Finite Group Databasebeta

← #877421

Subgroups — #877421

Order 512.

2354 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17077C214yesmaximal
2#16214C214yesmaximal, complemented
3#15636C214yesmaximal, complemented
4#15235C212yesmaximal, complemented
5#15125C214yesmaximal, complemented
6#14714C212yesmaximal, complemented
7#14471C214yesmaximal, complemented
8#13879C214yesmaximal, complemented
9#13871C212yesmaximal
10C4×#390C211yesmaximal, characteristic
11C22×C4⋊Q8C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14C2×#393C2218yes—
15C2×#391C2214yes—
16C2×#391C2214yes—
17C2×#390C2212yes—
18C2×#384C2212yes—
19C2×#382C2212yes—
20C2×#380C2212yes—
21C2×#380C2214yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2214yes—
24C2×C43C2211yescharacteristic
25#1446C2218yes—
26#1443C2218yes—
27#1442C2218yes—
28#1442C2218yes—
29#1440C2214yes—
30#1439C2218yes—
31#1439C2218yes—
32#1439C2218yes—
33#1438C2214yes—
34#1437C2212yes—
35#1437C2214yes—
36#1437C2212yes—
37#1436C2218yes—
38#1436C2214yes—
39#1436C2218yes—
40C4⋊#95C2212yes—
41C4⋊#95C2214yes—
42C4⋊#95C2214yes—
43C4⋊#95C2214yes—
44C4⋊#95C2214yes—
45C4⋊#95C2214yes—
46C4⋊#95C2212yes—
47C23×Q8C2314yes—
48C23×D4C2311yescharacteristic
49C24×C4C2312yes—
50C2×C4⋊Q8—28no—
51C2×C4⋊Q8—44no—
52C2×C4⋊Q8—44no—
53C2×C4×D4—42no—
54C2×C4×D4—42no—
55C2×C4×D4Q818yescomplemented
56C2×#117—48no—
57C2×#117—44no—
58C22×C4⋊C4C2314yes—
59C22×C4⋊C4C2314yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2318yes—
69C22×C22⋊C4C2314yes—
70C22×C22⋊C4C2312yes—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C22×C42C2314yes—
77C22×C42C2312yes—
78#393—416no—
79#391—216no—
80#391—48no—
81#390—28no—
82#384—44no—
83#382—44no—
84#380—48no—
85#380—44no—
86C4×C4⋊C4—42no—
87C4×C22⋊C4—216no—
88C2×#95C2318yes—
89C2×#95C2318yes—
90C2×#95C2318yes—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2318yes—
94C2×#95C2318yes—
95C2×#95C2318yes—
96C2×#95C2318yes—
97C2×#95C2318yes—
98C2×#95C2318yes—
99C2×#95C2314yes—
100C2×#95C2312yes—
101C43—42no—
102C25C2412yes—
103C22×Q8—28no—
104C22×Q8—28no—
105C22×Q8—28no—
106C22×Q8—44no—
107C22×Q8—44no—
108C22×Q8—44no—
109C22×Q8—44no—
110C22×D4—24no—
111C22×D4—41no—
112C22×D4—41no—
113C22×D4—42no—
114C22×D4C2×Q814yes—
115C23×C4—28no—
116C23×C4—24no—
117C23×C4—28no—
118C23×C4—24no—
119C23×C4C2×Q818yes—
120C23×C4C2418yes—
121C23×C4C2418yes—
122C23×C4C2414yes—
123C23×C4C2411yescharacteristic
124C23×C4C2412yes—
125C23×C4C2411yescharacteristic
126C23×C4C2412yes—
127C23×C4C2412yes—
128C23×C4C2411yescharacteristic
129C4⋊Q8—432no—
130C4×D4—416no—
131#117—432no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C2×Q814yes—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—48no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—48no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—48no—
168C2×C22⋊C4C2×Q818yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—48no—
178C2×C42—44no—
179C2×C42—42no—
180C2×C42—41no—
181C2×C42—41no—
182C2×C42—42no—
183C2×C42C2×Q814yes—
184C2×C42#4118yes—
185C2×C42#4118yes—
186#95—28no—
187#95—28no—
188#95—216no—
189#95—216no—
190#95—216no—
191#95—48no—
192#95—42no—
193#95—48no—
194#95—44no—
195#95—48no—
196#95—48no—
197#95—48no—
198#95—48no—
199C24—24no—
200C24—24no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—22no—
205C24—24no—
206C24—24no—
207C24C22×Q814yes—
208C24C2511yescharacteristic
209C2×Q8—416no—
210C2×Q8—416no—
211C2×Q8—416no—
212C2×Q8—88no—
213C2×Q8—88no—
214C2×Q8—88no—
215C2×Q8—88no—
216C2×D4—28no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—48no—
220C2×D4—48no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—216no—
241C22×C4—232no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—416no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4C22×Q812yes—
267C22×C4C22×Q814yes—
268C22×C4C22×Q812yes—
269C22×C4C22×Q812yes—
270C22×C4C2×#4114yes—
271C22×C4C2×#4114yes—
272C22×C4C2×#4114yes—
273C22×C4C2×#4114yes—
274C22×C4C2×#4114yes—
275C22×C4C2×#4114yes—
276C22×C4H(5,𝔽2)14yes—
277C22×C4H(5,𝔽2)12yes—
278C22×C4H(5,𝔽2)14yes—
279C22×C4H(5,𝔽2)12yes—
280C22×C42−1+414yes—
281C4⋊C4—44no—
282C4⋊C4—48no—
283C4⋊C4—416no—
284C4⋊C4—416no—
285C4⋊C4—416no—
286C4⋊C4—88no—
287C4⋊C4—88no—
288C22⋊C4—48no—
289C22⋊C4—416no—
290C22⋊C4—416no—
291C22⋊C4—416no—
292C22⋊C4—88no—
293C22⋊C4—816no—
294C42—216no—
295C42—28no—
296C42—48no—
297C42—44no—
298C42—44no—
299C42—416no—
300C42—48no—
301C23—24no—
302C23—24no—
303C23—24no—
304C23—28no—
305C23—28no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—42no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23C23×Q811yescharacteristic
323C23C22×#4112yes—
324C23C22×#4112yes—
325C23C2×H(5,𝔽2)11yescharacteristic
326C23C2×H(5,𝔽2)12yes—
327C23C2×H(5,𝔽2)11yescharacteristic
328C23C2×H(5,𝔽2)12yes—
329C23C2×2−1+412yes—
330C23#58512yes—
331Q8—832no—
332D4—416no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—432no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4#53314yes—
382C2×C4#53818yes—
383C2×C4#54314yes—
384C2×C4#55418yes—
385C2×C4#55518yes—
386C2×C4#55618yes—
387C2×C4#55914yes—
388C2×C4H(3,𝔽4)14yes—
389C22—44no—
390C22—48no—
391C22—48no—
392C22—44no—
393C22—44no—
394C22—84no—
395C22—84no—
396C22—84no—
397C22—84no—
398C22C2×#53311yescharacteristic
399C22C2×#53812yes—
400C22C2×#54311yescharacteristic
401C22#345811yescharacteristic
402C22C2×#55412yes—
403C22C2×#55512yes—
404C22C2×#55612yes—
405C22C2×#55911yescharacteristic
406C22#350312yes—
407C22#350512yes—
408C22#350612yes—
409C22#350612yes—
410C22#350712yes—
411C22#350812yes—
412C22#351112yes—
413C22C2×H(3,𝔽4)11yescharacteristic
414C22#353711yescharacteristic
415C22#353811yescharacteristic
416C22#354812yes—
417C22#355812yes—
418C22#356112yes—
419C4—28no—
420C4—28no—
421C4—24no—
422C4—44no—
423C4—42no—
424C4—88no—
425C4—88no—
426C4—84no—
427C4#280418yes—
428C2—84no—
429C2#3503311yescharacteristic
430C2#3619412yes—
431C2#3730711yescharacteristic
432C2#3949312yes—
433C2#3950312yes—
434C2#3976012yes—
435C2#4601612yes—
436C2C2×#280411yescharacteristic
437C2#5986312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #16214C2semidirect
3. #15636C2semidirect
4. #15235C2semidirect
5. #15125C2semidirect
6. #14714C2semidirect
7. #14471C2semidirect
8. #13879C2semidirect
55. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#170772
#162142
#156362
#152352
#151252
#147142
#144712
#138792
#138712
C4×#3902