Finite Group Databasebeta

← #471724

Subgroups — #471724

Order 512.

996 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic
3#14709C211yesmaximal, characteristic
4#14703C211yesmaximal, characteristic
5#11385C211yesmaximal, characteristic
6#11358C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11228C212yesmaximal
9#11190C212yesmaximal
10#11077C212yesmaximal
11#10832C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#2074—24no—
23#2026—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1436—22no—
34#1436—22no—
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—22no—
41C23×Q8C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—24no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—42no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42D412yes—
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—24no—
61C2×C4⋊C8—42no—
62C2×C4⋊C8—42no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#103—42no—
69#391—22no—
70#391—24no—
71#391—42no—
72#391D414yes—
73#389—24no—
74#382—22no—
75#382—22no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C43—42no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—42no—
93#340—42no—
94#340—42no—
95#336—22no—
96#336—22no—
97#336—42no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—22no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8C2×D412yes—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133#125—48no—
134#125—48no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4#4114yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42C2×D412yes—
180C2×C42#4112yes—
181C2×C42#4114yes—
182C4⋊C8—48no—
183#103—48no—
184#103—48no—
185#95—28no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—44no—
195#95—44no—
196#95—44no—
197C24C22×D411yescharacteristic
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4#12114yes—
270C22×C4#12412yes—
271C22×C4#12412yes—
272C22×C4#13514yes—
273C22×C4#13714yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4111yescharacteristic
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C4H(5,𝔽2)12yes—
281C22×C42−1+411yescharacteristic
282C22×C42−1+411yescharacteristic
283C22×C42−1+412yes—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—416no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—84no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—48no—
319C42—48no—
320C42—42no—
321C42—42no—
322C23C2×#12111yescharacteristic
323C23#52511yescharacteristic
324C23C2×#12411yescharacteristic
325C23#53411yescharacteristic
326C23#53611yescharacteristic
327C23#53711yescharacteristic
328C23#53711yescharacteristic
329C23C2×#13512yes—
330C23C2×#13712yes—
331C23#57512yes—
332C23#57512yes—
333Q8—44no—
334Q8—44no—
335Q8—82no—
336Q8—82no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#49114yes—
397C2×C4#53511yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4#53812yes—
400C2×C4#53912yes—
401C2×C4#54112yes—
402C2×C4#54212yes—
403C2×C4#54211yescharacteristic
404C2×C4#54211yescharacteristic
405C2×C4#54612yes—
406C2×C4#54712yes—
407C2×C4#55112yes—
408C2×C4#55612yes—
409C2×C4#55612yes—
410C2×C4#55912yes—
411C2×C4#56212yes—
412C2×C4#56312yes—
413C8—82no—
414C8—82no—
415C22#253411yescharacteristic
416C22#254711yescharacteristic
417C22#259211yescharacteristic
418C22#262011yescharacteristic
419C22#263711yescharacteristic
420C22#263911yescharacteristic
421C22#281411yescharacteristic
422C22#303512yes—
423C22#303712yes—
424C22#310212yes—
425C22#311612yes—
426C22#313111yescharacteristic
427C22C2×#49111yescharacteristic
428C22#314011yescharacteristic
429C22#314211yescharacteristic
430C22#319012yes—
431C22#319712yes—
432C22#323212yes—
433C22#323712yes—
434C22#324812yes—
435C22#325012yes—
436C22#325212yes—
437C22#325612yes—
438C4—24no—
439C4—44no—
440C4—44no—
441C4—42no—
442C4—82no—
443C4—82no—
444C4—84no—
445C4—162no—
446C4—161no—
447C4—161no—
448C4#268912yes—
449C4#271012yes—
450C4#275912yes—
451C4#279812yes—
452C2#1680511yescharacteristic
453C2#2812512yes—
454C2#2872011yescharacteristic
455C2#2872511yescharacteristic
456C2#2935111yescharacteristic
457C2#2935711yescharacteristic
458C2#2983312yes—
459C2#3015512yes—
460C2#3019011yescharacteristic
461C2#3019811yescharacteristic
462C2#3274812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#20742
C2×#20262
#147092
#147032
#113852
#113582
#113152
#112282
#111902
#110772
#108322