Finite Group Databasebeta

← #482429

Subgroups — #482429

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#11900C214yesmaximal, complemented
5#11812C214yesmaximal, complemented
6#11785C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#432—21no—
10#2857—21no—
11#2857—21no—
12#2628C412yescomplemented
13#2628C2212yes—
14#2568—21no—
15#2567C412yescomplemented
16#2563C2212yes—
17#2542—21no—
18#2487—21no—
19#2487—21no—
20#2409C412yescomplemented
21#2409C2212yes—
22#2317—21no—
23#2311—21no—
24#2285C411yescharacteristic, complemented
25#2283—21no—
26#1820C2212yes—
27#1799—21no—
28#1799C2212yes—
29#1782—21no—
30C4×#99—21no—
31C4×#99C2212yes—
32#1467—21no—
33#1466C2212yes—
34#1464—21no—
35#1464C2212yes—
36#1459—21no—
37#1459C2212yes—
38#1351C22116yes—
39C22×#41D411yescharacteristic
40C23×Q8C2×C412yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—41no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C4×#41—41no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72#432—22no—
73#432—22no—
74#432—41no—
75#432—41no—
76C2×C4⋊C8—41no—
77C2×#99—22no—
78C2×#99—22no—
79C2×C22⋊C8—22no—
80C2×#97—41no—
81#400—22no—
82#400—22no—
83#399—22no—
84#399—22no—
85#395C2×C414yes—
86#395C2×C414yes—
87#393C2×C414yes—
88#393C2314yes—
89#391—22no—
90#391—22no—
91#391—22no—
92#390—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#386—22no—
98#386—22no—
99#385—22no—
100#385—22no—
101#384—22no—
102#384—21no—
103#383—41no—
104#382—22no—
105#382—41no—
106#382C2×C412yes—
107#381—22no—
108#380—22no—
109#380—21no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—41no—
117C4×C4⋊C4C2×C412yes—
118C4×C22⋊C4—22no—
119#376—41no—
120#342—22no—
121#342C2314yes—
122#324—22no—
123#324—22no—
124#323—22no—
125#323C2314yes—
126C2×#41—24no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—42no—
130C22×Q8—22no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—84no—
134C22×Q8C22⋊C411yescharacteristic
135C22×Q8C22×C414yes—
136C22×D4C22⋊C411yescharacteristic
137C22×D4C2×D411yescharacteristic
138C22×D4C2×D411yescharacteristic
139C23×C4—22no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C22⋊Q8—84no—
161C22⋊Q8—84no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×D4—41no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172C4×D4—41no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—41no—
178#117—41no—
179#117—41no—
180#117—41no—
181#117—41no—
182#117—41no—
183#117—41no—
184#117—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4C22⋊C411yescharacteristic
210C2×C22⋊C4C22⋊C411yescharacteristic
211C2×C22⋊C4C22×C411yescharacteristic
212C2×C22⋊C4C2×D411yescharacteristic
213C2×C22⋊C4C2×D411yescharacteristic
214C2×C22⋊C4C2411yescharacteristic
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—42no—
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C4⋊C8—42no—
240C4⋊C8—42no—
241#99—44no—
242#99—44no—
243C22⋊C8—24no—
244C22⋊C8—42no—
245#97—42no—
246#97—42no—
247#95—22no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95C22×C414yes—
257C24C2×C22⋊C411yescharacteristic
258C24C2×C22⋊C411yescharacteristic
259C24C22×D411yescharacteristic
260#41—48no—
261#41—48no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—84no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4#13714yes—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—84no—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C2×C8—44no—
330C2×C8—84no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—44no—
379C42—84no—
380C42#9911yescharacteristic
381C42#9911yescharacteristic
382C42#9911yescharacteristic
383C42#9911yescharacteristic
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#41912yes—
397C23#42112yes—
398C23C22×C22⋊C411yescharacteristic
399C23C2×#13712yes—
400Q8—44no—
401Q8—84no—
402Q8—88no—
403Q8—84no—
404D4—44no—
405D4—44no—
406D4—44no—
407C2×C4—24no—
408C2×C4—24no—
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—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—88no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4C2×#9911yescharacteristic
460C2×C4C2×#9911yescharacteristic
461C2×C4C2×#9911yescharacteristic
462C2×C4C2×#9911yescharacteristic
463C2×C4C2×#9911yescharacteristic
464C2×C4C2×#9911yescharacteristic
465C2×C4#41912yes—
466C8—84no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—81no—
483C22—81no—
484C22C22×#9911yescharacteristic
485C22C2×#41911yescharacteristic
486C22C2×#42111yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—22no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—84no—
501C4—84no—
502C4—164no—
503C4#212212yes—
504C4#212912yes—
505C2—21no—
506C2—21no—
507C2—42no—
508C2—42no—
509C2—42no—
510C2#1757811yescharacteristic
511C2C2×#212211yescharacteristic
512C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11900C2semidirect
5. #11812C2semidirect
6. #11785C2semidirect
7. C22×C4⋊Q8C4semidirect
12. #2628C4semidirect
15. #2567C4semidirect
20. #2409C4semidirect
24. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#352792
#207232
#173632
#119002
#118122
#117852