Finite Group Databasebeta

← #206744

Subgroups — #206744

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#9867C214yesmaximal, complemented
5#9865C212yesmaximal, complemented
6#9864C212yesmaximal, complemented
7#7899C214yesmaximal
8C2×#431C2212yes—
9C2×#423C2212yes—
10C2×C4×M16—21no—
11#2864C412yes—
12#2863—21no—
13#2863—21no—
14#2862C411yescharacteristic
15#2859C411yescharacteristic
16#2857C411yescharacteristic
17#2857C411yescharacteristic
18#2855—21no—
19#2855—21no—
20#2851C412yes—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2295—21no—
27#2295—21no—
28#2283C2212yes—
29C2×C43C2211yescharacteristic
30#1842—21no—
31#1831—21no—
32#1797—21no—
33#1797—21no—
34#1767—21no—
35#1767C2212yes—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2212yes—
40#1728—21no—
41#1279C2218yes—
42#1279C2218yes—
43C22×M16—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#431—28no—
59#423—24no—
60#423Q818yescomplemented
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2314yes—
66C2×C22⋊C8—22no—
67C4×M16—44no—
68C2×#97—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2314yes—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#384C2×C411yescharacteristic
102#384C2×C411yescharacteristic
103#383—22no—
104#382C2×C412yes—
105#382C2×C412yes—
106#382C2×C412yes—
107#381—22no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111#380C2×C412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2312yes—
117C4×C4⋊C4C2312yes—
118C4×C22⋊C4—22no—
119#376—22no—
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—28no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—24no—
145C22×C8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148#125—44no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C4×Q8—44no—
152C4×Q8—42no—
153C4×Q8—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
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—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×Q814yes—
209C2×C42#4114yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—28no—
212C4⋊C8—24no—
213C4⋊C8—44no—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C4⋊C8C2×Q818yes—
217C22⋊C8—44no—
218#97—44no—
219C4×C8—28no—
220C4×C8—44no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235C24C2×C22⋊C411yescharacteristic
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C4×Q814yes—
277C22×C4C22⋊Q814yes—
278C22×C4#12412yes—
279C22×C4C4⋊Q812yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283M16—416no—
284C2×C8—28no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42C4×Q814yes—
335C42C22⋊Q814yes—
336C42#12412yes—
337C42C4⋊Q812yes—
338C42#13114yes—
339C42AGL(1,ℤ/8)12yes—
340C42#13712yes—
341C42C22×Q812yes—
342C42C2×#4112yes—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23#39112yes—
350C23C22×C22⋊C411yescharacteristic
351Q8—84no—
352Q8—84no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—28no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4#39112yes—
401C2×C4#39112yes—
402C2×C4#39112yes—
403C2×C4#40812yes—
404C2×C4#40812yes—
405C2×C4#41712yes—
406C2×C4#41712yes—
407C2×C4#42112yes—
408C2×C4#42112yes—
409C2×C4C2×C4×Q812yes—
410C2×C4C2×C22⋊Q812yes—
411C2×C4C2×#12411yescharacteristic
412C2×C4C2×C4⋊Q811yescharacteristic
413C2×C4C2×#13112yes—
414C2×C4#56812yes—
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C2×C4#57512yes—
418C8—48no—
419C8—84no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22C2×#39111yescharacteristic
428C22C2×#40811yescharacteristic
429C22#287911yescharacteristic
430C22C2×#41711yescharacteristic
431C22#289611yescharacteristic
432C22C2×#42111yescharacteristic
433C22C2×#42111yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4#149514yes—
451C4#296214yes—
452C4#307712yes—
453C4#307812yes—
454C2—42no—
455C2C2×#149511yescharacteristic
456C2#1748011yescharacteristic
457C2#1749311yescharacteristic
458C2#1749811yescharacteristic
459C2#2130211yescharacteristic
460C2#2130611yescharacteristic
461C2#2131211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9867C2semidirect
5. #9865C2semidirect
6. #9864C2semidirect
60. #423Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
#208082
C4×#4232
#98672
#98652
#98642
#78992