Finite Group Databasebeta

← #164574

Subgroups — #164574

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36445C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#20684C211yesmaximal, characteristic
4#7727C212yesmaximal, complemented
5#7707C212yesmaximal, complemented
6#7687C211yesmaximal, characteristic, complemented
7#7667C211yesmaximal, characteristic, complemented
8#7623C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic, complemented
10#7600C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic, complemented
12C2×#557C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#446—21no—
15C2×#408—21no—
16#2857C2211yescharacteristic
17#2857C2212yes—
18#2542C411yescharacteristic
19#2542C412yes—
20#2538C411yescharacteristic
21#2538C412yes—
22#2454—21no—
23#2454—22no—
24#2441—22no—
25#2441—21no—
26Q8×C4⋊C4—21no—
27Q8×C4⋊C4C411yescharacteristic
28#2341—21no—
29C4×C22⋊Q8C2211yescharacteristic
30C4×C22⋊Q8C2212yes—
31#1955—22no—
32#1954—21no—
33#1818—22no—
34#1817C2212yes—
35#1817C2212yes—
36#1797C2212yes—
37#1797C2211yescharacteristic
38#1749C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2212yes—
47#1272C2212yes—
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×#41D411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#557—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×#125—22no—
60C2×#125—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C2×C22⋊Q8C2×C412yes—
66C4×#41—44no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C412yes—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D412yes—
77C2×#117D412yes—
78C2×#117D411yescharacteristic
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#446—44no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84#408—44no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×#97—22no—
92C2×C4×C8—21no—
93#399—22no—
94#399—22no—
95#399—22no—
96#396—22no—
97#396—22no—
98#396—22no—
99#391C2312yes—
100#391C2311yescharacteristic
101#390—21no—
102#390—22no—
103#390—21no—
104#390—22no—
105#389—22no—
106#389—21no—
107#389—22no—
108#389—21no—
109#389—21no—
110#389—22no—
111#389—21no—
112#389—22no—
113#389C2311yescharacteristic
114#387—21no—
115#387—22no—
116#385—21no—
117#385—22no—
118#384—22no—
119#384—21no—
120#384C2×C411yescharacteristic
121#384C2×C412yes—
122#380—22no—
123#380—21no—
124#380—21no—
125#380—22no—
126#380C2×C411yescharacteristic
127#380C2×C412yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C412yes—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×Q8C22×C412yes—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22⋊C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—42no—
171C4⋊Q8—44no—
172C4⋊Q8—42no—
173#125—48no—
174#125—44no—
175C22⋊Q8—22no—
176C22⋊Q8—24no—
177C22⋊Q8—41no—
178C22⋊Q8—42no—
179C22⋊Q8—82no—
180C22⋊Q8—82no—
181C22⋊Q8—82no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—44no—
185C4×Q8—42no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×D4—44no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—44no—
192#117—42no—
193#117—44no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2411yescharacteristic
241C4⋊C8—42no—
242C4⋊C8—44no—
243C22⋊C8—44no—
244C22⋊C8—42no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—82no—
248C22⋊C8—81no—
249#97—44no—
250C4×C8—42no—
251#95—21no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95—21no—
258#95—21no—
259#95—22no—
260C24C2×C22⋊C412yes—
261C24C2×C22⋊C411yescharacteristic
262#41—48no—
263#41—48no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—21no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8#9914yes—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—42no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4#13712yes—
316C22×C4#13712yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320M16—84no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—84no—
340C4⋊C4—21no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4#13112yes—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—81no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C42—22no—
377C42—24no—
378C42—24no—
379C42—22no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—44no—
384C42—44no—
385C42—42no—
386C42—42no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23#41912yes—
401C23C22×C22⋊C411yescharacteristic
402Q8—28no—
403Q8—81no—
404Q8—82no—
405Q8—81no—
406Q8—82no—
407D4—48no—
408D4—44no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
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—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4C2×#9914yes—
460C2×C4C2×#9912yes—
461C2×C4#41912yes—
462C2×C4#42112yes—
463C2×C4C2×#13111yescharacteristic
464C2×C4#56811yescharacteristic
465C2×C4C2×#13711yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C2×C4#57711yescharacteristic
468C2×C4#57811yescharacteristic
469C8—42no—
470C8—81no—
471C8—82no—
472C8—81no—
473C8—81no—
474C8—82no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22#288011yescharacteristic
490C22C22×#9911yescharacteristic
491C22#288811yescharacteristic
492C22C2×#41911yescharacteristic
493C22#289811yescharacteristic
494C22C2×#42111yescharacteristic
495C22#290111yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—24no—
501C4—21no—
502C4—21no—
503C4—41no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—81no—
509C4—82no—
510C4—82no—
511C4—81no—
512C4—81no—
513C4#151114yes—
514C2—42no—
515C2—82no—
516C2#1741911yescharacteristic
517C2#1751411yescharacteristic
518C2#1752311yescharacteristic
519C2C2×#151111yescharacteristic
520C2#1760611yescharacteristic
521C2#1768411yescharacteristic
522C2#1776011yescharacteristic
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. #7727C2semidirect
5. #7707C2semidirect
6. #7687C2semidirect
7. #7667C2semidirect
8. #7623C2semidirect
9. #7617C2semidirect
10. #7600C2semidirect
11. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364452
#225322
#206842
#77272
#77072
#76872
#76672
#76232
#76172
#76002
#75662