Finite Group Databasebeta

← #190195

Subgroups — #190195

Order 512.

1085 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34379C211yesmaximal, characteristic
2#17396C211yesmaximal, characteristic
3#17396C211yesmaximal, characteristic
4#8759C214yesmaximal, complemented
5#8756C214yesmaximal, complemented
6#8108C211yesmaximal, characteristic, complemented
7#8107C211yesmaximal, characteristic, complemented
8#8104C211yesmaximal, characteristic, complemented
9C2×#1305C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C2×#407C2211yescharacteristic
15#2377—22no—
16#2361—22no—
17#2345C412yes—
18D4×C22⋊C4C412yes—
19D4×C22⋊C4C414yes—
20#2325—22no—
21#2324—22no—
22#2315C2212yes—
23#2312C2212yes—
24#2280C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1471—22no—
27#1471—22no—
28#1471—22no—
29#1471—22no—
30#1462C2214yes—
31#1462C2214yes—
32#1305C2212yes—
33#1305C2212yes—
34#1305C2214yes—
35#1305C2214yes—
36#1305C2212yes—
37#1305C2212yes—
38C23×D4C2×C412yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—21no—
42C22×M16—21no—
43C22×M16—21no—
44C22×M16—21no—
45C2×#121—21no—
46C2×#121—21no—
47C2×#121—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121D412yes—
54C2×#121D412yes—
55C2×#121D412yes—
56C2×#121D412yes—
57C2×C4×D4—22no—
58C2×C4×D4—24no—
59C2×#117—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2312yes—
65#407—21no—
66#407—21no—
67#407—21no—
68#407—21no—
69#407—21no—
70#407—21no—
71#407—21no—
72#407—21no—
73#407—21no—
74#407—21no—
75#407—21no—
76#407—21no—
77#407C2312yes—
78#407C2312yes—
79#407C2312yes—
80#407C2312yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89#390—22no—
90#390—24no—
91#389—22no—
92#388—24no—
93#388—24no—
94#387—24no—
95#386—24no—
96#386—24no—
97#386—24no—
98#385—24no—
99#385—24no—
100#381—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380C2×C414yes—
105#380C2×C412yes—
106#380C2311yescharacteristic
107#380C2312yes—
108#380C2311yescharacteristic
109#379C2×C414yes—
110#379C2×C412yes—
111C4×C4⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4C2×C414yes—
114C4×C22⋊C4C2×C412yes—
115C2×#95—21no—
116#323—28no—
117#323—28no—
118C25C22×C412yes—
119C25C22×C411yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—82no—
129C22×D4—84no—
130C22×D4C22⋊C414yes—
131C22×D4C22⋊C412yes—
132C22×D4C22⋊C412yes—
133C22×D4C22⋊C414yes—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—82no—
137C23×C4C22×C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4#4111yescharacteristic
146C23×C4#4111yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4#4111yescharacteristic
151C23×C4C2411yescharacteristic
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—41no—
157C2×M16—41no—
158C2×M16—41no—
159C2×M16—41no—
160C2×M16—82no—
161C2×M16—82no—
162C2×M16—82no—
163C2×M16—82no—
164C22×C8—41no—
165C22×C8—41no—
166C22×C8—41no—
167C22×C8—41no—
168#121—22no—
169#121—22no—
170#121—22no—
171#121—22no—
172#121—22no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182#121—22no—
183#121—22no—
184C4×D4—84no—
185C4×D4—88no—
186#117—82no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—84no—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C22×C414yes—
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C2×D414yes—
215C2×C22⋊C4C2×D414yes—
216C2×C42—22no—
217C2×C42—24no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—21no—
240#95—21no—
241C24—22no—
242C24—22no—
243C24—41no—
244C24—42no—
245C24—41no—
246C24—42no—
247C24—42no—
248C24—81no—
249C24—82no—
250C24—82no—
251C24C2×C22⋊C411yescharacteristic
252C24C2×C22⋊C414yes—
253C24C2×C22⋊C411yescharacteristic
254C24#11711yescharacteristic
255C24#11711yescharacteristic
256C24#11711yescharacteristic
257C24#11711yescharacteristic
258C24C23×C411yescharacteristic
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—44no—
268C2×D4—48no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—84no—
275C2×D4—84no—
276C2×D4—84no—
277C2×D4—84no—
278C2×D4—84no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—84no—
302C22×C4—82no—
303C22×C4—84no—
304C22×C4—84no—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C414yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4#11711yescharacteristic
309C22×C4#11711yescharacteristic
310C22×C4#11711yescharacteristic
311C22×C4#11711yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12311yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12311yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334M16—82no—
335M16—82no—
336M16—82no—
337M16—82no—
338M16—82no—
339M16—82no—
340M16—82no—
341M16—82no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—82no—
347C2×C8—82no—
348C2×C8—82no—
349C2×C8—82no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—84no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—82no—
368C22⋊C4—84no—
369C22⋊C4—82no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C42—44no—
373C42—84no—
374C23—22no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—22no—
381C23—21no—
382C23—21no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—84no—
410C23—84no—
411C23—164no—
412C23—164no—
413C23#38111yescharacteristic
414C23#38111yescharacteristic
415C23#38111yescharacteristic
416C23#38111yescharacteristic
417C23C22×C22⋊C411yescharacteristic
418C23C2×#11711yescharacteristic
419C23C2×#11711yescharacteristic
420D4—44no—
421D4—48no—
422D4—48no—
423D4—44no—
424D4—82no—
425D4—82no—
426D4—84no—
427D4—84no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—164no—
470C2×C4—168no—
471C2×C4#38111yescharacteristic
472C2×C4#38111yescharacteristic
473C2×C4#38111yescharacteristic
474C2×C4#38111yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12311yescharacteristic
478C2×C4C2×#12311yescharacteristic
479C8—82no—
480C8—82no—
481C8—82no—
482C8—82no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—81no—
510C22—81no—
511C22—82no—
512C22—82no—
513C22—84no—
514C22—84no—
515C22—82no—
516C22—82no—
517C22—164no—
518C22—164no—
519C22#182912yes—
520C22C2×#38111yescharacteristic
521C22#288811yescharacteristic
522C22#288911yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—44no—
531C4—82no—
532C4—82no—
533C4—84no—
534C4#182812yes—
535C2—21no—
536C2—21no—
537C2—22no—
538C2—41no—
539C2—41no—
540C2—41no—
541C2—41no—
542C2—41no—
543C2—84no—
544C2#2069511yescharacteristic
545C2C2×#182811yescharacteristic
546C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8759C2semidirect
5. #8756C2semidirect
6. #8108C2semidirect
7. #8107C2semidirect
8. #8104C2semidirect
9. C2×#1305C2semidirect

Maximal subgroups

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

subgroupindex
#343792
#173962
#87592
#87562
#81082
#81072
#81042
C2×#13052