Finite Group Databasebeta

← #189876

Subgroups — #189876

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#21742C211yesmaximal, characteristic
3#13940C211yesmaximal, characteristic
4#8715C214yesmaximal, complemented
5#8714C214yesmaximal, complemented
6#8019C211yesmaximal, characteristic, complemented
7#8019C211yesmaximal, characteristic, complemented
8#8015C211yesmaximal, characteristic, complemented
9#8015C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2353—22no—
14#2353—22no—
15#2352C414yescomplemented
16#2352C414yescomplemented
17#2313—21no—
18#2312—21no—
19#2308—21no—
20#2283C2212yes—
21#2280C2212yes—
22C2×#381C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#2273—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1870—22no—
28#1870—22no—
29#1817C2212yes—
30#1779C2212yes—
31#1440—21no—
32#1438—21no—
33#1437—21no—
34#1436—21no—
35#1434C2212yes—
36C22⋊#95C2212yes—
37#1294C2214yes—
38#1294C2214yes—
39#1294C2214yes—
40#1294C2214yes—
41C23×D4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—21no—
45C22×M16—21no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C412yes—
52C2×C4×D4D412yes—
53C2×C4×D4D412yes—
54C2×#117C2×C414yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#407—22no—
68#407—22no—
69#407—22no—
70#407—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76#388—24no—
77#388—24no—
78#387—24no—
79#387—24no—
80#386—24no—
81#386—24no—
82#383—22no—
83#382C2×C414yes—
84#381—22no—
85#381—22no—
86#381—22no—
87#381—22no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—21no—
92#380C2×C412yes—
93#380C2×C412yes—
94#380C2×C414yes—
95#380C2312yes—
96#380C2312yes—
97#379—22no—
98C4×C4⋊C4C2×C414yes—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106#376—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95C2312yes—
114C2×#95C2312yes—
115#336—24no—
116C25C2×D411yescharacteristic
117C25C2×D411yescharacteristic
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—42no—
121C22×D4—42no—
122C22×D4C22⋊C412yes—
123C22×D4C22⋊C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—82no—
152C2×M16—82no—
153C22×C8—22no—
154C22×C8—22no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163#117—84no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22×C414yes—
173C2×C4⋊C4C22×C414yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—42no—
204C2×C42—82no—
205C2×C42C22×C412yes—
206C2×C42C2×D412yes—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—42no—
212C22⋊C8—42no—
213C22⋊C8—84no—
214#95—22no—
215#95—22no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—82no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C411yescharacteristic
242C24C2×C22⋊C411yescharacteristic
243C24#12312yes—
244C24#12312yes—
245C24C22×D411yescharacteristic
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—48no—
253C2×D4—48no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4#11712yes—
312C22×C4#11712yes—
313C22×C4#11712yes—
314C22×C4#11712yes—
315C22×C4#12312yes—
316C22×C4#12312yes—
317C22×C4#12312yes—
318C22×C4#12312yes—
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327M16—84no—
328M16—84no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—48no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C22⋊C4—88no—
351C22⋊C4—84no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—84no—
357C42—84no—
358C42—84no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—81no—
393C23—81no—
394C23#38112yes—
395C23#38112yes—
396C23C22×C22⋊C411yescharacteristic
397C23C2×#12311yescharacteristic
398C23C2×#12311yescharacteristic
399C23#57512yes—
400C23#57512yes—
401D4—48no—
402D4—48no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—164no—
459C2×C4—164no—
460C2×C4#38112yes—
461C2×C4#38112yes—
462C2×C4C2×#11711yescharacteristic
463C2×C4C2×#11711yescharacteristic
464C2×C4C2×#12311yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C8—82no—
467C8—82no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—44no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—84no—
492C22—84no—
493C22—84no—
494C22—84no—
495C22C2×#38111yescharacteristic
496C22#288312yes—
497C22#289611yescharacteristic
498C22#289611yescharacteristic
499C22#290112yes—
500C22#309312yes—
501C22#309312yes—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—84no—
513C4—84no—
514C4—162no—
515C2—22no—
516C2—22no—
517C2—42no—
518C2—42no—
519C2—82no—
520C2#1762312yes—
521C2#1762512yes—
522C2#2068911yescharacteristic
523C2#2070111yescharacteristic
524C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8715C2semidirect
5. #8714C2semidirect
6. #8019C2semidirect
7. #8019C2semidirect
8. #8015C2semidirect
9. #8015C2semidirect
15. #2352C4semidirect
16. #2352C4semidirect

Maximal subgroups

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

subgroupindex
#341852
#217422
#139402
#87152
#87142
#80192
#80152