Finite Group Databasebeta

← #196393

Subgroups — #196393

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2#35249C211yesmaximal, characteristic
3C2×#1505C211yesmaximal, characteristic
4C2×#1502C211yesmaximal, characteristic
5#13786C211yesmaximal, characteristic, complemented
6#9118C214yesmaximal, complemented
7#9114C214yesmaximal, complemented
8#8021C211yesmaximal, characteristic, complemented
9#8013C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
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#2854C2214yes—
22#2632—22no—
23#2580C414yescomplemented
24#2567C411yescharacteristic, complemented
25#2567C411yescharacteristic, complemented
26#2548C411yescharacteristic, complemented
27#2548C411yescharacteristic, complemented
28#2501—21no—
29#2436—21no—
30#2406—22no—
31#2326—21no—
32#2326—21no—
33C2×#391—21no—
34#2283C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1505—24no—
38#1502—24no—
39C2×#327C2212yes—
40C2×#327C2212yes—
41C2×#327C2214yes—
42#1436—22no—
43C4⋊#95—21no—
44#1434C2214yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2214yes—
48C22×#41—21no—
49C23×Q8D411yescharacteristic
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57C2×#124—21no—
58C2×#124—21no—
59C2×#123—24no—
60C2×#121—21no—
61C2×#121C2312yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D412yes—
72C2×C4×D4—21no—
73C2×C4×D4C2×C412yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#446—216no—
82#446—28no—
83C8×Q8—28no—
84#432—22no—
85#432—24no—
86#432—24no—
87#432D414yes—
88#431—22no—
89#431—24no—
90#431—24no—
91#431D414yes—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8C2311yescharacteristic
97C2×C22⋊C8C2312yes—
98C2×C22⋊C8C2311yescharacteristic
99C2×#97—22no—
100C2×#97—21no—
101C2×C4×C8—22no—
102C2×C4×C8—21no—
103#399—24no—
104#395—22no—
105#395—22no—
106#394—22no—
107#394—22no—
108#393—24no—
109#391—28no—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390C2×C414yes—
115#390C2×C414yes—
116#389C2×C414yes—
117#389C2×C414yes—
118#388—24no—
119#388—24no—
120#386—24no—
121#385—24no—
122#385—24no—
123#384—21no—
124#384—21no—
125#383—21no—
126#382C2×C412yes—
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#327—24no—
143#327—24no—
144#327—28no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—82no—
148C2×#41—82no—
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—42no—
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22⋊C412yes—
156C22×Q8C22⋊C411yescharacteristic
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C412yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174#127—22no—
175#127—22no—
176#124—44no—
177#124—44no—
178#124—42no—
179#124—42no—
180#123—84no—
181#123—84no—
182#121—22no—
183#121—22no—
184#121—22no—
185#121—22no—
186#121—42no—
187#121—42no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—24no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—42no—
203C4×D4—42no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—82no—
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42#4112yes—
253C2×C42#4112yes—
254C4⋊C8—28no—
255C4⋊C8—28no—
256C4⋊C8—24no—
257C4⋊C8—24no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—22no—
261C22⋊C8C2×D412yes—
262C22⋊C8C2×D414yes—
263C22⋊C8C2×D412yes—
264#97—28no—
265#97—24no—
266C4×C8—28no—
267C4×C8—24no—
268#95—216no—
269#95—22no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—42no—
281#95—84no—
282C24—22no—
283C24C22×D411yescharacteristic
284#41—84no—
285#41—84no—
286#41—88no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—42no—
295C2×Q8—48no—
296C2×Q8—48no—
297C2×Q8—42no—
298C2×Q8—82no—
299C2×Q8#10414yes—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—82no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—84no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C2×C8—22no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—44no—
365C2×C8—42no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C22⋊C4—48no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—48no—
388C22⋊C4—84no—
389C22⋊C4—84no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—48no—
399C42—42no—
400C42—42no—
401C42—84no—
402C42C4×D412yes—
403C42C4×D412yes—
404C42C4×D412yes—
405C42C4×D412yes—
406C42#12012yes—
407C42#12012yes—
408C42#12312yes—
409C42#12312yes—
410C23—22no—
411C23—21no—
412C23—21no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23C22×C22⋊C411yescharacteristic
419C23C2×#12111yescharacteristic
420C23C2×#12111yescharacteristic
421C23#57612yes—
422C23#57712yes—
423Q8—28no—
424Q8—48no—
425Q8—44no—
426Q8—84no—
427D4—82no—
428D4—82no—
429D4—84no—
430D4—82no—
431D4—82no—
432D4—84no—
433D4—84no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—88no—
477C2×C4—82no—
478C2×C4—84no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4#38612yes—
487C2×C4#38612yes—
488C2×C4C2×#10414yes—
489C2×C4C2×#10412yes—
490C2×C4#42112yes—
491C2×C4C2×C4×D411yescharacteristic
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×#12011yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C8—42no—
496C8—44no—
497C8—42no—
498C22—22no—
499C22—22no—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—84no—
507C22—82no—
508C22—84no—
509C22—82no—
510C22—82no—
511C22C2×#38611yescharacteristic
512C22#288412yes—
513C22#289712yes—
514C22C22×#10411yescharacteristic
515C22C2×#42111yescharacteristic
516C22#305711yescharacteristic
517C22#305811yescharacteristic
518C22#306212yes—
519C4—22no—
520C4—24no—
521C4—22no—
522C4—22no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—84no—
531C4—84no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4#187114yes—
536C2—21no—
537C2—22no—
538C2—21no—
539C2—162no—
540C2#1755012yes—
541C2#1760212yes—
542C2C2×#187111yescharacteristic
543C2#2097511yescharacteristic
544C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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
1. C2×#2984C2semidirect
5. #13786C2semidirect
6. #9118C2semidirect
7. #9114C2semidirect
8. #8021C2semidirect
9. #8013C2semidirect
23. #2580C4semidirect
24. #2567C4semidirect
25. #2567C4semidirect
26. #2548C4semidirect
27. #2548C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#352492
C2×#15052
C2×#15022
#137862
#91182
#91142
#80212
#80132