Finite Group Databasebeta

← #413491

Subgroups — #413491

Order 512.

1208 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#1981C212yesmaximal
4C2×#1949C212yesmaximal
5#14274C211yesmaximal, characteristic
6#13779C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11122C212yesmaximal
9#11052C211yesmaximal, characteristic
10#11052C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×#376C2211yescharacteristic
29#2096—24no—
30#2095—24no—
31#1981—28no—
32#1949—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1445—24no—
39#1436—24no—
40C4⋊#95C2212yes—
41C23×Q8C2311yescharacteristic
42C23×Q8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8Q814yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—24no—
61C2×#107—42no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8D418yes—
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—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—42no—
85#391—42no—
86#384—22no—
87#384D414yes—
88#382—24no—
89#382—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92#376—24no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—24no—
101C2×#95D414yes—
102#340—24no—
103#340—24no—
104#336—22no—
105#336—22no—
106#336—42no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8C2×Q812yes—
126C22×Q8C2×Q812yes—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
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—22no—
149C22×C8C2×D414yes—
150C22×C8C2×D414yes—
151C4⋊Q8—216no—
152C4⋊Q8—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4#4114yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—44no—
185C2×C42—84no—
186C2×C42C2×D412yes—
187C2×C42C2×D414yes—
188C2×C42C2×Q812yes—
189C2×C42#4112yes—
190C2×C42#4114yes—
191C2×C42#4112yes—
192#107—48no—
193#106—48no—
194C4⋊C8—216no—
195#103—48no—
196#103—48no—
197#103—48no—
198#103—48no—
199#95—28no—
200#95—28no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—84no—
208C24C22×D411yescharacteristic
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
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—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4#12114yes—
285C22×C4C22⋊Q814yes—
286C22×C4#12314yes—
287C22×C4#12714yes—
288C22×C4C22×D412yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C4H(5,𝔽2)12yes—
297C22×C42−1+411yescharacteristic
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—416no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—88no—
326C42—24no—
327C42—24no—
328C42—42no—
329C42—42no—
330C42—88no—
331C42C22⋊Q818yes—
332C42#12318yes—
333C23C2×#12111yescharacteristic
334C23C2×C22⋊Q811yescharacteristic
335C23C2×#12311yescharacteristic
336C23#52511yescharacteristic
337C23C2×#12711yescharacteristic
338C23#53511yescharacteristic
339C23#53711yescharacteristic
340C23#57714yes—
341C23#57814yes—
342Q8—44no—
343Q8—44no—
344Q8—44no—
345Q8—44no—
346Q8—82no—
347Q8—82no—
348Q8—82no—
349Q8—82no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4#39918yes—
397C2×C4C2×C22⋊Q812yes—
398C2×C4C2×#12312yes—
399C2×C4#52912yes—
400C2×C4#53212yes—
401C2×C4#54012yes—
402C2×C4#54212yes—
403C2×C4D4212yes—
404C2×C4#54612yes—
405C2×C4#54812yes—
406C2×C4D4×Q811yescharacteristic
407C2×C4#55011yescharacteristic
408C2×C4#55012yes—
409C2×C4#55412yes—
410C2×C4#55411yescharacteristic
411C2×C4#55611yescharacteristic
412C2×C4#55912yes—
413C8—44no—
414C8—44no—
415C22C2×#39911yescharacteristic
416C22#253811yescharacteristic
417C22#254011yescharacteristic
418C22#254111yescharacteristic
419C22#258011yescharacteristic
420C22#262411yescharacteristic
421C22#285411yescharacteristic
422C22#304212yes—
423C22#304412yes—
424C22#308214yes—
425C22#309714yes—
426C22#312014yes—
427C22#315412yes—
428C22#315612yes—
429C22#322112yes—
430C22#322312yes—
431C22#325814yes—
432C4—28no—
433C4—44no—
434C4—84no—
435C4—82no—
436C4—82no—
437C4—84no—
438C4—82no—
439C4—161no—
440C4—161no—
441C4#244512yes—
442C4#244812yes—
443C4#245212yes—
444C4#245412yes—
445C2#1514811yescharacteristic
446C2#2373612yes—
447C2#2829112yes—
448C2#2839012yes—
449C2#2845511yescharacteristic
450C2#2845611yescharacteristic
451C2#2926612yes—
452C2#2993412yes—
453C2#3311912yes—
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×#20962
C2×#20952
C2×#19812
C2×#19492
#142742
#137792
#111242
#111222
#110522
#108332