Finite Group Databasebeta

← #391871

Subgroups — #391871

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2#14682C212yesmaximal, complemented
3#11381C212yesmaximal, complemented
4#11043C218yesmaximal, complemented
5#10980C212yesmaximal
6C22×#124C411yescharacteristic, complemented
7C22×C22⋊Q8C2211yescharacteristic
8C22×#121C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C22×#99C2218yes—
11#2850—22no—
12#2850—22no—
13#2850—22no—
14#2850—22no—
15C2×#391C411yescharacteristic, complemented
16C2×#390C411yescharacteristic, complemented
17C2×#388C414yescomplemented
18C2×#386C2214yes—
19C2×#382—22no—
20C2×#382C411yescharacteristic, complemented
21C2×#380C2214yes—
22C2×#342C2218yes—
23C4⋊#95—22no—
24#1434—22no—
25#1434—22no—
26C22⋊#95C2214yes—
27C22⋊#95C2214yes—
28C23×Q8C2×C411yescharacteristic
29C23×D4C2×C411yescharacteristic
30C23×D4C2312yes—
31C24×C4C2311yescharacteristic
32C24×C4C2312yes—
33C2×#124—21no—
34C2×#124—22no—
35C2×#124—21no—
36C2×#124—44no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—44no—
40C2×#121—22no—
41C2×#121—22no—
42C2×#121—44no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4C2×C414yes—
49C22×C22⋊C4C2×C414yes—
50C22×C22⋊C4C2318yes—
51C22×C22⋊C4C2312yes—
52C22×C42—44no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C412yes—
55C22×C42C2×C411yescharacteristic
56C22×C42D412yes—
57C22×C42D412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C2×#99—48no—
61C2×#99—416no—
62#391—22no—
63#391—21no—
64#391—21no—
65#390—22no—
66#390—21no—
67#390—21no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#386—24no—
73#386—24no—
74#386—24no—
75#386—24no—
76#382—21no—
77#382—22no—
78#382—21no—
79#382—48no—
80#380—48no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95C2×C414yes—
87C2×#95C2×C414yes—
88#342—48no—
89C25C22×C412yes—
90C25C2411yescharacteristic
91C22×Q8—21no—
92C22×Q8—21no—
93C22×Q8—42no—
94C22×Q8—42no—
95C22×Q8—42no—
96C22×D4—24no—
97C22×D4—28no—
98C22×D4—28no—
99C22×D4—28no—
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—42no—
103C22×D4—42no—
104C22×D4—42no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—28no—
109C23×C4—44no—
110C23×C4—42no—
111C23×C4—44no—
112C23×C4—42no—
113C23×C4C22×C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C412yes—
116C23×C4C22×C412yes—
117C23×C4C22×C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126#124—44no—
127#124—48no—
128#124—44no—
129C22⋊Q8—48no—
130C22⋊Q8—48no—
131#121—48no—
132#121—48no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—48no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—216no—
152C2×C22⋊C4—216no—
153C2×C22⋊C4—28no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—48no—
157C2×C22⋊C4—48no—
158C2×C42—24no—
159C2×C42—28no—
160C2×C42—42no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C412yes—
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C412yes—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175#99—432no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—48no—
181#95—48no—
182#95—48no—
183#95—48no—
184#95—48no—
185C24—22no—
186C24—24no—
187C24—28no—
188C24—41no—
189C24—41no—
190C24—42no—
191C24—42no—
192C24—44no—
193C24—42no—
194C24C23×C411yescharacteristic
195C24C22×D412yes—
196C24H(5,𝔽2)14yes—
197C24H(5,𝔽2)14yes—
198C242−1+414yes—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—88no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—48no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—48no—
214C2×D4—48no—
215C2×D4—416no—
216C2×D4—48no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—88no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—416no—
260C22×C4—416no—
261C22×C4—416no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C4⋊C4—816no—
285C22⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—416no—
292C22⋊C4—416no—
293C42—24no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—22no—
298C42—41no—
299C42—42no—
300C42—41no—
301C42—42no—
302C42#9914yescomplemented
303C23—22no—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—22no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—42no—
317C23—44no—
318C23—42no—
319C23—44no—
320C23—42no—
321C23—48no—
322C23—84no—
323C23—84no—
324C23—88no—
325C23—88no—
326C23C22×C22⋊C411yescharacteristic
327C23#51812yes—
328C23#52014yes—
329C23#53512yes—
330C23#53514yes—
331C23#53612yes—
332C23#53714yes—
333C23H(3,𝔽4)14yes—
334C23#56218yes—
335C23#56314yes—
336Q8—84no—
337Q8—84no—
338D4—84no—
339D4—84no—
340D4—88no—
341D4—88no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—88no—
382C2×C4—88no—
383C2×C4—88no—
384C2×C4—88no—
385C2×C4—88no—
386C2×C4—88no—
387C2×C4—88no—
388C2×C4C2×#9914yes—
389C2×C4C2×#9912yes—
390C2×C4#41014yes—
391C2×C4#41012yes—
392C22—22no—
393C22—24no—
394C22—22no—
395C22—24no—
396C22—22no—
397C22—22no—
398C22—24no—
399C22—24no—
400C22—22no—
401C22—24no—
402C22—24no—
403C22—22no—
404C22—44no—
405C22—48no—
406C22—44no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22—84no—
411C22—84no—
412C22—168no—
413C22#230911yescharacteristic
414C22#231112yes—
415C22#237914yes—
416C22#281914yes—
417C22#282014yes—
418C22C22×#9911yescharacteristic
419C22C2×#41011yescharacteristic
420C22C2×#41012yes—
421C4—22no—
422C4—22no—
423C4—24no—
424C4—24no—
425C4—42no—
426C4—42no—
427C4—42no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—168no—
432C2—24no—
433C2—24no—
434C2—24no—
435C2—162no—
436C2#1466211yescharacteristic
437C2#2179311yescharacteristic
438C2#2179411yescharacteristic
439C2#2187412yes—
440C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14682C2semidirect
3. #11381C2semidirect
4. #11043C2semidirect
6. C22×#124C4semidirect
15. C2×#391C4semidirect
16. C2×#390C4semidirect
17. C2×#388C4semidirect
20. C2×#382C4semidirect
302. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#146822
#113812
#110432
#109802