Finite Group Databasebeta

← #164338

Subgroups — #164338

Order 512.

905 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36453C211yesmaximal, characteristic
2#20567C211yesmaximal, characteristic, complemented
3#20558C211yesmaximal, characteristic, complemented
4#7683C211yesmaximal, characteristic, complemented
5#7681C212yesmaximal, complemented
6#7680C211yesmaximal, characteristic, complemented
7#7675C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic, complemented
9#7633C212yesmaximal, complemented
10#7567C211yesmaximal, characteristic, complemented
11#7565C212yesmaximal, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14#2859C2211yescharacteristic
15#2855C2212yes—
16#2576—22no—
17#2538C411yescharacteristic
18#2537C411yescharacteristic
19#2494—21no—
20#2494C412yes—
21#2460—22no—
22#2427C412yes—
23#2420—21no—
24#2361—21no—
25#2355C411yescharacteristic
26D4×C4⋊C4C411yescharacteristic
27#2345—21no—
28C4×#124C2211yescharacteristic
29C4×#121C2212yes—
30C42×D4C2211yescharacteristic
31#1770—21no—
32C8×C4⋊C4—21no—
33#1767C2211yescharacteristic
34#1767C2211yescharacteristic
35#1753—22no—
36#1752—22no—
37#1749C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40#1744C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55C2×#125—21no—
56C2×#124C2×C411yescharacteristic
57C2×C22⋊Q8C2×C412yes—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C42C2312yes—
69#436—28no—
70C8×D4—28no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97—22no—
76C2×#97—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81#401—22no—
82#397—22no—
83#397—22no—
84#396—22no—
85#394—22no—
86#394—22no—
87#391C2×C412yes—
88#391C2311yescharacteristic
89#390—21no—
90#390C2311yescharacteristic
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2312yes—
97#388—22no—
98#388—21no—
99#388—21no—
100#388—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#386—22no—
106#386—22no—
107#385—22no—
108#385—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382—21no—
112#382—21no—
113#382C2×C411yescharacteristic
114#381—22no—
115#380—21no—
116#380—21no—
117#380—22no—
118#380C2×C412yes—
119#380C2×C412yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4—22no—
144C22×D4C22⋊C411yescharacteristic
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C2×M16—28no—
152C2×M16—82no—
153C2×M16—82no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—28no—
161C22×C8—82no—
162C22×C8—82no—
163#125—81no—
164#125—81no—
165#124—22no—
166#124—41no—
167C22⋊Q8—82no—
168#121—24no—
169#121—42no—
170#121—44no—
171C4×Q8—42no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—82no—
177C4×D4—82no—
178C4×D4—82no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—82no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2411yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—24no—
224C22⋊C8—28no—
225C22⋊C8—28no—
226#97—81no—
227#97—81no—
228#97—82no—
229C4×C8—24no—
230C4×C8—24no—
231C4×C8—82no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24—21no—
248C24C2×C22⋊C412yes—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—82no—
261C2×D4#10414yes—
262C2×D4#13112yes—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296M16—88no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—88no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—82no—
330C4⋊C4—81no—
331C4⋊C4—82no—
332C4⋊C4—81no—
333C4⋊C4#10414yes—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—42no—
359C42—42no—
360C42—41no—
361C42—42no—
362C42—42no—
363C42—82no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23C2×#10414yes—
378C23C22×C22⋊C411yescharacteristic
379Q8—81no—
380Q8—81no—
381D4—28no—
382D4—42no—
383D4—42no—
384D4—88no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—84no—
430C2×C4—164no—
431C2×C4#41312yes—
432C2×C4C2×#10414yes—
433C2×C4C2×#10412yes—
434C2×C4C2×#10412yes—
435C2×C4C2×#13111yescharacteristic
436C2×C4#56811yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57811yescharacteristic
441C8—42no—
442C8—42no—
443C8—44no—
444C8—42no—
445C8—44no—
446C8—42no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#288011yescharacteristic
462C22#288311yescharacteristic
463C22C2×#41311yescharacteristic
464C22#289711yescharacteristic
465C22#289811yescharacteristic
466C22C22×#10411yescharacteristic
467C22C22×#10411yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—24no—
473C4—21no—
474C4—21no—
475C4—42no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—81no—
481C4—81no—
482C4—82no—
483C4—82no—
484C4—81no—
485C4#152414yes—
486C2—44no—
487C2—82no—
488C2#1742411yescharacteristic
489C2#1751711yescharacteristic
490C2#1751811yescharacteristic
491C2#1754711yescharacteristic
492C2#1754811yescharacteristic
493C2C2×#152411yescharacteristic
494C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20567C2semidirect
3. #20558C2semidirect
4. #7683C2semidirect
5. #7681C2semidirect
6. #7680C2semidirect
7. #7675C2semidirect
8. #7635C2semidirect
9. #7633C2semidirect
10. #7567C2semidirect
11. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364532
#205672
#205582
#76832
#76812
#76802
#76752
#76352
#76332
#75672
#75652