Finite Group Databasebeta

← #164149

Subgroups — #164149

Order 512.

999 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36405C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic
3#20378C211yesmaximal, characteristic
4#7685C211yesmaximal, characteristic, complemented
5#7675C212yesmaximal, complemented
6#7668C211yesmaximal, characteristic, complemented
7#7661C212yesmaximal, complemented
8#7617C211yesmaximal, characteristic, complemented
9#7613C212yesmaximal, complemented
10#7566C211yesmaximal, characteristic, complemented
11#7564C212yesmaximal, complemented
12C2×#556C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×Q8—21no—
15C2×#408—21no—
16C2×#405—21no—
17#2857C2211yescharacteristic
18#2855C2212yes—
19#2582C411yescharacteristic
20#2546C411yescharacteristic
21#2485C412yes—
22#2478—22no—
23#2468—22no—
24#2447—21no—
25#2432—21no—
26#2415C412yes—
27#2365—21no—
28#2362—21no—
29#2354C411yescharacteristic
30C4×C22⋊Q8C2211yescharacteristic
31#2296C2212yes—
32#1954—21no—
33#1954—22no—
34#1797C2212yes—
35#1797C2211yescharacteristic
36#1750—22no—
37#1749C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#556—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#126—22no—
60C2×#124—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C412yes—
64C4×#41—44no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D412yes—
73C2×#117C2×C412yes—
74C2×#117C2×C412yes—
75C2×#117D411yescharacteristic
76C2×#117D412yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C8×Q8—44no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82#408—28no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89#405—44no—
90C2×#97—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—21no—
94#398—22no—
95#398—22no—
96#397—22no—
97#397—22no—
98#394—22no—
99#394—22no—
100#391—21no—
101#391—21no—
102#391—21no—
103#391C2×C412yes—
104#391C2311yescharacteristic
105#390—21no—
106#389—21no—
107#389—22no—
108#389—21no—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2312yes—
113#389C2311yescharacteristic
114#388—22no—
115#388—22no—
116#388—22no—
117#387—22no—
118#387—22no—
119#387—22no—
120#387—21no—
121#385—22no—
122#385—22no—
123#385—22no—
124#385—21no—
125#384C2×C411yescharacteristic
126#381—22no—
127#380—21no—
128#380C2×C412yes—
129#380C2×C412yes—
130#380C2×C411yescharacteristic
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C22⋊C4—21no—
137#336—21no—
138#336—21no—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145C2×#41—24no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C22×D4C22⋊C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C2×M16—24no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—24no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—42no—
171C22×C8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—44no—
174#126—48no—
175#124—44no—
176C22⋊Q8—22no—
177C22⋊Q8—41no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—44no—
181C4×Q8—24no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—44no—
185C4×D4—24no—
186C4×D4—44no—
187C4×D4—42no—
188C4×D4—44no—
189#117—24no—
190#117—24no—
191#117—44no—
192#117—42no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2411yescharacteristic
237C4⋊C8—44no—
238C4⋊C8—42no—
239C22⋊C8—28no—
240C22⋊C8—28no—
241C22⋊C8—42no—
242C22⋊C8—44no—
243C22⋊C8—81no—
244C22⋊C8—82no—
245#97—44no—
246C4×C8—44no—
247C4×C8—44no—
248C4×C8—42no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—21no—
262C24C2×C22⋊C412yes—
263C24C2×C22⋊C411yescharacteristic
264#41—48no—
265#41—48no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8#10414yes—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—42no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4#13512yes—
317C22×C4#13512yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—48no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—48no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—82no—
362C4⋊C4—81no—
363C4⋊C4#9914yes—
364C4⋊C4#13112yes—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—81no—
374C22⋊C4—82no—
375C42—24no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—41no—
380C42—42no—
381C42—41no—
382C42—44no—
383C42—44no—
384C42—42no—
385C42—42no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23#41612yes—
400C23C22×C22⋊C411yescharacteristic
401Q8—28no—
402Q8—81no—
403Q8—81no—
404Q8—84no—
405D4—48no—
406D4—44no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4C2×#9914yes—
457C2×C4C2×#9912yes—
458C2×C4#41612yes—
459C2×C4C2×#10414yes—
460C2×C4C2×#10412yes—
461C2×C4C2×#13111yescharacteristic
462C2×C4#56811yescharacteristic
463C2×C4C2×#13511yescharacteristic
464C2×C4C2×#13511yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57811yescharacteristic
467C8—42no—
468C8—42no—
469C8—42no—
470C8—44no—
471C8—82no—
472C8—81no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22#288011yescharacteristic
488C22C22×#9911yescharacteristic
489C22#288911yescharacteristic
490C22C2×#41611yescharacteristic
491C22#289811yescharacteristic
492C22C22×#10411yescharacteristic
493C22#290111yescharacteristic
494C4—21no—
495C4—21no—
496C4—24no—
497C4—21no—
498C4—21no—
499C4—24no—
500C4—21no—
501C4—21no—
502C4—41no—
503C4—42no—
504C4—41no—
505C4—42no—
506C4—81no—
507C4—82no—
508C4—82no—
509C4—81no—
510C4—81no—
511C4#150114yes—
512C2—42no—
513C2—82no—
514C2#1741611yescharacteristic
515C2#1751011yescharacteristic
516C2#1751811yescharacteristic
517C2C2×#150111yescharacteristic
518C2#1760611yescharacteristic
519C2#1765411yescharacteristic
520C2#1772511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7685C2semidirect
5. #7675C2semidirect
6. #7668C2semidirect
7. #7661C2semidirect
8. #7617C2semidirect
9. #7613C2semidirect
10. #7566C2semidirect
11. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#364052
#225302
#203782
#76852
#76752
#76682
#76612
#76172
#76132
#75662
#75642