Finite Group Databasebeta

← #380327

Subgroups — #380327

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34059C211yesmaximal, characteristic
2#21171C212yesmaximal
3C2×#1801C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#11342C212yesmaximal
6#11296C212yesmaximal
7#11136C214yesmaximal
8#10852C212yesmaximal
9C22×C4×Q8C2211yescharacteristic
10C2×#428C2214yes—
11C2×#423C2212yes—
12C22×#101C2212yes—
13#2320—22no—
14#2320—22no—
15#2316C2211yescharacteristic
16#2316C2211yescharacteristic
17#2313—22no—
18#2310C2212yes—
19#2295—22no—
20#2295—22no—
21C2×#391C2212yes—
22C2×#384C2211yescharacteristic
23#2283—22no—
24C4×#117C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1948—22no—
27#1948—22no—
28#1947—22no—
29#1947—22no—
30#1801C418yescomplemented
31C2×#343C2212yes—
32#1736C2214yes—
33C2×#339C2214yes—
34C2×#339C2214yes—
35#1441—22no—
36C4⋊#95—22no—
37C4⋊#95C2212yes—
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40C2×C4×Q8—22no—
41C2×C4×Q8—22no—
42C2×C4×Q8—24no—
43C2×C4×Q8C2×C414yes—
44C2×C4×Q8Q814yes—
45C2×#117C2312yes—
46C2×#117C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C42D412yes—
54C22×C42C2312yes—
55C22×C42C2311yescharacteristic
56#428—44no—
57#428—44no—
58C2×#107—24no—
59C2×#106—24no—
60#423—24no—
61#423C2×C418yes—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×#101—42no—
65C2×#101—42no—
66C2×#101C2×C418yes—
67#391—22no—
68#391—22no—
69#391—22no—
70#391—22no—
71#391—44no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#384—22no—
81#384—22no—
82#383—24no—
83#382—22no—
84#382—22no—
85#381—24no—
86#380—22no—
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4D412yes—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97#376—24no—
98#376—22no—
99C2×#95—24no—
100C2×#95—21no—
101C2×#95—24no—
102C43—22no—
103#343C2×C418yes—
104#340—24no—
105#340—24no—
106#339—44no—
107#339—44no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8C22×C412yes—
117C22×Q8C2×Q812yes—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16C22×C418yes—
131C22×C8—24no—
132C4×Q8—28no—
133C4×Q8—28no—
134C4×Q8—28no—
135C4×Q8—28no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
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—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C4⋊C418yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×Q812yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C4⋊C418yes—
187C2×C42C22×C412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×Q812yes—
191C2×C42#4114yes—
192#107—48no—
193#106—48no—
194C4⋊C8—28no—
195C4⋊C8—28no—
196#101—48no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—48no—
208#95—42no—
209#95—84no—
210C24C22×D411yescharacteristic
211C2×Q8—24no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C22×C4—22no—
228C22×C4—22no—
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—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
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—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C2×C4⋊C414yes—
258C22×C4C2×C4⋊C412yes—
259C22×C4C2×C4⋊C414yes—
260C22×C4C2×C4⋊C412yes—
261C22×C4#12011yescharacteristic
262C22×C4#12012yes—
263C22×C4#12011yescharacteristic
264C22×C4#12312yes—
265C22×C4#12312yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C42−1+411yescharacteristic
274C22×C42−1+411yescharacteristic
275C22×C42−1+412yes—
276M16—28no—
277M16—28no—
278M16—44no—
279M16—44no—
280C2×C8—28no—
281C2×C8—44no—
282C2×C8—44no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—216no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—48no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—82no—
298C4⋊C4—84no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C42—28no—
309C42—28no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—44no—
315C42—44no—
316C42—44no—
317C42—44no—
318C42—82no—
319C42—82no—
320C42C4×D414yes—
321C42C4×D414yes—
322C42C22⋊Q814yes—
323C42C22⋊Q814yes—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23C22×C4⋊C411yescharacteristic
329C23C2×#12011yescharacteristic
330C23C2×#12311yescharacteristic
331C23#53511yescharacteristic
332C23#53611yescharacteristic
333C23#53711yescharacteristic
334C23#53711yescharacteristic
335Q8—44no—
336Q8—44no—
337Q8—44no—
338Q8—44no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#38512yes—
384C2×C4#38514yes—
385C2×C4#38512yes—
386C2×C4#45412yes—
387C2×C4#45612yes—
388C2×C4C2×C4×D412yes—
389C2×C4#51811yescharacteristic
390C2×C4#51911yescharacteristic
391C2×C4#52012yes—
392C2×C4C2×C22⋊Q812yes—
393C2×C4#53512yes—
394C2×C4#53612yes—
395C2×C4#53712yes—
396C2×C4#53712yes—
397C2×C4#54012yes—
398C2×C4#54112yes—
399C2×C4#54212yes—
400C2×C4#54212yes—
401C2×C4#54411yescharacteristic
402C2×C4#54412yes—
403C2×C4#54411yescharacteristic
404C8—48no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22C2×#38511yescharacteristic
418C22#231611yescharacteristic
419C22#231711yescharacteristic
420C22#261511yescharacteristic
421C22#261611yescharacteristic
422C22#262411yescharacteristic
423C22#262511yescharacteristic
424C22C2×#45411yescharacteristic
425C22C2×#45611yescharacteristic
426C22#302111yescharacteristic
427C22#302211yescharacteristic
428C4—22no—
429C4—24no—
430C4—44no—
431C4—44no—
432C4—84no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4—162no—
438C4#185714yes—
439C4#232312yes—
440C4#232412yes—
441C4#262812yes—
442C4#262912yes—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2#1426411yescharacteristic
448C2C2×#185711yescharacteristic
449C2#2085811yescharacteristic
450C2#2967911yescharacteristic
451C2#2968111yescharacteristic
452C2#2970411yescharacteristic
453C2#2970611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1801C4semidirect

Maximal subgroups

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

subgroupindex
#340592
#211712
C2×#18012
#139262
#113422
#112962
#111362
#108522