Finite Group Databasebeta

← #196373

Subgroups — #196373

Order 512.

1547 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36626C211yesmaximal, characteristic
3C2×#1501C212yesmaximal
4C4×#381C211yesmaximal, characteristic, complemented
5#9118C214yesmaximal, complemented
6#9116C214yesmaximal, complemented
7#8012C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432—22no—
11C2×#432—28no—
12C2×#432C2214yes—
13C22×C4⋊C8—21no—
14C22×C22⋊C8C2212yes—
15C22×#97—21no—
16#2855C411yescharacteristic, complemented
17#2855C411yescharacteristic, complemented
18#2855C2212yes—
19#2854C411yescharacteristic, complemented
20#2854C411yescharacteristic, complemented
21#2854C2212yes—
22#2853—22no—
23#2849—21no—
24#2849—21no—
25C4×#123—21no—
26C4×#123—21no—
27C4×#123—22no—
28#2297C412yescomplemented
29#2296C412yescomplemented
30#2280C2211yescharacteristic
31C2×#381—21no—
32C2×C4×C22⋊C4C2212yes—
33#1501—28no—
34C2×#328C2214yes—
35C2×#327C2214yes—
36#1434C2214yes—
37C22⋊#95—21no—
38C4×#95—22no—
39#1294C2218yes—
40C22×#41C2×C411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45#525—24no—
46#525—24no—
47C2×#123—21no—
48C2×#123—21no—
49C2×#123—22no—
50C2×#123—21no—
51C2×C22⋊Q8C2×C411yescharacteristic
52C2×C22⋊Q8C2311yescharacteristic
53C2×#121C2×C411yescharacteristic
54C2×#121C2311yescharacteristic
55C2×#120—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C412yes—
60C22×C22⋊C4—22no—
61C22×C42—24no—
62C22×C42—24no—
63C22×C42—22no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2312yes—
72#432—216no—
73#432—216no—
74C2×C4⋊C8—28no—
75C2×C4⋊C8—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8D418yes—
79C2×C22⋊C8C2314yes—
80C2×#97—28no—
81C2×#97—24no—
82#391C2×C412yes—
83#391C2×C412yes—
84#390C2×C412yes—
85#390C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386—21no—
101#386—21no—
102#386—22no—
103#385—22no—
104#385—22no—
105#385—21no—
106#385—21no—
107#385—22no—
108#381—28no—
109#381—21no—
110#381—21no—
111#381—22no—
112#381—21no—
113#380C2×C411yescharacteristic
114#380C2×C411yescharacteristic
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—
123C4×C22⋊C4—24no—
124C4×C22⋊C4—24no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95C2314yes—
128#328—28no—
129#327—28no—
130C25C2×D411yescharacteristic
131C2×#41—24no—
132C2×#41—44no—
133C22×Q8C22×C411yescharacteristic
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C411yescharacteristic
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—44no—
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C411yescharacteristic
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—48no—
165#123—48no—
166#123—44no—
167#123—44no—
168#123—42no—
169#123—42no—
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—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
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×C42—24no—
209C2×C42—28no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—28no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—48no—
223C2×C42—44no—
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—
236#97—216no—
237#95—216no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—24no—
250#95—24no—
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—42no—
264C2×D4—22no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—416no—
309C22×C4—416no—
310C22×C4—48no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—48no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4#10414yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C4×D414yes—
324C22×C4C4×D414yes—
325C22×C4#12012yes—
326C22×C4#12012yes—
327C22×C4#12114yes—
328C22×C4#12312yes—
329C22×C4#12312yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C2×#4112yes—
334C2×C8—28no—
335C2×C8—48no—
336C2×C8—416no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—416no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C42—24no—
364C42—24no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—24no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—42no—
374C42—44no—
375C42—88no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—28no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—28no—
387C23—44no—
388C23—48no—
389C23—48no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23#38612yes—
396C23#38612yes—
397C23C2×#10412yes—
398C23#42112yes—
399C23C22×C22⋊C411yescharacteristic
400C23C2×#12112yes—
401C23#57514yes—
402Q8—82no—
403D4—82no—
404D4—84no—
405D4—84no—
406D4—84no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—416no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—84no—
454C2×C4—88no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4#38612yes—
458C2×C4#38612yes—
459C2×C4#41014yes—
460C2×C4C2×#10414yes—
461C2×C4C2×C4×D412yes—
462C2×C4C2×#12011yescharacteristic
463C2×C4C2×#12311yescharacteristic
464C8—48no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—416no—
477C22—42no—
478C22—42no—
479C22—82no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22#187114yes—
485C22C2×#38611yescharacteristic
486C22C2×#41012yes—
487C22#289612yes—
488C22C22×#10411yescharacteristic
489C22C2×#42111yescharacteristic
490C22#305414yes—
491C4—24no—
492C4—21no—
493C4—22no—
494C4—21no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—88no—
500C4—82no—
501C4—82no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4#189914yes—
506C2—24no—
507C2—24no—
508C2—42no—
509C2—161no—
510C2#1754312yes—
511C2#1756012yes—
512C2C2×#187111yescharacteristic
513C2#2097011yescharacteristic
514C2C2×#189911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#432C2semidirect
4. C4×#381C2semidirect
5. #9118C2semidirect
6. #9116C2semidirect
7. #8012C2semidirect
16. #2855C4semidirect
17. #2855C4semidirect
19. #2854C4semidirect
20. #2854C4semidirect
28. #2297C4semidirect
29. #2296C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366262
C2×#15012
C4×#3812
#91182
#91162
#80122