Finite Group Databasebeta

← #191120

Subgroups — #191120

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13949C211yesmaximal, characteristic
4#8717C211yesmaximal, characteristic
5#8717C211yesmaximal, characteristic
6#8578C212yesmaximal
7#8572C212yesmaximal
8#8333C211yesmaximal, characteristic
9#8333C211yesmaximal, characteristic
10#8221C212yesmaximal
11C4×#328C212yesmaximal
12C2×#533C2211yescharacteristic
13#2370—22no—
14#2369C412yescomplemented
15#2365C412yescomplemented
16#2355—22no—
17#2355C412yescomplemented
18Q8×C4⋊C4C412yescomplemented
19#2329—21no—
20#2329—21no—
21#2317—21no—
22C4×C22⋊Q8C2212yes—
23C4×C22⋊Q8C2212yes—
24C2×#387—21no—
25#2285C2211yescharacteristic
26#2274—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1916—22no—
29#1916—22no—
30#1817C2211yescharacteristic
31#1817C2211yescharacteristic
32#1797C2212yes—
33C4×C22⋊C8C2212yes—
34C2×#328C2212yes—
35C2×#328C2212yes—
36C2×#328C2212yes—
37C2×#328C2212yes—
38#1442—21no—
39#1441—21no—
40#1436—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43#1434C2211yescharacteristic
44#1434C2211yescharacteristic
45#1434C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#533—24no—
52#533—24no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2312yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C4×C8—22no—
76#391—22no—
77#391—22no—
78#391C2×C412yes—
79#391C2×C412yes—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389C2×C412yes—
85#389C2×C412yes—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—44no—
93#385—22no—
94#385—22no—
95#384—22no—
96#384C2×C411yescharacteristic
97#384C2×C411yescharacteristic
98#382—22no—
99#382—21no—
100#380—21no—
101#380C2312yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—22no—
139#328—24no—
140#328—24no—
141#328—24no—
142#328—24no—
143C22×Q8—21no—
144C22×Q8C22×C412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160#125—28no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8#4114yes—
166C22⋊Q8#4114yes—
167C4×Q8—28no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—82no—
171C4×Q8—82no—
172#117—24no—
173#117—24no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4SD1614yes—
202C2×C4⋊C4Q1614yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—81no—
219C2×C42—81no—
220C2×C42—81no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C4⋊C8—82no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C4×C8—82no—
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—
247#95—22no—
248#95—22no—
249#95—42no—
250#95—42no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—81no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24C22×D411yescharacteristic
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—81no—
268C2×Q8—81no—
269C2×Q8#11714yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
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—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×SD1612yes—
333C22×C4C2×SD1612yes—
334C22×C4C2×SD1612yes—
335C22×C4C2×Q1612yes—
336C22×C4C2×Q1612yes—
337C22×C4C2×Q1612yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4#10318yes—
370C4⋊C4#10318yes—
371C4⋊C4#11714yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C42—28no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—44no—
386C42—82no—
387C42—82no—
388C42—82no—
389C42—82no—
390C42—84no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23C22×C22⋊C411yescharacteristic
396C23#52511yescharacteristic
397C23#52511yescharacteristic
398C23C22×SD1611yescharacteristic
399C23C22×Q1611yescharacteristic
400C23#57511yescharacteristic
401C23#57511yescharacteristic
402Q8—44no—
403Q8—44no—
404Q8—84no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
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—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4C2×#10314yes—
486C2×C4C2×#10314yes—
487C2×C4C2×#10312yes—
488C2×C4C2×#10312yes—
489C2×C4C2×#11712yes—
490C2×C4#52512yes—
491C8—82no—
492C8—82no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22#228011yescharacteristic
506C22#288311yescharacteristic
507C22#288411yescharacteristic
508C22C22×#10311yescharacteristic
509C22#289611yescharacteristic
510C22#290111yescharacteristic
511C22#290111yescharacteristic
512C22#310311yescharacteristic
513C22#310311yescharacteristic
514C22#310411yescharacteristic
515C22#310411yescharacteristic
516C4—24no—
517C4—24no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—44no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—161no—
533C4—161no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2#1761411yescharacteristic
539C2#1761911yescharacteristic
540C2#1762411yescharacteristic
541C2#1762911yescharacteristic
542C2#2072511yescharacteristic
543C2#2073411yescharacteristic
544C2#2074911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2369C4semidirect
15. #2365C4semidirect
17. #2355C4semidirect
18. Q8×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#219122
#139492
#87172
#85782
#85722
#83332
#82212
C4×#3282