Finite Group Databasebeta

← #241248

Subgroups — #241248

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37561C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic
5#10586C214yesmaximal, complemented
6#10584C214yesmaximal, complemented
7#8034C211yesmaximal, characteristic
8#8033C211yesmaximal, characteristic
9#8029C211yesmaximal, characteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2845C2212yes—
14#2844C2212yes—
15#2666—22no—
16#2660—21no—
17#2659—21no—
18#2647—21no—
19#2646—21no—
20#2570—21no—
21#2550—21no—
22#2442—21no—
23#2440—21no—
24#2438—22no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#2283C2211yescharacteristic
28C2×#382C2211yescharacteristic
29#1948—21no—
30#1947—21no—
31#1853—21no—
32#1852—21no—
33#1847C2212yes—
34#1836—22no—
35#1836C2214yes—
36#1836C2214yes—
37#1836C2212yes—
38#1834—21no—
39#1834—21no—
40#1736—22no—
41C2×#340—21no—
42C2×#340—21no—
43#1295C2212yes—
44#1295C2214yes—
45#1295C2214yes—
46#1295C2212yes—
47C22×M16—21no—
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×#125—21no—
51C2×C22⋊Q8—22no—
52C2×#117D412yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C42C2311yescharacteristic
59#432—22no—
60#432Q814yescomplemented
61C2×#107—22no—
62C2×#106—22no—
63C2×C4⋊C8—21no—
64C2×C22⋊C8—21no—
65C4×M16—22no—
66C4×M16Q814yescomplemented
67C2×#97—21no—
68C2×#97—22no—
69C2×#97Q814yescomplemented
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×#97C2312yes—
73C2×C4×C8—21no—
74#400—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#398—22no—
85#398—22no—
86#395—22no—
87#395—22no—
88#395—22no—
89#395—22no—
90#391—22no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#384—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#384C2312yes—
107#383—21no—
108#382—21no—
109#382—21no—
110#382—21no—
111#382D412yes—
112#382C2311yescharacteristic
113#382C2312yes—
114#382C2312yes—
115#382C2311yescharacteristic
116#381—21no—
117#380—22no—
118#380—21no—
119#380—21no—
120#380D411yescharacteristic
121#380D411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127#376—21no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—22no—
135#340—22no—
136C22×Q8—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×M16—24no—
144C2×M16—42no—
145C2×M16—42no—
146C22×C8—24no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—42no—
152C4⋊Q8—82no—
153#125—82no—
154C22⋊Q8—84no—
155#117—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
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—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q812yes—
200C2×C42C2×Q812yes—
201C2×C42C2×Q812yes—
202C2×C42#4112yes—
203C2×C42C2411yescharacteristic
204#107—82no—
205#106—82no—
206C4⋊C8—24no—
207C22⋊C8—24no—
208#97—24no—
209#97—22no—
210#97—22no—
211#97—22no—
212#97C2×Q814yes—
213#97C2×Q814yes—
214#97C2×Q814yes—
215C4×C8—24no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236C24#12011yescharacteristic
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—82no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4C22⋊Q812yes—
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4C4⋊Q812yes—
271C22×C4C4⋊Q812yes—
272C22×C4C4⋊Q812yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—48no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—48no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C42—42no—
321C42—42no—
322C42—82no—
323C42C22⋊Q812yes—
324C42C22⋊Q812yes—
325C42C22⋊Q812yes—
326C42C4⋊Q812yes—
327C42C4⋊Q812yes—
328C42C4⋊Q812yes—
329C42C22×Q811yescharacteristic
330C42C22×Q811yescharacteristic
331C42C22×Q811yescharacteristic
332C42C2×#4111yescharacteristic
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23#39512yes—
338C23#45412yes—
339C23#45512yes—
340C23C2×#12011yescharacteristic
341Q8—164no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—162no—
386C2×C4—162no—
387C2×C4—164no—
388C2×C4#39512yes—
389C2×C4#39512yes—
390C2×C4#39512yes—
391C2×C4#45412yes—
392C2×C4#45512yes—
393C2×C4C2×C22⋊Q811yescharacteristic
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C4⋊Q811yescharacteristic
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4C2×C4⋊Q811yescharacteristic
399C8—44no—
400C8—44no—
401C8—44no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22#206114yes—
409C22C2×#39511yescharacteristic
410C22C2×#45511yescharacteristic
411C22C2×#45511yescharacteristic
412C22C2×#45411yescharacteristic
413C22C2×#45411yescharacteristic
414C22#302111yescharacteristic
415C22#302211yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—82no—
425C4—82no—
426C4—81no—
427C4—82no—
428C4—81no—
429C4—81no—
430C4—82no—
431C4—81no—
432C4—162no—
433C4#206114yes—
434C2—24no—
435C2#1902011yescharacteristic
436C2#1902311yescharacteristic
437C2#1902711yescharacteristic
438C2#1902811yescharacteristic
439C2C2×#206111yescharacteristic
440C2C2×#206111yescharacteristic
441C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
5. #10586C2semidirect
6. #10584C2semidirect
60. #432Q8semidirect
66. C4×M16Q8semidirect
69. C2×#97Q8semidirect

Maximal subgroups

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

subgroupindex
#375612
#219872
#207642
C2×#18362
#105862
#105842
#80342
#80332
#80292