Finite Group Databasebeta

← #193502

Subgroups — #193502

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic, complemented
2#34963C211yesmaximal, characteristic
3#20718C211yesmaximal, characteristic, complemented
4C2×#1486C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic, complemented
6#8943C214yesmaximal, complemented
7#8939C214yesmaximal, complemented
8#7899C211yesmaximal, characteristic, complemented
9#7894C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#2990—24no—
13#2990—24no—
14C2×#446—21no—
15C2×#435—22no—
16C2×C8×D4—21no—
17C2×#432C2212yescomplemented
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2212yescomplemented
23#2598—22no—
24#2580C412yes—
25#2535C412yes—
26#2487C412yes—
27#2482—21no—
28#2408C412yes—
29#2404—21no—
30#2398—21no—
31#2397—21no—
32C4×#121C2214yes—
33C4×#120—22no—
34#2283C2211yescharacteristic
35#1818—21no—
36#1798—22no—
37#1780—21no—
38C4×C22⋊C8C2214yescomplemented
39#1486—24no—
40#1478—24no—
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2214yes—
44#1279C2212yescomplemented
45#1279C2212yescomplemented
46#1279C2214yescomplemented
47C22×#41D411yescharacteristic
48C23×D4C2×C411yescharacteristic
49C24×C4—21no—
50C22×M16—21no—
51C23×C8—22no—
52C2×#121C2×C414yes—
53C2×#120—22no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41—22no—
57C4×#41—22no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4—24no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D412yes—
72C2×C4×D4C2312yes—
73C2×#117D412yes—
74C2×#117D411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#446—28no—
80#435—216no—
81C8×D4—28no—
82#432—24no—
83#432—24no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89#431D414yescomplemented
90#423—22no—
91#423D414yescomplemented
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2311yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—22no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109C2×C4×C8C2312yes—
110#400—22no—
111#399—22no—
112#399—22no—
113#397—22no—
114#394—22no—
115#393—22no—
116#392—22no—
117#392—22no—
118#390C2×C414yes—
119#389C2×C414yes—
120#388—24no—
121#386—22no—
122#386—24no—
123#386—24no—
124#385—24no—
125#385—22no—
126#383—21no—
127#382C2×C412yes—
128#381—21no—
129#380C2×C412yes—
130#380C2×C414yes—
131#379—22no—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—24no—
135#376—21no—
136#336—24no—
137D4⋊C8—24no—
138D4⋊C8—24no—
139D4⋊C8—28no—
140C25—21no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—44no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22⋊C412yes—
153C22×D4C22×C412yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×M16—24no—
168C2×M16—82no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—28no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—84no—
176#121—28no—
177#121—44no—
178#120—48no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—84no—
202C4×D4—84no—
203#117—22no—
204#117—22no—
205#117—24no—
206#117—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C414yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C412yes—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4112yes—
243C2×C42#4112yes—
244C2×C42C2411yescharacteristic
245C4⋊C8—28no—
246C4⋊C8—24no—
247C4⋊C8—21no—
248C4⋊C8—22no—
249C4⋊C8—21no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8C2×D412yes—
253C4⋊C8C2×D414yes—
254C4⋊C8C2×D412yes—
255C22⋊C8—28no—
256C22⋊C8—24no—
257C22⋊C8—48no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—28no—
261C4×C8—24no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266#95—22no—
267#95—22no—
268#95—24no—
269#95—22no—
270#95—24no—
271#95—22no—
272C24—22no—
273C24—44no—
274C24—44no—
275C24—42no—
276C24—82no—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C412yes—
279#41—48no—
280#41—48no—
281C2×Q8—22no—
282C2×Q8—42no—
283C2×D4—24no—
284C2×D4—28no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—48no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—42no—
300C2×D4—84no—
301C2×D4#13114yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4—82no—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12012yes—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4#12312yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347M16—84no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—44no—
353C2×C8—42no—
354C2×C8—44no—
355C2×C8—44no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—88no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—44no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4—84no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—42no—
387C42—42no—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12012yes—
391C42#12112yes—
392C42#12112yes—
393C42#12312yes—
394C42C22×D411yescharacteristic
395C42C22×D411yescharacteristic
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C23—24no—
399C23—22no—
400C23—24no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—48no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—84no—
412C23—84no—
413C23—816no—
414C23#38612yes—
415C23#42114yes—
416C23C22×C22⋊C411yescharacteristic
417Q8—44no—
418D4—28no—
419D4—24no—
420D4—24no—
421D4—48no—
422D4—48no—
423D4—42no—
424D4—42no—
425D4—816no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—88no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#38612yes—
464C2×C4#38612yes—
465C2×C4#38612yes—
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×#12011yescharacteristic
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4C2×#13112yes—
473C2×C4#56812yes—
474C2×C4#57612yes—
475C2×C4#57712yes—
476C8—42no—
477C8—44no—
478C8—42no—
479C8—82no—
480C8—82no—
481C22—28no—
482C22—22no—
483C22—24no—
484C22—22no—
485C22—22no—
486C22—44no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—48no—
491C22—44no—
492C22—816no—
493C22—84no—
494C22—82no—
495C22—82no—
496C22C2×#38611yescharacteristic
497C22#288012yes—
498C22#289712yes—
499C22C2×#42112yes—
500C4—22no—
501C4—21no—
502C4—21no—
503C4—22no—
504C4—22no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—84no—
509C4—82no—
510C4#292911yescharacteristic
511C4#293011yescharacteristic
512C4#293312yes—
513C4#300811yescharacteristic
514C4#301011yescharacteristic
515C4#309612yes—
516C2—44no—
517C2—42no—
518C2—84no—
519C2#1752114yes—
520C2#2089911yescharacteristic
521C2#2090211yescharacteristic
522C2#2091411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#34963C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2990C2semidirect
3. #20718C2semidirect
4. C2×#1486C2semidirect
5. C2×#1478C2semidirect
6. #8943C2semidirect
7. #8939C2semidirect
8. #7899C2semidirect
9. #7894C2semidirect
17. C2×#432C22semidirect
22. C22×C4⋊C8C22semidirect
38. C4×C22⋊C8C22semidirect
44. #1279C22semidirect
45. #1279C22semidirect
46. #1279C22semidirect
89. #431D4semidirect
91. #423D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#349632
#207182
C2×#14862
C2×#14782
#89432
#89392
#78992
#78942