Finite Group Databasebeta

← #464770

Subgroups — #464770

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2642C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#2095C211yesmaximal, characteristic
4C2×#2059C212yesmaximal
5C2×#2043C212yesmaximal
6#14277C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11122C212yesmaximal
9#11052C211yesmaximal, characteristic
10#11052C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20#2642—24no—
21#2642—24no—
22C2×#400—22no—
23C2×#400—22no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2212yes—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#2096—24no—
33#2095—24no—
34#2059—28no—
35#2043—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—22no—
43#1442—22no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—42no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#125—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—42no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#106—42no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88#400—216no—
89#400—216no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#391—22no—
95#391—22no—
96#391D414yes—
97#391D414yes—
98#389—24no—
99#389—24no—
100#384—22no—
101#384—22no—
102#382—24no—
103#376—22no—
104#376—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—24no—
112C2×#95D412yes—
113C2×#95D412yes—
114#340—24no—
115#340—24no—
116#336—22no—
117#336—22no—
118#336—42no—
119#336D418yes—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8C2×D412yes—
139C22×Q8C2×D412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C2×D414yes—
164C22×C8C2×D414yes—
165C4⋊Q8—48no—
166C4⋊Q8—48no—
167#125—48no—
168#125—48no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—84no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4#4114yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D414yes—
219#107—48no—
220#106—48no—
221#103—48no—
222#103—48no—
223#103—48no—
224#103—48no—
225#95—28no—
226#95—28no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—44no—
233#95—84no—
234C24C22×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
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—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4#12014yes—
315C22×C4#12712yes—
316C22×C4#12712yes—
317C22×C4C22×D412yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+412yes—
326C22×C42−1+411yescharacteristic
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—48no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—88no—
358C4⋊C4—88no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C23C2×#12011yescharacteristic
368C23#52511yescharacteristic
369C23C2×#12711yescharacteristic
370C23#53511yescharacteristic
371C23#53611yescharacteristic
372C23#53611yescharacteristic
373C23#53711yescharacteristic
374C23#57714yes—
375C23#57814yes—
376Q8—44no—
377Q8—44no—
378Q8—44no—
379Q8—44no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4#53612yes—
435C2×C4#53612yes—
436C2×C4#53611yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#53711yescharacteristic
439C2×C4#53711yescharacteristic
440C2×C4#54212yes—
441C2×C4#54212yes—
442C2×C4D4212yes—
443C2×C4D4212yes—
444C2×C4#54612yes—
445C2×C4#54712yes—
446C2×C4#54712yes—
447C2×C4#54812yes—
448C2×C4#56212yes—
449C2×C4#56312yes—
450C8—44no—
451C8—44no—
452C22#240611yescharacteristic
453C22#243311yescharacteristic
454C22#254011yescharacteristic
455C22#261611yescharacteristic
456C22#263211yescharacteristic
457C22#265811yescharacteristic
458C22#282011yescharacteristic
459C22#301012yes—
460C22#301112yes—
461C22#312014yes—
462C22#315412yes—
463C22#315612yes—
464C22#322112yes—
465C22#322312yes—
466C22#324214yes—
467C22#324214yes—
468C22#325814yes—
469C4—44no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—84no—
478C4—162no—
479C4#267212yes—
480C4#267212yes—
481C4#267412yes—
482C4#267412yes—
483C2#1651011yescharacteristic
484C2#2407112yes—
485C2#2573311yescharacteristic
486C2#2573511yescharacteristic
487C2#2839012yes—
488C2#2969912yes—
489C2#3007312yes—
490C2#3028812yes—
491C2#3283212yes—
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×#26422
C2×#20962
C2×#20952
C2×#20592
C2×#20432
#142772
#111242
#111222
#110522
#107962