes—
191C2×C42#4112yes—
192C4⋊C8—28no—
193#103—28no—
194#103—48no—
195#103—44no—
196C4×C8—28no—
197C4×C8—48no—
198#95—24no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—42no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—42no—
210#95—81no—
211#95—81no—
212#95—82no—
213C24C22×D411yescharacteristic
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4C2×SD1614yes—
289C22×C4C2×SD1612yes—
290C22×C4#13514yes—
291C22×C4#13512yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4C2×#4112yes—
298C22×C4C2×#4111yescharacteristic
299C22×C4H(5,𝔽2)11yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+412yes—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8#12314yes—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—88no—
340C4⋊C4—84no—
341C42—22no—
342C42—22no—
343C42—24no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C23C2×#12111yescharacteristic
349C23#53212yes—
350C23#54611yescharacteristic
351C23#54611yescharacteristic
352C23#54812yes—
353C23C22×SD1611yescharacteristic
354C23C2×#13512yes—
355C23C2×#13511yescharacteristic
356C23#57611yescharacteristic
357C23#57812yes—
358C23#57811yescharacteristic
359Q8—44no—
360Q8—42no—
361Q8—42no—
362Q8—82no—
363Q8—81no—
364Q8—81no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4#46514yes—
425C2×C4C2×#12311yescharacteristic
426C2×C4#52511yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4#52912yes—
429C2×C4#52912yes—
430C2×C4#52912yes—
431C2×C4#53211yescharacteristic
432C2×C4#53511yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#53912yes—
435C2×C4#53912yes—
436C2×C4#55112yes—
437C2×C4#55212yes—
438C2×C4#55311yescharacteristic
439C2×C4#55311yescharacteristic
440C2×C4#55312yes—
441C2×C4#55612yes—
442C2×C4#55612yes—
443C8—24no—
444C8—44no—
445C8—42no—
446C22#253511yescharacteristic
447C22#254511yescharacteristic
448C22#255812yes—
449C22#256012yes—
450C22#285411yescharacteristic
451C22C2×#46511yescharacteristic
452C22#305011yescharacteristic
453C22#305911yescharacteristic
454C22#306111yescharacteristic
455C22#316712yes—
456C22#316812yes—
457C22#317512yes—
458C22#317712yes—
459C22#329511yescharacteristic
460C22#329811yescharacteristic
461C22#330012yes—
462C22#330012yes—
463C22#330211yescharacteristic
464C22#330511yescharacteristic
465C22#335912yes—
466C22#336112yes—
467C22#336412yes—
468C22#336412yes—
469C4—24no—
470C4—42no—
471C4—44no—
472C4—44no—
473C4—82no—
474C4—84no—
475C4—82no—
476C4—162no—
477C4—161no—
478C4—161no—
479C4#239511yescharacteristic
480C4#257712yes—
481C4#259412yes—
482C4#259911yescharacteristic
483C4#260311yescharacteristic
484C4#262311yescharacteristic
485C2#1517411yescharacteristic
486C2#2814111yescharacteristic
487C2#2814511yescharacteristic
488C2#2863911yescharacteristic
489C2#2864111yescharacteristic
490C2#2901012yes—
491C2#2901312yes—
492C2#2907412yes—
493C2#2907512yes—
494C2#3312811yescharacteristic
495C2#3313911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#19332
C2×C8×C4⋊C42
#151922
#151492
#113152
#112272
#110772
#110672
#109262
#108802
#108202