Finite Group Databasebeta

← #239928

Subgroups — #239928

Order 512.

932 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
1C22×#475C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17746C211yesmaximal, characteristic
5#17742C211yesmaximal, characteristic
6#17449C211yesmaximal, characteristic
7#10451C214yesmaximal, complemented
8#10450C214yesmaximal, complemented
9#8082C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#475—22no—
13C2×#475—22no—
14C2×#475—22no—
15C2×#475—22no—
16C2×#475—22no—
17C2×#475C2214yes—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C22×#106—21no—
23C22×C4⋊C8C2211yescharacteristic
24C22×#103—21no—
25#1821—21no—
26#1801C2212yes—
27#1801C2212yes—
28#1784—21no—
29C4×#101—22no—
30#1573—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1565—22no—
34#1558—22no—
35#1556—22no—
36#1549C2214yes—
37#1548—22no—
38#1548C2212yes—
39#1548C2212yes—
40#1546—22no—
41#1491—22no—
42#1487—22no—
43#1479C2214yes—
44#1301C2214yes—
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C23×C8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—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⋊Q8D412yes—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8D412yes—
71C2×C4×Q8C2312yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#475—28no—
79#475—28no—
80#475—28no—
81#475—28no—
82#432—21no—
83#432—21no—
84#432—21no—
85#432—21no—
86#432—21no—
87#432—21no—
88#432—21no—
89#432D412yes—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431C2312yes—
97#431C2312yes—
98C2×#106—24no—
99C2×#106—24no—
100C2×#106—24no—
101#423—21no—
102#423D412yes—
103#423C2311yescharacteristic
104#423C2311yescharacteristic
105#423C2312yes—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8D412yes—
110C2×C4⋊C8D412yes—
111C2×C4⋊C8C2312yes—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#101—22no—
116C2×#101—22no—
117#407—21no—
118#407—21no—
119#407—21no—
120#407—21no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16—21no—
125#343—22no—
126#343—22no—
127#335—24no—
128#334—24no—
129#333—24no—
130#333—24no—
131#332—28no—
132Q8⋊C8—28no—
133C22×Q8—21no—
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—42no—
143C22×Q8—41no—
144C22×Q8—41no—
145C22×Q8—41no—
146C22×Q8—84no—
147C22×Q8C2×D412yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—41no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—44no—
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—22no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8C2×D414yes—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C2×D412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42—41no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42C2411yescharacteristic
237#106—216no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—22no—
249C4⋊C8—41no—
250C4⋊C8C2×D414yes—
251C4⋊C8C2×D414yes—
252#103—216no—
253#101—44no—
254#101—44no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263#97—22no—
264#97—41no—
265#97—41no—
266#97—41no—
267C4×C8—22no—
268C4×C8—22no—
269C4×C8—41no—
270C4×C8—41no—
271C24#12311yescharacteristic
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—44no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8#13714yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—84no—
312C22×C4—82no—
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12411yescharacteristic
327C22×C4#12411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331M16—44no—
332M16—42no—
333M16—42no—
334M16—42no—
335M16—42no—
336M16—82no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—44no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42#12011yescharacteristic
367C42#12011yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12111yescharacteristic
373C42#12111yescharacteristic
374C42#12311yescharacteristic
375C42#12311yescharacteristic
376C42#12411yescharacteristic
377C42#12411yescharacteristic
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C42C22×D411yescharacteristic
381C42C2×#4111yescharacteristic
382C23—22no—
383C23—22no—
384C23—22no—
385C23#39411yescharacteristic
386C23#39411yescharacteristic
387C23C2×#12311yescharacteristic
388Q8—24no—
389Q8—24no—
390Q8—44no—
391Q8—44no—
392Q8—42no—
393Q8—42no—
394Q8—84no—
395Q8—84no—
396Q8—84no—
397Q8—84no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4#39411yescharacteristic
430C2×C4#39411yescharacteristic
431C2×C4#39411yescharacteristic
432C2×C4#39411yescharacteristic
433C2×C4#39411yescharacteristic
434C2×C4#39411yescharacteristic
435C2×C4#47114yes—
436C2×C4C2×#12011yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C2×#13712yes—
443C2×C4#57512yes—
444C8—44no—
445C8—44no—
446C8—82no—
447C8—81no—
448C8—81no—
449C8—82no—
450C22—28no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22#204612yes—
458C22#204912yes—
459C22C2×#39411yescharacteristic
460C22#309512yes—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—84no—
471C4—82no—
472C4#204612yes—
473C4#300411yescharacteristic
474C4#300511yescharacteristic
475C4#305612yes—
476C4C2×#47112yes—
477C2—22no—
478C2—22no—
479C2—22no—
480C2#2348111yescharacteristic
481C2C2×#204611yescharacteristic
482C2C2×#204911yescharacteristic
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10451C2semidirect
8. #10450C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
#207272
#178072
#177462
#177422
#174492
#104512
#104502
#80822