Finite Group Databasebeta

← #478355

Subgroups — #478355

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15196C211yesmaximal, characteristic
2#15128C211yesmaximal, characteristic
3#11384C212yesmaximal
4#11377C212yesmaximal
5#11346C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11219C212yesmaximal
8#11219C212yesmaximal
9#11207C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#107C2212yes—
12C2×#395—21no—
13C2×#389C2212yes—
14C2×#389C2212yes—
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×#340C2212yes—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#336C2211yescharacteristic
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28#1446—21no—
29#1444—21no—
30#1444—21no—
31#1444—21no—
32#1441—21no—
33#1441—21no—
34#1441—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C23×Q8—21no—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—42no—
61C2×#107—42no—
62C2×#107Q818yescomplemented
63#395—44no—
64#389—24no—
65#389—42no—
66#389—42no—
67#389—42no—
68#384—41no—
69#384—41no—
70C4×C4⋊C4—41no—
71C4×C4⋊C4—41no—
72C2×#95—22no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95D411yescharacteristic
83C2×#95D411yescharacteristic
84#340—24no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—42no—
89#340—42no—
90#340—42no—
91#336—42no—
92#336—41no—
93#336—41no—
94#336—42no—
95C22×Q8—44no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8—82no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2411yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8C2×Q814yes—
125C4⋊Q8—44no—
126C4⋊Q8—48no—
127C4⋊Q8—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
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—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4Q1614yes—
167C2×C4⋊C4C2×Q814yes—
168C2×C4⋊C4C2×Q814yes—
169C2×C4⋊C4#4114yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182#107—48no—
183#95—41no—
184#95—41no—
185#95—44no—
186#95—42no—
187#95—42no—
188#95—44no—
189#95—44no—
190#95—42no—
191#95—42no—
192#95—82no—
193#95—82no—
194#95—82no—
195C24C22×D411yescharacteristic
196C2×Q8—48no—
197C2×Q8—88no—
198C2×Q8—84no—
199C2×Q8—88no—
200C2×Q8—84no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4C2×Q1612yes—
257C22×C4C2×Q1612yes—
258C22×C4C2×Q1612yes—
259C22×C4AGL(1,ℤ/8)12yes—
260C22×C4#13712yes—
261C22×C4C22×Q812yes—
262C22×C4C2×#4112yes—
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C4H(5,𝔽2)12yes—
265C22×C4H(5,𝔽2)12yes—
266C22×C4H(5,𝔽2)11yescharacteristic
267C22×C42−1+412yes—
268C22×C42−1+412yes—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—88no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C42—42no—
307C42—42no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—42no—
313C42—42no—
314C23#53411yescharacteristic
315C23#53511yescharacteristic
316C23#53611yescharacteristic
317C23#54612yes—
318C23D4×Q812yes—
319C23C22×Q1611yescharacteristic
320C23C2×AGL(1,ℤ/8)11yescharacteristic
321C23C2×#13711yescharacteristic
322C23#57511yescharacteristic
323C23#57612yes—
324C23#57712yes—
325Q8—816no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#53812yes—
387C2×C4#54112yes—
388C2×C4#54212yes—
389C2×C4#54312yes—
390C2×C4#54412yes—
391C2×C4#54412yes—
392C2×C4#55111yescharacteristic
393C2×C4#55412yes—
394C2×C4#55712yes—
395C2×C4Q8211yescharacteristic
396C2×C4#56011yescharacteristic
397C2×C4#56212yes—
398C2×C4#56212yes—
399C2×C4#56212yes—
400C2×C4#56311yescharacteristic
401C2×C4#56312yes—
402C8—44no—
403C8—82no—
404C22#269212yes—
405C22#269711yescharacteristic
406C22#271312yes—
407C22#273411yescharacteristic
408C22#281611yescharacteristic
409C22#318911yescharacteristic
410C22#319311yescharacteristic
411C22#319911yescharacteristic
412C22#320011yescharacteristic
413C22#320911yescharacteristic
414C22#321411yescharacteristic
415C22#322412yes—
416C22#323011yescharacteristic
417C22#323711yescharacteristic
418C22#324112yes—
419C22#329912yes—
420C22#331412yes—
421C22#331812yes—
422C22#332612yes—
423C22Q8×Q1612yes—
424C22#338412yes—
425C22#338712yes—
426C22#338912yes—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—81no—
433C4—84no—
434C4—84no—
435C4—84no—
436C4—162no—
437C4#275512yes—
438C4#280612yes—
439C4#283112yes—
440C4#284112yes—
441C2#1714311yescharacteristic
442C2#3152412yes—
443C2#3154212yes—
444C2#3166911yescharacteristic
445C2#3168011yescharacteristic
446C2#3213612yes—
447C2#3215412yes—
448C2#3268411yescharacteristic
449C2#3269411yescharacteristic
450C2#3277411yescharacteristic
451C2#3277911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
62. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
#151962
#151282
#113842
#113772
#113462
#112932
#112192
#112072