Finite Group Databasebeta

← #479276

Subgroups — #479276

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15184C212yesmaximal
2#11357C212yesmaximal
3#11346C211yesmaximal, characteristic
4#11328C211yesmaximal, characteristic
5#11124C212yesmaximal
6#11067C212yesmaximal
7#11050C212yesmaximal
8#11024C212yesmaximal
9#11024C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C2×#391C2212yes—
16C2×#391C2212yes—
17C2×#391C2212yes—
18C2×#389C2212yes—
19C2×#384C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#340C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1441—22no—
31#1441—22no—
32#1441—22no—
33#1441—22no—
34C4⋊#95C2212yes—
35C23×Q8C2312yes—
36C22×Q16—41no—
37C22×Q16—42no—
38C22×Q16—42no—
39C22×Q16—41no—
40C22×Q16Q814yes—
41C23×C8C2312yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D411yescharacteristic
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42C2312yes—
53C22×C42C2311yescharacteristic
54C2×#107—42no—
55C2×#107—41no—
56C2×#103—24no—
57C2×#103—24no—
58C2×#103—42no—
59C2×#103—42no—
60C2×#103—42no—
61C2×#103—42no—
62C2×#103—42no—
63C2×#103—42no—
64C2×#103—42no—
65#391—24no—
66#391—42no—
67#391—42no—
68#391—42no—
69#391—42no—
70#391—42no—
71#389—42no—
72#389—42no—
73#384—24no—
74#384—42no—
75C4×C4⋊C4—41no—
76C4×C4⋊C4—41no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95D412yes—
87#340—22no—
88#340—42no—
89#340—42no—
90#340—41no—
91#340—42no—
92#336—41no—
93#336—41no—
94#336—42no—
95#336—42no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8C2×Q814yes—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2411yescharacteristic
119C2×Q16—48no—
120C2×Q16—44no—
121C2×Q16—44no—
122C2×Q16—48no—
123C2×Q16—44no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×Q812yes—
140C22×C8#4112yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
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—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4Q1614yescomplemented
172C2×C4⋊C4#4114yes—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183#107—44no—
184#103—48no—
185#103—48no—
186#103—48no—
187#95—42no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—82no—
196#95—82no—
197C24C22×D411yescharacteristic
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
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—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
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×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C2×Q1612yes—
276C22×C4C2×Q1614yes—
277C22×C4#13714yes—
278C22×C4C22×Q811yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)12yes—
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+411yescharacteristic
288Q16—416no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C4⋊C4—88no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—41no—
328C42—42no—
329C42—41no—
330C23#53411yescharacteristic
331C23#53512yes—
332C23#54612yes—
333C23D4×Q811yescharacteristic
334C23#55011yescharacteristic
335C23C22×Q1611yescharacteristic
336C23C2×#13712yes—
337C23#57511yescharacteristic
338C23#57611yescharacteristic
339C23#57712yes—
340C23#57811yescharacteristic
341Q8—44no—
342Q8—44no—
343Q8—44no—
344Q8—44no—
345Q8—82no—
346Q8—82no—
347Q8—82no—
348Q8—82no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
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×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#53812yes—
402C2×C4#53812yes—
403C2×C4#53911yescharacteristic
404C2×C4#54012yes—
405C2×C4#54212yes—
406C2×C4#54312yes—
407C2×C4#54311yescharacteristic
408C2×C4#55312yes—
409C2×C4#55411yescharacteristic
410C2×C4#55412yes—
411C2×C4#55511yescharacteristic
412C2×C4#56011yescharacteristic
413C2×C4#56011yescharacteristic
414C2×C4#56012yes—
415C2×C4#56012yes—
416C2×C4#56011yescharacteristic
417C2×C4#56212yes—
418C2×C4#56211yescharacteristic
419C8—42no—
420C8—42no—
421C8—82no—
422C22#269212yes—
423C22#269911yescharacteristic
424C22#271412yes—
425C22#273211yescharacteristic
426C22#281511yescharacteristic
427C22#318911yescharacteristic
428C22#319511yescharacteristic
429C22#320311yescharacteristic
430C22#320411yescharacteristic
431C22#320912yes—
432C22#321612yes—
433C22#322412yes—
434C22#322512yes—
435C22#329912yes—
436C22#331212yes—
437C22#331412yes—
438C22#332312yes—
439C22Q8×Q1611yescharacteristic
440C22#338712yes—
441C22#339011yescharacteristic
442C22#339511yescharacteristic
443C22#340012yes—
444C22#340311yescharacteristic
445C4—44no—
446C4—44no—
447C4—42no—
448C4—81no—
449C4—84no—
450C4—84no—
451C4—162no—
452C4—162no—
453C4#278012yes—
454C4#278411yescharacteristic
455C4#279511yescharacteristic
456C4#282611yescharacteristic
457C4#282912yes—
458C4#284211yescharacteristic
459C2#1719611yescharacteristic
460C2#3152412yes—
461C2#3154812yes—
462C2#3172211yescharacteristic
463C2#3173211yescharacteristic
464C2#3216812yes—
465C2#3218612yes—
466C2#3265011yescharacteristic
467C2#3266711yescharacteristic
468C2#3276211yescharacteristic
469C2#3276911yescharacteristic
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
171. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
#151842
#113572
#113462
#113282
#111242
#110672
#110502
#110242