Finite Group Databasebeta

← #196849

Subgroups — #196849

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#9153C214yesmaximal
7#9145C214yesmaximal
8#8026C211yesmaximal, characteristic
9#8014C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2855C2214yes—
15#2627—22no—
16#2582C414yescomplemented
17#2581—22no—
18#2568C412yescomplemented
19#2567C412yescomplemented
20#2565—22no—
21#2356—24no—
22#2356—24no—
23#2317—21no—
24#2310—21no—
25C2×#391—21no—
26C2×#387—22no—
27#2285C2211yescharacteristic
28C2×#383C2212yes—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2212yes—
33#1984—22no—
34#1984—22no—
35#1817C2214yes—
36#1518—24no—
37#1503—24no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×Q8D411yescharacteristic
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2312yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D412yes—
63C2×C4×Q8D412yes—
64C2×#117—21no—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C42—22no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2312yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#400—24no—
85#398—28no—
86#395—24no—
87#391—28no—
88#391—22no—
89#391—22no—
90#391—24no—
91#391C2×C418yes—
92#389—24no—
93#389C2×C418yes—
94#387—216no—
95#387—28no—
96#385—24no—
97#384—22no—
98#384C2×C412yes—
99#383—24no—
100#383—24no—
101#382—22no—
102#382—21no—
103#382—22no—
104#382D414yes—
105#381—24no—
106#380—21no—
107#380C2312yes—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4D414yes—
113#376—24no—
114#376—24no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95C2311yescharacteristic
120C2×#95C2312yes—
121C2×#95C2311yescharacteristic
122#336—28no—
123#328—24no—
124#328—24no—
125#328—28no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22⋊C412yes—
134C22×Q8C22⋊C412yes—
135C22×Q8C22×C412yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150#125—42no—
151#125—42no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—42no—
171#117—42no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C414yes—
194C2×C4⋊C4C22⋊C414yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C414yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—28no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—48no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22×C414yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D414yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C4⋊C8—42no—
221C4⋊C8—42no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C4×C8—42no—
227C4×C8—42no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—22no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—24no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95C2×D412yes—
241#95C2×D414yes—
242#95C2×D412yes—
243C24C22×D411yescharacteristic
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—44no—
253C2×Q8—42no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—82no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4C2×C22⋊C414yes—
289C22×C4C2×C22⋊C414yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4#12012yes—
295C22×C4#12012yes—
296C22×C4#12312yes—
297C22×C4#12312yes—
298C22×C4#13512yes—
299C22×C4#13512yes—
300C22×C4#13712yes—
301C22×C4#13712yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—48no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—48no—
328C4⋊C4—82no—
329C4⋊C4—88no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—416no—
339C42—44no—
340C42—44no—
341C42—48no—
342C42C4×D412yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42#12112yes—
347C42#12112yes—
348C42#12112yes—
349C42#12112yes—
350C23—22no—
351C23—22no—
352C23C22×C22⋊C411yescharacteristic
353C23C2×#12011yescharacteristic
354C23C2×#12311yescharacteristic
355C23C2×#13511yescharacteristic
356C23C2×#13511yescharacteristic
357C23C2×#13711yescharacteristic
358C23C2×#13711yescharacteristic
359Q8—48no—
360Q8—48no—
361Q8—82no—
362Q8—82no—
363Q8—84no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—28no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—84no—
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4#38612yes—
410C2×C4#38612yes—
411C2×C4#41612yes—
412C2×C4#41612yes—
413C2×C4#41912yes—
414C2×C4#41912yes—
415C2×C4#45214yes—
416C2×C4#45214yes—
417C2×C4#48514yes—
418C2×C4#48514yes—
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C8—82no—
424C8—82no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22C2×#38611yescharacteristic
434C22#288412yes—
435C22C2×#41611yescharacteristic
436C22C2×#41911yescharacteristic
437C22#290112yes—
438C22C2×#45211yescharacteristic
439C22C2×#45211yescharacteristic
440C22C2×#48511yescharacteristic
441C22C2×#48511yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—44no—
449C4—44no—
450C4—44no—
451C4—82no—
452C4—84no—
453C4—84no—
454C4—82no—
455C4—82no—
456C4#187314yes—
457C4#187314yes—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2#1761612yes—
463C2#1763212yes—
464C2C2×#187311yescharacteristic
465C2#2098011yescharacteristic
466C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2582C4semidirect
18. #2568C4semidirect
19. #2567C4semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#23562
#218182
C2×#15182
C2×#15032
#91532
#91452
#80262
#80142