Finite Group Databasebeta

← #414569

Subgroups — #414569

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4C2×#1928C212yesmaximal
5#14261C211yesmaximal, characteristic
6#11219C212yesmaximal
7#11208C211yesmaximal, characteristic
8#11067C212yesmaximal
9#11052C211yesmaximal, characteristic
10#10806C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2331—28no—
17C2×#391—22no—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#387—22no—
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27#2057—24no—
28#2024—24no—
29#1928—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1443—22no—
40#1442—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8D414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72#391—216no—
73#391—22no—
74#391—22no—
75#391—24no—
76#391—22no—
77#391—22no—
78#389—24no—
79#389—24no—
80#389—22no—
81#389—22no—
82#387—216no—
83#382—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4D418yes—
86#376—24no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D412yes—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#336—24no—
106#336—22no—
107#336—24no—
108#336—22no—
109C22×Q8—24no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—82no—
122C22×Q8—82no—
123C22×Q8#4112yes—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8#4114yes—
149C4⋊Q8—216no—
150C4⋊Q8—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4#4112yes—
184C2×C4⋊C4#4114yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—28no—
195C2×C42—82no—
196C2×C42—82no—
197C2×C42C2×D414yes—
198C2×C42C2×D412yes—
199C2×C42#4112yes—
200#107—48no—
201#103—216no—
202#103—48no—
203#95—24no—
204#95—24no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—24no—
209#95—24no—
210#95—44no—
211#95—42no—
212#95—42no—
213#95—42no—
214#95—82no—
215#95—82no—
216C24C22×D411yescharacteristic
217C2×Q8—216no—
218C2×Q8—24no—
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—88no—
237C2×Q8—88no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—22no—
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—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4#12014yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D412yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4112yes—
304C22×C4C2×#4111yescharacteristic
305C22×C42−1+411yescharacteristic
306C22×C42−1+411yescharacteristic
307C22×C42−1+412yes—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—216no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—84no—
339C4⋊C4—88no—
340C4⋊C4#12118yes—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—42no—
346C42—42no—
347C42—88no—
348C23C2×#12011yescharacteristic
349C23C2×#12111yescharacteristic
350C23#52511yescharacteristic
351C23#52511yescharacteristic
352C23#53211yescharacteristic
353C23#53611yescharacteristic
354C23#53611yescharacteristic
355C23#57612yes—
356C23#57712yes—
357C23#57812yes—
358C23#57812yes—
359Q8—44no—
360Q8—44no—
361Q8—816no—
362Q8—82no—
363Q8—82no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—81no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
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—81no—
423C2×C4—81no—
424C2×C4C2×#12112yes—
425C2×C4#52512yes—
426C2×C4#52512yes—
427C2×C4#53212yes—
428C2×C4#53612yes—
429C2×C4#53612yes—
430C2×C4#54811yescharacteristic
431C2×C4#54812yes—
432C2×C4#54811yescharacteristic
433C2×C4#55211yescharacteristic
434C2×C4#55212yes—
435C2×C4#55211yescharacteristic
436C2×C4#55212yes—
437C2×C4#55612yes—
438C2×C4#55612yes—
439C2×C4#55612yes—
440C8—44no—
441C8—82no—
442C22#238811yescharacteristic
443C22#253611yescharacteristic
444C22#254511yescharacteristic
445C22#259911yescharacteristic
446C22#260411yescharacteristic
447C22#261411yescharacteristic
448C22#284711yescharacteristic
449C22#301112yes—
450C22#301212yes—
451C22#304111yescharacteristic
452C22#304211yescharacteristic
453C22#304411yescharacteristic
454C22#304411yescharacteristic
455C22#311812yes—
456C22#312112yes—
457C22#312412yes—
458C22#312712yes—
459C22#317512yes—
460C22#317812yes—
461C22#324212yes—
462C22#324212yes—
463C22#324312yes—
464C22#324312yes—
465C4—42no—
466C4—42no—
467C4—48no—
468C4—44no—
469C4—82no—
470C4—84no—
471C4—82no—
472C4—82no—
473C4—161no—
474C4—161no—
475C4#253612yes—
476C4#254512yes—
477C4#259912yes—
478C4#260412yes—
479C2#1517611yescharacteristic
480C2#2358512yes—
481C2#2821111yescharacteristic
482C2#2821311yescharacteristic
483C2#2866311yescharacteristic
484C2#2866611yescharacteristic
485C2#2940212yes—
486C2#2952612yes—
487C2#2964912yes—
488C2#3291511yescharacteristic
489C2#3291711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#23312
C2×#20572
C2×#20242
C2×#19282
#142612
#112192
#112082
#110672
#110522
#108062
#107962