Finite Group Databasebeta

← #240330

Subgroups — #240330

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1570C211yesmaximal, characteristic
6C2×#1542C211yesmaximal, characteristic
7#10486C214yesmaximal, complemented
8#10482C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3078—24no—
13#3078—24no—
14C2×#479—22no—
15C2×#477C2212yes—
16C2×#475C2212yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#419—21no—
23C22×#103—21no—
24C2×#384C2211yescharacteristic
25#1990C2212yes—
26#1984—22no—
27#1984C2214yes—
28#1956C2212yes—
29#1872—22no—
30#1817—22no—
31#1803—22no—
32#1803—22no—
33#1803C2214yes—
34C2×#340—21no—
35#1570—24no—
36C2×#334C2214yes—
37#1542—24no—
38C2×#333C2212yes—
39C2×#332C2212yes—
40#1296C2214yes—
41C23×Q8D411yescharacteristic
42C22×M16—21no—
43C23×C8—21no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533D414yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2312yes—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#479—216no—
72#477—24no—
73#477—24no—
74#475—24no—
75#475—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#428—28no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423D414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93#419—28no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—44no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8D412yes—
101C2×C4×C8—21no—
102C2×C4×C8C2312yes—
103#384—21no—
104#384—21no—
105#384—21no—
106#384C2312yes—
107#340—24no—
108#340—24no—
109#336—28no—
110#334—28no—
111#333—24no—
112#332—24no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8SD1614yes—
122C22×Q8C2×D412yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—24no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8C2×D414yes—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8C2×D412yes—
146C4⋊Q8C2×D412yes—
147#125—24no—
148#125—42no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×Q8—42no—
167#117—24no—
168#117—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—41no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4111yescharacteristic
206C2×C42C2411yescharacteristic
207#107—28no—
208#106—28no—
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—42no—
217C4⋊C8C2×D414yes—
218#103—28no—
219#103—28no—
220#103—48no—
221#103—84no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—42no—
227C24#12111yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—81no—
247C2×Q8—81no—
248C2×Q8C2×SD1614yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12012yes—
268C22×C4#12112yes—
269C22×C4#12112yes—
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4#12412yes—
276C22×C4C2×SD1612yes—
277C22×C4#13512yes—
278C22×C4#13712yes—
279C22×C4#13712yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283M16—48no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—84no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—44no—
321C42—42no—
322C42—42no—
323C42—44no—
324C42—44no—
325C42#12012yes—
326C42#12112yes—
327C42#12112yes—
328C42#12312yes—
329C42#12312yes—
330C42#12412yes—
331C42C22×D411yescharacteristic
332C42C22×D411yescharacteristic
333C42C2×#4111yescharacteristic
334C42C2×#4111yescharacteristic
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#39412yes—
340C23#46512yes—
341C23#47012yes—
342C23C2×#12111yescharacteristic
343Q8—24no—
344Q8—24no—
345Q8—42no—
346Q8—42no—
347Q8—48no—
348Q8—44no—
349Q8—82no—
350Q8—82no—
351Q8—82no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—164no—
380C2×C4#39412yes—
381C2×C4#39412yes—
382C2×C4#39412yes—
383C2×C4#44814yes—
384C2×C4#45214yes—
385C2×C4#46512yes—
386C2×C4#47012yes—
387C2×C4#48314yes—
388C2×C4#48514yes—
389C2×C4C2×#12011yescharacteristic
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×#12111yescharacteristic
392C2×C4C2×#12311yescharacteristic
393C2×C4C2×#12311yescharacteristic
394C2×C4C2×#12411yescharacteristic
395C2×C4C22×SD1611yescharacteristic
396C2×C4C2×#13511yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4C2×#13711yescharacteristic
399C2×C4#57712yes—
400C2×C4#57812yes—
401C8—44no—
402C8—44no—
403C8—82no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#203514yes—
412C22C2×#39411yescharacteristic
413C22#304311yescharacteristic
414C22#304411yescharacteristic
415C22C2×#46511yescharacteristic
416C22C2×#47011yescharacteristic
417C22#306811yescharacteristic
418C22#306911yescharacteristic
419C4—21no—
420C4—21no—
421C4—22no—
422C4—22no—
423C4—44no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—162no—
428C4#203514yes—
429C4C2×#44811yescharacteristic
430C4C2×#45211yescharacteristic
431C4#304612yes—
432C4#306312yes—
433C4C2×#48311yescharacteristic
434C4C2×#48511yescharacteristic
435C2—24no—
436C2#1939611yescharacteristic
437C2#1940211yescharacteristic
438C2#1943511yescharacteristic
439C2#1944111yescharacteristic
440C2C2×#203511yescharacteristic
441C2#2353311yescharacteristic
442C2#2354511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10486C2semidirect
8. #10482C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#218602
#218182
C2×#18032
C2×#15702
C2×#15422
#104862
#104822
#80432