Finite Group Databasebeta

← #382237

Subgroups — #382237

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2074C211yesmaximal, characteristic
2#14707C211yesmaximal, characteristic
3#14703C211yesmaximal, characteristic
4#11358C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11268C211yesmaximal, characteristic
7#11067C212yesmaximal
8#11052C211yesmaximal, characteristic
9#10928C212yesmaximal
10#10881C212yesmaximal
11#10693C211yesmaximal, characteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#97C2212yes—
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2212yes—
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#2074C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1436—22no—
34#1436—22no—
35#1436—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39#1434—22no—
40C23×Q8C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#103—24no—
56C2×#103—42no—
57C2×#103—42no—
58C2×#103—42no—
59C2×#103—42no—
60C2×#103C2×C418yes—
61C2×#97—24no—
62C2×#97—42no—
63C2×#97—42no—
64#391—22no—
65#391—24no—
66#391—42no—
67#391C2×C414yes—
68#389—24no—
69#389—42no—
70#389—42no—
71#382—22no—
72#382—42no—
73#382C2×C414yes—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—21no—
86C43—42no—
87#340—24no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340C2×C418yes—
93#336—24no—
94#336—22no—
95#336C2×C414yes—
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—21no—
105C22×Q8C22×C412yes—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
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×C8C22×C414yes—
130C22×C8#4114yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C22×C414yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—22no—
164C2×C42—21no—
165C2×C42—24no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C22×C412yes—
176C2×C42#4112yes—
177C2×C42#4114yes—
178#103—48no—
179#103—48no—
180#97—48no—
181#95—28no—
182#95—28no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—42no—
191#95—42no—
192#95—42no—
193C24C22×D411yescharacteristic
194C2×Q8—24no—
195C2×Q8—22no—
196C2×Q8—24no—
197C2×Q8—22no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4C4×D414yes—
267C22×C4#12412yes—
268C22×C4#12412yes—
269C22×C4#13714yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4111yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)12yes—
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C42−1+411yescharacteristic
279C22×C42−1+412yes—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—84no—
309C4⋊C4—88no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—24no—
314C42—48no—
315C42—48no—
316C42—42no—
317C42—42no—
318C23C2×C4×D411yescharacteristic
319C23C2×#12411yescharacteristic
320C23#53211yescharacteristic
321C23#53411yescharacteristic
322C23#53511yescharacteristic
323C23#53611yescharacteristic
324C23#53711yescharacteristic
325C23C2×#13712yes—
326C23#57512yes—
327C23#57512yes—
328C23#57512yes—
329Q8—44no—
330Q8—44no—
331Q8—82no—
332Q8—82no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
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—44no—
359C2×C4—42no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4C4×#4112yes—
394C2×C4#51811yescharacteristic
395C2×C4#52011yescharacteristic
396C2×C4#52012yes—
397C2×C4#52912yes—
398C2×C4#53812yes—
399C2×C4#53812yes—
400C2×C4#53912yes—
401C2×C4#53912yes—
402C2×C4#53912yes—
403C2×C4#54012yes—
404C2×C4#54011yescharacteristic
405C2×C4#54112yes—
406C2×C4#54212yes—
407C2×C4#54212yes—
408C2×C4#54211yescharacteristic
409C8—44no—
410C8—44no—
411C22C4×#12411yescharacteristic
412C22#232211yescharacteristic
413C22#232411yescharacteristic
414C22#263911yescharacteristic
415C22#264011yescharacteristic
416C22#266111yescharacteristic
417C22#266411yescharacteristic
418C22#294312yes—
419C22C4×#13712yes—
420C22#313911yescharacteristic
421C22#314011yescharacteristic
422C22#314211yescharacteristic
423C22#314211yescharacteristic
424C22#318112yes—
425C22#318312yes—
426C22#319612yes—
427C22#319712yes—
428C22#321512yes—
429C22#321912yes—
430C22#323712yes—
431C22#324012yes—
432C22#325212yes—
433C22#325612yes—
434C4—44no—
435C4—44no—
436C4—42no—
437C4—42no—
438C4—84no—
439C4—82no—
440C4—82no—
441C4—161no—
442C4—162no—
443C4—161no—
444C4#233412yes—
445C4#233512yes—
446C4#264912yes—
447C4#265412yes—
448C2#1432411yescharacteristic
449C2#2164811yescharacteristic
450C2#2165711yescharacteristic
451C2#2208312yes—
452C2#2215612yes—
453C2#3019811yescharacteristic
454C2#3020211yescharacteristic
455C2#3021511yescharacteristic
456C2#3021711yescharacteristic
457C2#3032312yes—
458C2#3038712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
23. #2074C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#20742
#147072
#147032
#113582
#112932
#112682
#110672
#110522
#109282
#108812
#106932