Finite Group Databasebeta

← #482436

Subgroups — #482436

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#11908C214yesmaximal, complemented
5#11815C214yesmaximal
6#11795C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2920—21no—
10#2920—21no—
11#2567—21no—
12#2567—21no—
13#2567—21no—
14#2567—21no—
15#2567C412yes—
16#2567C412yes—
17#2567C2212yes—
18#2567C2212yes—
19C2×#395C412yes—
20C2×#395C2212yes—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#384—21no—
25C2×C4×C4⋊C4C411yescharacteristic
26#1821C2212yes—
27#1801—21no—
28#1801C2212yes—
29#1784—21no—
30C4×#101—21no—
31C4×#101C2212yes—
32#1467—21no—
33#1467C2212yes—
34#1463—21no—
35#1463C2212yes—
36#1461—21no—
37#1461C2212yes—
38#1354C22116yes—
39C23×Q8C2×C412yes—
40C23×Q8D411yescharacteristic
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C4×Q8—41no—
50C2×C4×Q8—41no—
51C2×C4×Q8—41no—
52C2×C4×Q8—41no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73#432—22no—
74#432—41no—
75#432—41no—
76#432—41no—
77#431—22no—
78#431—41no—
79#431—41no—
80#431—41no—
81#423—22no—
82#423—41no—
83C2×#101—22no—
84C2×#101—22no—
85#407—22no—
86#407—22no—
87C4×M16—22no—
88C4×M16—41no—
89#395—22no—
90#395—22no—
91#395—22no—
92#395—22no—
93#395C2×C414yes—
94#395C2×C414yes—
95#395C2×C414yes—
96#395C2314yes—
97#391—22no—
98#391—22no—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#384—22no—
110#384—22no—
111#384—22no—
112#384—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2×C412yes—
120#343—22no—
121#343C2314yes—
122#324—22no—
123#324—22no—
124#324—22no—
125#324C2314yes—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—84no—
134C22×Q8C22⋊C411yescharacteristic
135C22×Q8C22⋊C411yescharacteristic
136C22×Q8C22×C414yes—
137C22×Q8C2×D411yescharacteristic
138C22×Q8C2×D411yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×Q8—41no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×Q8—41no—
172C4×Q8—41no—
173C4×Q8—41no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—84no—
198C2×C4⋊C4—84no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2411yescharacteristic
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242#101—44no—
243#101—44no—
244C22⋊C8—24no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—42no—
248#97—42no—
249#97—42no—
250C4×C8—42no—
251C4×C8—42no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—24no—
257#95C22×C414yes—
258C24C2×C22⋊C411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×Q8#13714yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4#9911yescharacteristic
311C22×C4#9911yescharacteristic
312C22×C4#9911yescharacteristic
313C22×C4#9911yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328M16—44no—
329M16—44no—
330M16—84no—
331C2×C8—44no—
332C2×C8—44no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—44no—
382C42—44no—
383C23—22no—
384C23—21no—
385C23—22no—
386C23—21no—
387C23C2×#9911yescharacteristic
388C23C2×#9911yescharacteristic
389C23C22×C22⋊C411yescharacteristic
390Q8—44no—
391Q8—44no—
392Q8—44no—
393Q8—44no—
394Q8—88no—
395Q8—88no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—28no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—416no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4#35614yes—
449C2×C4C2×#9911yescharacteristic
450C2×C4C2×#9911yescharacteristic
451C2×C4C2×#9911yescharacteristic
452C2×C4C2×#9911yescharacteristic
453C2×C4#41912yes—
454C2×C4#41912yes—
455C2×C4#42112yes—
456C2×C4C2×#13712yes—
457C8—84no—
458C8—84no—
459C22—24no—
460C22—24no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22#212212yes—
468C22C2×#35612yes—
469C22C22×#9911yescharacteristic
470C22C2×#41911yescharacteristic
471C22C2×#42111yescharacteristic
472C4—21no—
473C4—21no—
474C4—22no—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—48no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—84no—
488C4C2×#35612yes—
489C4C2×#35612yes—
490C2—21no—
491C2—22no—
492C2—21no—
493C2—22no—
494C2#1757811yescharacteristic
495C2C2×#212211yescharacteristic
496C2C22×#35611yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11908C2semidirect
6. #11795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
#207272
#173652
#119082
#118152
#117952