Finite Group Databasebeta

← #196403

Subgroups — #196403

Order 512.

1531 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2#36626C211yesmaximal, characteristic
3C2×#1516C212yesmaximal
4#14684C211yesmaximal, characteristic, complemented
5#9114C214yesmaximal, complemented
6#9112C214yesmaximal, complemented
7#8024C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#431—22no—
11C2×#431—28no—
12C2×#431C2214yes—
13C22×C4⋊C8—21no—
14C22×C22⋊C8C2212yes—
15C22×C4×C8—21no—
16#2855C412yescomplemented
17#2855C2212yes—
18#2854C412yescomplemented
19#2854C2212yes—
20#2853—21no—
21#2853—21no—
22#2849—21no—
23#2849—21no—
24C4×#123—22no—
25C4×#123—22no—
26#2297C411yescharacteristic, complemented
27#2297C411yescharacteristic, complemented
28#2296C411yescharacteristic, complemented
29#2296C411yescharacteristic, complemented
30#2280C2211yescharacteristic
31C2×#381C2212yes—
32#1516—28no—
33C2×#328C2214yes—
34C2×#327C2214yes—
35#1434—22no—
36#1434C2214yes—
37C22⋊#95—21no—
38C22⋊#95—21no—
39#1294C2218yes—
40C22×#41C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45#525—24no—
46#525—24no—
47C2×#123—22no—
48C2×#123—22no—
49C2×#123—21no—
50C2×C22⋊Q8C2×C411yescharacteristic
51C2×C22⋊Q8C2311yescharacteristic
52C2×#121C2×C411yescharacteristic
53C2×#121C2311yescharacteristic
54C2×#120—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C42—24no—
62C22×C42—24no—
63C22×C42C2×C412yes—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D412yes—
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D412yes—
70C22×C42C2312yes—
71#431—216no—
72#431—216no—
73C2×C4⋊C8—28no—
74C2×C4⋊C8—24no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8D418yes—
78C2×C22⋊C8C2314yes—
79C2×C4×C8—28no—
80C2×C4×C8—24no—
81#391C2×C412yes—
82#391C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#386—21no—
98#386—21no—
99#386—21no—
100#386—21no—
101#386—22no—
102#386—22no—
103#385—21no—
104#385—21no—
105#385—21no—
106#385—21no—
107#385—22no—
108#385—22no—
109#381—22no—
110#381—22no—
111#381—21no—
112#381—24no—
113#381—24no—
114#380C2×C412yes—
115#380C2312yes—
116#379—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95—22no—
126C2×#95C2314yes—
127#328—28no—
128#327—28no—
129C25C2×D411yescharacteristic
130C2×#41—24no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8C22×C411yescharacteristic
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C411yescharacteristic
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—24no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C412yes—
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D414yes—
160C23×C4#4114yes—
161C23×C4C2411yescharacteristic
162C22×C8—28no—
163C22×C8—24no—
164C22×C8—44no—
165C22×C8—44no—
166#123—48no—
167#123—42no—
168#123—42no—
169#123—48no—
170C22⋊Q8—24no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173#121—24no—
174#121—22no—
175#121—22no—
176#120—44no—
177C4×D4—42no—
178C4×D4—44no—
179C4×D4—42no—
180#117—24no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—28no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—28no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22⋊C412yes—
227C2×C42C22⋊C412yes—
228C2×C42C22×C412yes—
229C2×C42C22×C412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C4⋊C8—216no—
233C22⋊C8—28no—
234C22⋊C8—24no—
235C22⋊C8C2×D418yes—
236C4×C8—216no—
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—
247#95—24no—
248#95—24no—
249#95—48no—
250#95—44no—
251#95—48no—
252C24—24no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—42no—
257C24C2×C22⋊C411yescharacteristic
258C24C2×C22⋊C411yescharacteristic
259C24#12114yes—
260C24C22×D411yescharacteristic
261#41—88no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×D4—22no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C22×C4—232no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—416no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—48no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—48no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C4×D414yes—
330C22×C4C4×D414yes—
331C22×C4#12012yes—
332C22×C4#12012yes—
333C22×C4#12114yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#13518yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D412yes—
340C22×C4C2×#4112yes—
341C2×C8—28no—
342C2×C8—44no—
343C2×C8—44no—
344C2×C8—416no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—88no—
371C42—24no—
372C42—24no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—42no—
382C42—42no—
383C42—42no—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—22no—
393C23—28no—
394C23—28no—
395C23—44no—
396C23—48no—
397C23—48no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23#38612yes—
404C23#38612yes—
405C23#41012yes—
406C23#41012yes—
407C23C22×C22⋊C411yescharacteristic
408C23C2×#12112yes—
409C23C2×#13514yes—
410Q8—82no—
411D4—82no—
412D4—84no—
413D4—84no—
414D4—84no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—216no—
418C2×C4—28no—
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—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—216no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—88no—
465C2×C4—84no—
466C2×C4—88no—
467C2×C4—84no—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4#41614yes—
471C2×C4#41614yes—
472C2×C4#42114yes—
473C2×C4C2×C4×D412yes—
474C2×C4C2×#12011yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C8—48no—
477C22—24no—
478C22—24no—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—24no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—416no—
489C22—42no—
490C22—42no—
491C22—82no—
492C22—81no—
493C22—82no—
494C22—81no—
495C22—81no—
496C22#189914yes—
497C22C2×#38611yescharacteristic
498C22C2×#41011yescharacteristic
499C22C2×#41011yescharacteristic
500C22C2×#41612yes—
501C22C2×#42112yes—
502C22#305014yes—
503C4—24no—
504C4—24no—
505C4—24no—
506C4—21no—
507C4—22no—
508C4—22no—
509C4—21no—
510C4—22no—
511C4—42no—
512C4—82no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4—81no—
517C4—164no—
518C2—24no—
519C2—24no—
520C2—42no—
521C2—161no—
522C2#1757112yes—
523C2#1757612yes—
524C2#2090211yescharacteristic
525C2#2096711yescharacteristic
526C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
4. #14684C2semidirect
5. #9114C2semidirect
6. #9112C2semidirect
7. #8024C2semidirect
16. #2855C4semidirect
18. #2854C4semidirect
26. #2297C4semidirect
27. #2297C4semidirect
28. #2296C4semidirect
29. #2296C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366262
C2×#15162
#146842
#91142
#91122
#80242