Finite Group Databasebeta

← #387971

Subgroups — #387971

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2#21049C212yesmaximal
3#11354C212yesmaximal, complemented
4#11045C212yesmaximal, complemented
5#10968C211yesmaximal, characteristic
6#10968C211yesmaximal, characteristic
7#10963C212yesmaximal, complemented
8#10857C212yesmaximal, complemented
9#10727C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#423C2212yes—
13C22×#101C2211yescharacteristic
14C22×#101C2212yes—
15C22×#101C2211yescharacteristic
16C2×#407C2212yes—
17C2×#407C2212yes—
18C2×C4×M16C2212yes—
19#2502—24no—
20#2502—24no—
21C2×#398C414yes—
22C2×#391C2212yes—
23C2×#385C414yes—
24C2×#384C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1924—24no—
27#1923—24no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2211yescharacteristic
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2211yescharacteristic
34C2×#343C2212yes—
35#1728C2212yes—
36#1728C2212yes—
37C23×Q8C2311yescharacteristic
38C24×C4C2×C412yes—
39C24×C4C2311yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—42no—
46C2×C22⋊Q8—41no—
47C2×C22⋊Q8—41no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—42no—
50C22×C4⋊C4—41no—
51C22×C4⋊C4—41no—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C418yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C42C2312yes—
60#423—24no—
61#423—42no—
62C2×C4⋊C8—24no—
63C2×#101—24no—
64C2×#101—24no—
65C2×#101—41no—
66C2×#101—42no—
67C2×#101—42no—
68C2×#101—41no—
69C2×#101—42no—
70#407—24no—
71#407—24no—
72#407—42no—
73#407—42no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C4×M16—24no—
77C4×M16—42no—
78C2×#97—22no—
79C2×C4×C8—22no—
80#398—24no—
81#398—28no—
82#398—24no—
83#391—22no—
84#391—22no—
85#391—24no—
86#385—24no—
87#385—28no—
88#385—24no—
89#384—22no—
90#384—22no—
91#384—24no—
92C4×C22⋊C4D414yes—
93C4×C22⋊C4D414yes—
94C2×#95C2×C418yes—
95C2×#95C2×C412yes—
96#343—42no—
97#343—42no—
98#343—42no—
99#343—42no—
100#343—41no—
101#343—42no—
102#343—41no—
103#336—22no—
104#336—24no—
105#336—22no—
106C25C22×C411yescharacteristic
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—84no—
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C414yes—
129C23×C4C22×C412yes—
130C23×C4C22×C414yes—
131C23×C4C2×D414yes—
132C23×C4C2411yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16#4114yes—
156C2×M16#4114yes—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—88no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C414yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D414yes—
198C4⋊C8—48no—
199#101—44no—
200#101—48no—
201#101—44no—
202C22⋊C8—48no—
203C22⋊C8—48no—
204#97—44no—
205C4×C8—44no—
206#95—28no—
207#95—28no—
208#95C22⋊C412yes—
209#95C22⋊C414yes—
210#95C22⋊C412yes—
211C24—21no—
212C24—21no—
213C24—24no—
214C24—22no—
215C24—22no—
216C24—42no—
217C24C2×C22⋊C412yes—
218C24C23×C411yescharacteristic
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—41no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—48no—
271C22×C4—44no—
272C22×C4—48no—
273C22×C4—88no—
274C22×C4—88no—
275C22×C4—88no—
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C414yes—
279C22×C4C22×D412yes—
280C22×C4C2×#4112yes—
281C22×C4C2×#4112yes—
282C22×C4H(5,𝔽2)12yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C42−1+412yes—
287M16—24no—
288M16—24no—
289M16—24no—
290M16—24no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—88no—
324C4⋊C4—88no—
325C22⋊C4—24no—
326C22⋊C4—28no—
327C22⋊C4—48no—
328C22⋊C4#10118yescomplemented
329C42—22no—
330C42—22no—
331C42—24no—
332C42—22no—
333C42—24no—
334C42—22no—
335C23—22no—
336C23—22no—
337C23—24no—
338C23—24no—
339C23—22no—
340C23—24no—
341C23—22no—
342C23—24no—
343C23—48no—
344C23—48no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23C2×#10114yes—
351C23C22×C22⋊C411yescharacteristic
352C23C4×#4112yes—
353C23#51811yescharacteristic
354C23#51811yescharacteristic
355C23#52012yes—
356Q8—82no—
357Q8—82no—
358Q8—82no—
359Q8—82no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—816no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4C2×#10118yes—
391C2×C4#53512yes—
392C2×C4#53512yes—
393C2×C4#53812yes—
394C2×C4#53912yes—
395C2×C4#54012yes—
396C2×C4#54212yes—
397C2×C4#54612yes—
398C2×C4#54612yes—
399C2×C4#54712yes—
400C2×C4#54712yes—
401C2×C4#55111yescharacteristic
402C2×C4#55312yes—
403C2×C4#55911yescharacteristic
404C2×C4#56011yescharacteristic
405C2×C4#56212yes—
406C2×C4#56311yescharacteristic
407C8—44no—
408C8—44no—
409C8—82no—
410C8—82no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—24no—
417C22—24no—
418C22—28no—
419C22—48no—
420C22—44no—
421C22—44no—
422C22—42no—
423C22—42no—
424C22—44no—
425C22#230911yescharacteristic
426C22#233512yes—
427C22#233912yes—
428C22#237411yescharacteristic
429C22#237611yescharacteristic
430C22C22×#10112yes—
431C22#288812yes—
432C4—28no—
433C4—42no—
434C4—44no—
435C4—82no—
436C4—84no—
437C4—82no—
438C4#270312yes—
439C4#270612yes—
440C4#271012yes—
441C4#271812yes—
442C2—22no—
443C2—21no—
444C2—21no—
445C2—44no—
446C2#1450611yescharacteristic
447C2#2179911yescharacteristic
448C2#2180111yescharacteristic
449C2#2248114yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11354C2semidirect
4. #11045C2semidirect
7. #10963C2semidirect
8. #10857C2semidirect
328. C22⋊C4#101semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#210492
#113542
#110452
#109682
#109632
#108572
#107272