Finite Group Databasebeta

← #196392

Subgroups — #196392

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#13786C211yesmaximal, characteristic
6#9116C214yesmaximal
7#9112C214yesmaximal
8#8021C211yesmaximal, characteristic
9#8013C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2984—24no—
14C2×#446—21no—
15C2×#446—22no—
16C2×C8×Q8—21no—
17C2×#432C2212yes—
18C2×#432C2211yescharacteristic
19C2×#431C2212yes—
20C2×#431C2211yescharacteristic
21#2855C2214yes—
22#2634—22no—
23#2581C414yescomplemented
24#2568C411yescharacteristic, complemented
25#2568C411yescharacteristic, complemented
26#2546C411yescharacteristic, complemented
27#2546C411yescharacteristic, complemented
28#2503—21no—
29#2484—22no—
30#2434—21no—
31#2327—21no—
32#2327—21no—
33C2×#391—21no—
34#2283C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1506—24no—
38#1503—24no—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#328C2214yes—
42#1436—22no—
43C4⋊#95—21no—
44#1434C2214yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2214yes—
48C23×Q8D411yescharacteristic
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2312yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D412yes—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#446—216no—
81#446—28no—
82C8×Q8—28no—
83#432—22no—
84#432—24no—
85#432—24no—
86#432D414yes—
87#431—22no—
88#431—24no—
89#431—24no—
90#431D414yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2311yescharacteristic
98C2×#97—22no—
99C2×#97—21no—
100C2×C4×C8—22no—
101C2×C4×C8—21no—
102#400—22no—
103#400—22no—
104#400—24no—
105#398—22no—
106#398—22no—
107#395—24no—
108#391—28no—
109#391—21no—
110#391—21no—
111#391—21no—
112#391—21no—
113#391C2×C414yes—
114#391C2×C414yes—
115#389C2×C414yes—
116#389C2×C414yes—
117#387—24no—
118#387—24no—
119#387—24no—
120#387—24no—
121#385—24no—
122#383—21no—
123#382—21no—
124#382—21no—
125#382C2×C412yes—
126#381—24no—
127#381—21no—
128#380C2312yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—21no—
134#376—21no—
135#376—22no—
136#376—22no—
137C2×#95—24no—
138C2×#95—21no—
139C2×#95—24no—
140C2×#95C2312yes—
141C2×#95C2312yes—
142#328—24no—
143#328—24no—
144#328—28no—
145C22×Q8—24no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—42no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22⋊C412yes—
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170#125—42no—
171#125—42no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—24no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—42no—
193C4×Q8—42no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—82no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C4⋊C8—28no—
247C4⋊C8—28no—
248C4⋊C8—24no—
249C4⋊C8—24no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—22no—
253C22⋊C8C2×D412yes—
254C22⋊C8C2×D414yes—
255C22⋊C8C2×D412yes—
256#97—28no—
257#97—24no—
258C4×C8—28no—
259C4×C8—24no—
260#95—216no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—42no—
277#95—84no—
278C24C22×D411yescharacteristic
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—48no—
290C2×Q8—48no—
291C2×Q8—44no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8#10414yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—44no—
348C2×C8—42no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—48no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—48no—
382C42—44no—
383C42—42no—
384C42—42no—
385C42—84no—
386C42C4×D412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12012yes—
391C42#12012yes—
392C42#12312yes—
393C42#12312yes—
394C23—22no—
395C23—21no—
396C23—21no—
397C23C22×C22⋊C411yescharacteristic
398C23C2×#12111yescharacteristic
399C23C2×#12111yescharacteristic
400C23#57712yes—
401C23#57812yes—
402Q8—28no—
403Q8—48no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407Q8—84no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—48no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#38612yes—
461C2×C4#38612yes—
462C2×C4#38612yes—
463C2×C4#38612yes—
464C2×C4C2×#10414yes—
465C2×C4C2×#10412yes—
466C2×C4#42112yes—
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C2×#12011yescharacteristic
470C2×C4C2×#12311yescharacteristic
471C8—42no—
472C8—44no—
473C8—42no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—21no—
481C22—21no—
482C22C2×#38611yescharacteristic
483C22#288412yes—
484C22#289812yes—
485C22C22×#10411yescharacteristic
486C22C2×#42111yescharacteristic
487C22#305711yescharacteristic
488C22#305911yescharacteristic
489C22#306312yes—
490C4—22no—
491C4—24no—
492C4—22no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—84no—
502C4—84no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4#187114yes—
508C2—21no—
509C2—22no—
510C2—21no—
511C2#1754912yes—
512C2#1760112yes—
513C2C2×#187111yescharacteristic
514C2#2097411yescharacteristic
515C2#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
23. #2581C4semidirect
24. #2568C4semidirect
25. #2568C4semidirect
26. #2546C4semidirect
27. #2546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#352482
C2×#15062
C2×#15032
#137862
#91162
#91122
#80212
#80132