Finite Group Databasebeta

← #207000

Subgroups — #207000

Order 512.

996 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#35079C211yesmaximal, characteristic
2#20525C211yesmaximal, characteristic
3#14355C211yesmaximal, characteristic
4#9888C212yesmaximal, complemented
5#9887C212yesmaximal, complemented
6#9886C212yesmaximal, complemented
7#9885C212yesmaximal, complemented
8#8024C211yesmaximal, characteristic
9#8023C211yesmaximal, characteristic
10#8016C211yesmaximal, characteristic
11#8015C211yesmaximal, characteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2857C2212yes—
15#2851C2212yes—
16#2667—21no—
17#2665—21no—
18#2569—21no—
19#2549—21no—
20#2442C412yescomplemented
21#2441C412yescomplemented
22#2440C412yescomplemented
23#2439C412yescomplemented
24#2437—22no—
25#2329—22no—
26#2280C2211yescharacteristic
27C2×#381C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1753—22no—
32#1752—21no—
33C8×C22⋊C4—21no—
34#1749C2212yes—
35C4×C22⋊C8C2212yes—
36#1444—21no—
37#1442—21no—
38#1439—22no—
39C4⋊#95C2212yes—
40#1434C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2211yescharacteristic
48#1294C2211yescharacteristic
49#1294C2211yescharacteristic
50#1294C2211yescharacteristic
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C23×C8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125C2×C412yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×#117C2×C412yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4D412yes—
66C22×C42C2312yes—
67#407—22no—
68#407Q814yescomplemented
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8Q814yescomplemented
73C2×C22⋊C8C2311yescharacteristic
74C2×C22⋊C8C2311yescharacteristic
75C2×C22⋊C8C2312yes—
76C2×#97—22no—
77C2×C4×C8—22no—
78#400—22no—
79#399—22no—
80#398—22no—
81#398—22no—
82#396—22no—
83#396—22no—
84#395—22no—
85#393—22no—
86#391—22no—
87#391—22no—
88#389—22no—
89#389—22no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#384C2×C412yes—
103#382C2×C412yes—
104#381—21no—
105#381—22no—
106#381—22no—
107#380—22no—
108#380—21no—
109#380—21no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380C2×C411yescharacteristic
114#380C2×C412yes—
115#380C2×C411yescharacteristic
116#380C2311yescharacteristic
117#380C2312yes—
118#380C2311yescharacteristic
119#379—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4C2×C411yescharacteristic
124C4×C22⋊C4C2×C411yescharacteristic
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D411yescharacteristic
127C4×C22⋊C4D411yescharacteristic
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138#336—22no—
139#336—22no—
140C25C2×D411yescharacteristic
141C22×Q8—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—82no—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×Q812yes—
155C23×C4C2×Q812yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C2×M16—24no—
160C2×M16—82no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—24no—
165C22×C8—82no—
166C4⋊Q8—82no—
167#125—82no—
168C22⋊Q8—48no—
169C22⋊Q8—82no—
170C22⋊Q8—82no—
171#117—82no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—81no—
203C2×C22⋊C4—81no—
204C2×C22⋊C4—81no—
205C2×C22⋊C4—81no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—22no—
212C2×C42—82no—
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—22no—
222C22⋊C8C2×Q812yes—
223C22⋊C8C2×Q812yes—
224C22⋊C8C2×Q814yes—
225#97—82no—
226C4×C8—82no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—44no—
246#95—42no—
247#95—82no—
248#95—82no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C411yescharacteristic
252#95C22⋊C411yescharacteristic
253#95C22⋊C411yescharacteristic
254#95C22⋊C411yescharacteristic
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—42no—
260C24—42no—
261C24C2×C22⋊C411yescharacteristic
262C24C2×C22⋊C411yescharacteristic
263C24#12412yes—
264C24C4⋊Q812yes—
265C24C22×D411yescharacteristic
266C2×Q8—44no—
267C2×Q8—82no—
268C2×Q8—82no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C4×Q812yes—
318C22×C4C4×Q812yes—
319C22×C4C4×Q812yes—
320C22×C4C4×Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4#12412yes—
326C22×C4C4⋊Q812yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333M16—84no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—84no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—84no—
357C4⋊C4—88no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—42no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—81no—
370C22⋊C4—81no—
371C22⋊C4—82no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—82no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23#39112yes—
402C23#39112yes—
403C23C22×C22⋊C411yescharacteristic
404C23C2×#12411yescharacteristic
405C23C2×C4⋊Q811yescharacteristic
406C23#57611yescharacteristic
407C23#57711yescharacteristic
408C23#57711yescharacteristic
409C23#57811yescharacteristic
410Q8—164no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
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—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—164no—
474C2×C4#39112yes—
475C2×C4#39112yes—
476C2×C4#42114yes—
477C2×C4C2×C4×Q811yescharacteristic
478C2×C4C2×C4×Q811yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×C22⋊Q811yescharacteristic
481C8—42no—
482C8—42no—
483C8—44no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22C2×#39111yescharacteristic
507C22#288312yes—
508C22#289711yescharacteristic
509C22#289811yescharacteristic
510C22C2×#42112yes—
511C22#313611yescharacteristic
512C22#313711yescharacteristic
513C22#316311yescharacteristic
514C22#316411yescharacteristic
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—24no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—161no—
530C4—162no—
531C4—161no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—42no—
537C2—42no—
538C2#1759712yes—
539C2#1760012yes—
540C2#2130611yescharacteristic
541C2#2132611yescharacteristic
542C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9888C2semidirect
5. #9887C2semidirect
6. #9886C2semidirect
7. #9885C2semidirect
20. #2442C4semidirect
21. #2441C4semidirect
22. #2440C4semidirect
23. #2439C4semidirect
68. #407Q8semidirect
72. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#350792
#205252
#143552
#98882
#98872
#98862
#98852
#80242
#80232
#80162
#80152