Finite Group Databasebeta

← #207265

Subgroups — #207265

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C4×#382C211yesmaximal, characteristic
4#9913C214yesmaximal
5#9909C212yesmaximal
6#9908C212yesmaximal
7#8017C214yesmaximal
8C2×#432C2212yes—
9#2864C412yescomplemented
10#2863—21no—
11#2863—21no—
12#2862C411yescharacteristic, complemented
13#2859C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2857C2211yescharacteristic
16#2855—21no—
17#2855—21no—
18#2851C2212yes—
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×C4⋊Q8C411yescharacteristic, complemented
21C4×#125C412yescomplemented
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24#2295—21no—
25#2295—21no—
26#2283C2212yes—
27C2×#382C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C43C2211yescharacteristic
30#1842—21no—
31#1831—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1767—21no—
35#1767—21no—
36C4⋊#95—21no—
37#1434—21no—
38#1434C2212yes—
39C4×#95—21no—
40C4×#95C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C22×C4⋊C4—22no—
51C22×C42—22no—
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#432—28no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8C2314yes—
64C2×#97—22no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#383—22no—
98#382—24no—
99#382C2×C412yes—
100#382C2×C412yes—
101#382C2×C412yes—
102#382Q818yescomplemented
103#381—22no—
104#380—21no—
105#380—21no—
106#380C2312yes—
107#380C2312yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—44no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C22⋊C4—22no—
116#376—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95C2314yes—
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43C2×C411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129C43D411yescharacteristic
130#336—22no—
131#336—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C23×C4—24no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—24no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145#125—44no—
146C22⋊Q8—44no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C4×Q8—44no—
150C4×Q8—44no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
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—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42—44no—
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×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q814yes—
213C2×C42#4114yes—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C4⋊C8—44no—
217C22⋊C8—28no—
218#97—44no—
219#95—22no—
220#95—22no—
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—28no—
232#95—22no—
233#95—22no—
234#95—24no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95C2×Q818yes—
239C24C22×D411yescharacteristic
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C22⋊Q814yes—
289C22×C4C22⋊Q814yes—
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4#13712yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×Q812yes—
295C22×C4C2×#4112yes—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—416no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—44no—
344C42—44no—
345C42—88no—
346C42C4×Q814yes—
347C42C4×Q814yes—
348C42#12412yes—
349C42#12412yes—
350C42C4⋊Q812yes—
351C42C4⋊Q812yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23C22×C22⋊C411yescharacteristic
357C23C2×C22⋊Q812yes—
358C23C2×AGL(1,ℤ/8)11yescharacteristic
359C23C2×#13711yescharacteristic
360C23#57512yes—
361Q8—84no—
362Q8—84no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
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—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4#39112yes—
424C2×C4#39112yes—
425C2×C4#39112yes—
426C2×C4#39112yes—
427C2×C4#41012yes—
428C2×C4#41012yes—
429C2×C4#41712yes—
430C2×C4#41712yes—
431C2×C4#42112yes—
432C2×C4#42112yes—
433C2×C4C2×C4×Q812yes—
434C2×C4C2×#12411yescharacteristic
435C2×C4C2×C4⋊Q811yescharacteristic
436C8—84no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22C2×#39111yescharacteristic
446C22C2×#41011yescharacteristic
447C22C2×#41011yescharacteristic
448C22C2×#41711yescharacteristic
449C22#289611yescharacteristic
450C22C2×#42111yescharacteristic
451C22C2×#42111yescharacteristic
452C22#307712yes—
453C22#307812yes—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—22no—
458C4—21no—
459C4—22no—
460C4—21no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—88no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4#151214yes—
474C2—22no—
475C2—22no—
476C2C2×#151211yescharacteristic
477C2#1756311yescharacteristic
478C2#1757911yescharacteristic
479C2#1759111yescharacteristic
480C2#2131211yescharacteristic
481C2#2132911yescharacteristic
482C2#2133211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2864C4semidirect
12. #2862C4semidirect
13. #2859C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect
21. C4×#125C4semidirect
102. #382Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
#208102
C4×#3822
#99132
#99092
#99082
#80172