Finite Group Databasebeta

← #201314

Subgroups — #201314

Order 512.

1187 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#34365C211yesmaximal, characteristic
3#21799C211yesmaximal, characteristic
4#17590C211yesmaximal, characteristic
5#17562C211yesmaximal, characteristic
6#9462C218yesmaximal
7#7941C211yesmaximal, characteristic
8#7940C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2868C412yescomplemented
14#2862C412yescomplemented
15#2660—22no—
16#2567—22no—
17#2546C414yescomplemented
18#2502—22no—
19#2434C2214yes—
20#2381—22no—
21#2369—22no—
22#2344—22no—
23#2344—22no—
24#2331—22no—
25#2331—22no—
26#2317—22no—
27#2310C2214yes—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1966—22no—
32#1966—22no—
33#1784C2214yes—
34#1518—22no—
35#1515—22no—
36#1511C2214yes—
37#1511C2214yes—
38#1506—22no—
39#1503—22no—
40#1284C2214yes—
41#1284C2214yes—
42C23×Q8C2312yes—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C418yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2312yes—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C412yes—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—24no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#432—21no—
78#432—21no—
79#432—21no—
80#432—21no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#423—21no—
86#423—21no—
87C2×#101—28no—
88#407C2312yes—
89#407C2312yes—
90C4×M16—21no—
91C4×M16—21no—
92#398—28no—
93#395—28no—
94#391—24no—
95#391—28no—
96#391—28no—
97#391C2×C418yes—
98#389—24no—
99#389—24no—
100#389—28no—
101#387—28no—
102#387—28no—
103#385—28no—
104#384—24no—
105#384—24no—
106#383—21no—
107#383D412yes—
108#382—24no—
109#382—21no—
110#382D412yes—
111#381—24no—
112#381C2312yes—
113#381C2312yes—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2312yes—
126#376—21no—
127#376—21no—
128#376D412yes—
129C43C2×C412yes—
130#328—28no—
131#328—28no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—84no—
138C22×Q8C22×C412yes—
139C22×Q8C22×C414yes—
140C22×Q8C2×D414yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D414yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—88no—
172#125—41no—
173#125—41no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—28no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—88no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4C22⋊C414yes—
217C2×C4⋊C4C22⋊C414yes—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C22⋊C414yes—
220C2×C4⋊C4C22×C412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—28no—
231C2×C42C22×C414yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248#101—88no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253#97—41no—
254#97—41no—
255C4×C8—41no—
256C4×C8—41no—
257#95—28no—
258#95—28no—
259#95—22no—
260#95—216no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95C2×D414yes—
269#95C2×D414yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—82no—
284C2×Q8—84no—
285C2×Q8—82no—
286C2×Q8—84no—
287C2×Q8—84no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4C2×C22⋊C418yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318M16—44no—
319M16—44no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—44no—
345C4⋊C4—816no—
346C4⋊C4—88no—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C42—28no—
353C42—44no—
354C42—44no—
355C42—44no—
356C42—42no—
357C42—42no—
358C42—44no—
359C42—88no—
360C42C4×D411yescharacteristic
361C42C4×D411yescharacteristic
362C42C4×D411yescharacteristic
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42C4×D411yescharacteristic
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12111yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C23—22no—
377C23—22no—
378C23—41no—
379C23C22×C22⋊C411yescharacteristic
380C23C2×#12011yescharacteristic
381C23C2×#12311yescharacteristic
382Q8—84no—
383Q8—84no—
384Q8—84no—
385Q8—84no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—28no—
401C2×C4—216no—
402C2×C4—28no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#45614yes—
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C8—82no—
439C8—82no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—44no—
448C22C2×#38611yescharacteristic
449C22#288312yes—
450C22C2×#45612yes—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—48no—
461C4—48no—
462C4—84no—
463C4—84no—
464C4—82no—
465C4—82no—
466C4#188612yes—
467C4#188612yes—
468C4#191212yes—
469C2—22no—
470C2—22no—
471C2—41no—
472C2C2×#188611yescharacteristic
473C2#2098511yescharacteristic
474C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2868C4semidirect
14. #2862C4semidirect
17. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352812
#343652
#217992
#175902
#175622
#94622
#79412
#79402