Finite Group Databasebeta

← #6490388

Subgroups — #6490388

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#33102C214yesmaximal, complemented
4#30102C212yesmaximal, complemented
5#30069C212yesmaximal, complemented
6#29524C214yesmaximal, complemented
7#29340C212yesmaximal
8#29307C212yesmaximal, complemented
9#29275C212yesmaximal, complemented
10#28202C212yesmaximal, complemented
11#27679C212yesmaximal, complemented
12#24299C212yesmaximal, complemented
13#22117C212yesmaximal, complemented
14#22111C212yesmaximal, complemented
15#20786C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#476C2212yes—
18C2×#476C2212yes—
19#2651—22no—
20#2648—22no—
21#2647C2211yescharacteristic
22#2644C2212yes—
23#2641C2211yescharacteristic
24#2629C2211yescharacteristic
25#2624C2212yes—
26#2619C2211yescharacteristic
27#2395—22no—
28#2392—22no—
29#2369—22no—
30#2365—22no—
31#2345—22no—
32Q8×C22⋊C4—22no—
33#2331C2212yes—
34#2331C2212yes—
35#2316C2212yes—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×#127C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×#121C2212yes—
41#2293C2212yes—
42#2285C2211yescharacteristic
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2101C2214yes—
46#2101C2214yes—
47#2092C2214yes—
48#2084C2214yes—
49#2059C2214yes—
50#2058C2214yes—
51#2041C2214yes—
52#2023C2214yes—
53#1987C2212yes—
54#1987C2212yes—
55#1948C2212yes—
56#1947C2212yes—
57#1944C2214yes—
58#1941C2214yes—
59#1936C2214yes—
60#1934C2214yes—
61#1926C2214yes—
62#1926C2214yes—
63#1926C2214yes—
64#1925C2214yes—
65#1920C2214yes—
66#1920C2214yes—
67#1919C2214yes—
68#1918C2214yes—
69#1875C2214yes—
70#1874C2214yes—
71#1854C2214yes—
72#1853C2214yes—
73#1840C2212yes—
74#1831C2212yes—
75#1818C2212yes—
76#1802C2212yes—
77#1802C2212yes—
78#1797C2212yes—
79#1767C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81#1750C2212yes—
82#1746C2212yes—
83C23×Q8—21no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#121C2312yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×D4—22no—
95C2×C4×D4—22no—
96C2×C4×D4C2312yes—
97C2×#117—22no—
98C2×#117C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2312yes—
103C22×C42C2311yescharacteristic
104#476—28no—
105#476—28no—
106C2×#107C2314yes—
107C2×#106C2314yes—
108C2×#106C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2312yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×C22⋊C8C2314yes—
117C2×#97C2312yes—
118C2×C4×C8C2312yes—
119#400—22no—
120#399—22no—
121#398—22no—
122#398—22no—
123#397—22no—
124#397—22no—
125#397—22no—
126#397—22no—
127#396—22no—
128#396—22no—
129#394—22no—
130#394—22no—
131#394—22no—
132#394—22no—
133#394—22no—
134#392—22no—
135#391—22no—
136#391—22no—
137#391—24no—
138#391—24no—
139#390—22no—
140#390C2312yes—
141#390C2312yes—
142#390C2312yes—
143#390C2312yes—
144#389—24no—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2314yes—
149#389C2314yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#388—22no—
154#388—22no—
155#388—22no—
156#387—22no—
157#387—24no—
158#387—24no—
159#386—22no—
160#384C2311yescharacteristic
161#384C2312yes—
162#382C2311yescharacteristic
163#382C2312yes—
164#380—21no—
165#380—21no—
166#380—24no—
167#380C2312yes—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—24no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4D412yes—
184#376C2311yescharacteristic
185#376C2311yescharacteristic
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201C22×Q8—28no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—84no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2412yes—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2411yescharacteristic
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—28no—
216C4⋊Q8—28no—
217C4⋊Q8—28no—
218C4⋊Q8—24no—
219C4⋊Q8#4118yes—
220#127—41no—
221#127—41no—
222#125—41no—
223#125—41no—
224#121—24no—
225#121—24no—
226C4×Q8—84no—
227C4×Q8—84no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—84no—
231C4×D4—84no—
232#117—24no—
233#117—41no—
234#117—41no—
235#117—42no—
236#117—42no—
237#117—84no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—44no—
292#106—28no—
293#106—28no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297#102—28no—
298#102—28no—
299#102—44no—
300#102—44no—
301C22⋊C8—28no—
302#97—42no—
303C4×C8—42no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95C2×D412yes—
318#95C2×D412yes—
319#95C2×D412yes—
320#95C2×D412yes—
321C24—22no—
322C24C22×D411yescharacteristic
323C2×Q8—28no—
324C2×Q8—28no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—84no—
328C2×Q8—816no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—82no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—84no—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C42−1+414yes—
368C22×C4C2511yescharacteristic
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4C2×#4118yes—
400C22⋊C4—24no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C42—22no—
415C42—22no—
416C42—24no—
417C42—24no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—42no—
426C42—42no—
427C42—44no—
428C42—44no—
429C42—84no—
430C42—84no—
431C42C2×#4114yes—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—22no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#53612yes—
444C23C23×D411yescharacteristic
445Q8—416no—
446Q8—88no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—88no—
452D4—88no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—88no—
492C2×C4—88no—
493C2×C4#52512yes—
494C2×C4#52512yes—
495C2×C4#53612yes—
496C2×C4#57714yes—
497C2×C4C22×#4112yes—
498C2×C4C2×2−1+412yes—
499C2×C4#58511yescharacteristic
500C2×C4#58511yescharacteristic
501C8—44no—
502C8—44no—
503C22—22no—
504C22—24no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22C2×#52511yescharacteristic
516C22C2×#53611yescharacteristic
517C22#344911yescharacteristic
518C22C2×#57712yes—
519C22#358512yes—
520C4—22no—
521C4—22no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—48no—
527C4—42no—
528C4—42no—
529C4—84no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4#312614yes—
534C4#351612yes—
535C4#351612yes—
536C2—42no—
537C2—162no—
538C2#3524511yescharacteristic
539C2C2×#312611yescharacteristic
540C2#6056411yescharacteristic
541C2#6089212yes—
542C2#6098912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #33102C2semidirect
4. #30102C2semidirect
5. #30069C2semidirect
6. #29524C2semidirect
8. #29307C2semidirect
9. #29275C2semidirect
10. #28202C2semidirect
11. #27679C2semidirect
12. #24299C2semidirect
13. #22117C2semidirect
14. #22111C2semidirect
15. #20786C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#342162
#331022
#301022
#300692
#295242
#293402
#293072
#292752
#282022
#276792
#242992
#221172
#221112
#207862