Finite Group Databasebeta

← #189892

Subgroups — #189892

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic
4#8722C212yesmaximal, complemented
5#8721C212yesmaximal, complemented
6#8720C214yesmaximal, complemented
7#8029C214yesmaximal, complemented
8C23×C42C2211yescharacteristic
9C2×#407—21no—
10C22×C22⋊C8—21no—
11C2×C4×M16C2212yes—
12C22×#97C2212yes—
13#2285C411yescharacteristic
14#2285C411yescharacteristic
15#2285C411yescharacteristic
16#2285C411yescharacteristic
17#2285C2211yescharacteristic
18#2285C2211yescharacteristic
19#2283—21no—
20#2283—21no—
21#2283—21no—
22#2283—21no—
23#2283C2212yes—
24#2283C2212yes—
25#2280—21no—
26#2280—21no—
27#2280C412yes—
28C4×#117—21no—
29C4×#117—21no—
30C4×#117C412yes—
31#1749—21no—
32#1749—21no—
33#1749C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39#1728—21no—
40C2×#336—21no—
41#1295C2218yescomplemented
42#1295C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—22no—
47C23×C8—22no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117C2×C412yes—
53C2×#117C2×C412yes—
54C22×C42—28no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#407—28no—
68C2×C22⋊C8—28no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C4×M16—28no—
74C2×#97—28no—
75C2×#97—22no—
76C2×#97C2314yes—
77C2×#97C2314yes—
78C2×C4×C8—22no—
79#384C2×C411yescharacteristic
80#384C2×C411yescharacteristic
81#384C2×C411yescharacteristic
82#384C2×C411yescharacteristic
83#383—22no—
84#383—22no—
85#382—21no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382C2312yes—
90#382C2312yes—
91#381—22no—
92#381—22no—
93#381—22no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380C2×C412yes—
103#380C2×C412yes—
104#380C2×C412yes—
105#380C2×C412yes—
106#379—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125#376—22no—
126#376—22no—
127#376—22no—
128C43—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C25C22⋊C411yescharacteristic
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—28no—
162C2×M16—44no—
163C2×M16—44no—
164C22×C8—28no—
165C22×C8—24no—
166C22×C8—24no—
167C22×C8—44no—
168C22×C8—44no—
169#117—44no—
170#117—44no—
171#117—44no—
172#117—44no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4118yes—
226C2×C42C2411yescharacteristic
227C22⋊C8—28no—
228C22⋊C8—28no—
229C22⋊C8—28no—
230C22⋊C8—28no—
231#97—28no—
232#97—28no—
233#97—44no—
234C4×C8—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—41no—
253C24—41no—
254C24C2×C22⋊C411yescharacteristic
255C24C2×C22⋊C411yescharacteristic
256C24C2×C22⋊C411yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—416no—
311C22×C4—416no—
312C22×C4—416no—
313C22×C4#10414yes—
314C22×C4#10414yes—
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×C4#11714yes—
325C22×C4#12314yes—
326C22×C4#12314yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330M16—416no—
331C2×C8—28no—
332C2×C8—28no—
333C2×C8—44no—
334C2×C8—416no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—44no—
369C42—44no—
370C42—44no—
371C42—44no—
372C42—44no—
373C42—44no—
374C42#11714yes—
375C42#12314yes—
376C42#12314yes—
377C42C2×#4114yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—28no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23#38112yes—
402C23#41312yes—
403C23C2×#10412yes—
404C23C2×#10412yes—
405C23#42112yes—
406C23C22×C22⋊C411yescharacteristic
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—21no—
411C2×C4—21no—
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—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4#38112yes—
462C2×C4#38112yes—
463C2×C4#38112yes—
464C2×C4#41312yes—
465C2×C4C2×#10412yes—
466C2×C4C2×#10412yes—
467C2×C4C2×#10412yes—
468C2×C4C2×#10412yes—
469C2×C4#42112yes—
470C2×C4C2×#11712yes—
471C2×C4C2×#12312yes—
472C2×C4C2×#12312yes—
473C8—48no—
474C8—48no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—416no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22#182614yes—
497C22C2×#38111yescharacteristic
498C22C2×#41311yescharacteristic
499C22C2×#41311yescharacteristic
500C22C22×#10411yescharacteristic
501C22C22×#10411yescharacteristic
502C22C2×#42111yescharacteristic
503C22C2×#42111yescharacteristic
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4#182614yes—
520C2—24no—
521C2—42no—
522C2—42no—
523C2#1763811yescharacteristic
524C2#1764611yescharacteristic
525C2#1765111yescharacteristic
526C2#1766211yescharacteristic
527C2#2069411yescharacteristic
528C2C2×#182611yescharacteristic
529C2C2×#182611yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8722C2semidirect
5. #8721C2semidirect
6. #8720C2semidirect
7. #8029C2semidirect
41. #1295C22semidirect
42. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#340122
C4×#4072
C2×#17492
#87222
#87212
#87202
#80292