Finite Group Databasebeta

← #202237

Subgroups — #202237

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21973C211yesmaximal, characteristic
2#21361C211yesmaximal, characteristic
3#20535C211yesmaximal, characteristic
4#9540C211yesmaximal, characteristic, complemented
5#9539C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9535C211yesmaximal, characteristic, complemented
10#9534C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#7898C211yesmaximal, characteristic
13C2×#1279C211yesmaximal, characteristic
14#7879C211yesmaximal, characteristic
15#7849C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2283C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1949C411yescharacteristic
24#1949C411yescharacteristic
25#1949C411yescharacteristic
26#1949C411yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1853C411yescharacteristic
32#1853C411yescharacteristic
33#1852C411yescharacteristic
34#1852C411yescharacteristic
35#1850—21no—
36#1850—21no—
37#1843—21no—
38#1832—21no—
39C4×#107C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1771—21no—
44#1771—21no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49#1749—21no—
50C4×C22⋊C8—21no—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54#1279C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432Q814yes—
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8D411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8C2×C411yescharacteristic
104C2×C22⋊C8C2×C411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×C22⋊C8D411yescharacteristic
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#383—21no—
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#381—21no—
125#380C2×C411yescharacteristic
126#380C2×C411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136#376—21no—
137C2×#95—21no—
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4111yescharacteristic
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#107—22no—
212#107—22no—
213#107—41no—
214#107—41no—
215#106—22no—
216#106—22no—
217#106—41no—
218#106—41no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230C4⋊C8—81no—
231C4⋊C8—81no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8C2×Q812yes—
235C4⋊C8C2×Q812yes—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—24no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#97—24no—
242#97—81no—
243#97—81no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—42no—
257C4×C8—81no—
258C4×C8—81no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271C24C4×D411yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4#11712yes—
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4#12312yes—
303C22×C4#12512yes—
304C22×C4#12612yes—
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—48no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—84no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4#13112yes—
369C4⋊C4#13112yes—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—81no—
381C42—81no—
382C42#11712yes—
383C42C4×Q812yes—
384C42C22⋊Q812yes—
385C42#12312yes—
386C42#12512yes—
387C42#12612yes—
388C42C22×Q811yescharacteristic
389C42C2×#4111yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#38712yes—
397C23#44312yes—
398C23#44412yes—
399C23C2×C4×D411yescharacteristic
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
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—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4#38712yes—
445C2×C4#38712yes—
446C2×C4#38712yes—
447C2×C4C2×#11711yescharacteristic
448C2×C4C2×C4×Q811yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C2×C4C2×#12511yescharacteristic
452C2×C4C2×#12611yescharacteristic
453C2×C4C2×#13111yescharacteristic
454C2×C4C2×#13111yescharacteristic
455C2×C4#56811yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4#57611yescharacteristic
458C2×C4#57711yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57811yescharacteristic
461C8—42no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—81no—
474C8—81no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22#192214yes—
482C22C2×#38711yescharacteristic
483C22#293311yescharacteristic
484C22#293311yescharacteristic
485C22#294611yescharacteristic
486C22#294711yescharacteristic
487C22C2×#44311yescharacteristic
488C22C2×#44411yescharacteristic
489C4—22no—
490C4—22no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4#292211yescharacteristic
503C4#292311yescharacteristic
504C4#296211yescharacteristic
505C4#296211yescharacteristic
506C4#308311yescharacteristic
507C4#308411yescharacteristic
508C4#309611yescharacteristic
509C4#309711yescharacteristic
510C2—24no—
511C2#1801411yescharacteristic
512C2#1801711yescharacteristic
513C2#1802011yescharacteristic
514C2#1802311yescharacteristic
515C2#2102211yescharacteristic
516C2#2103511yescharacteristic
517C2C2×#192211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9540C2semidirect
5. #9539C2semidirect
6. #9538C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9535C2semidirect
10. #9534C2semidirect
11. #9533C2semidirect

Maximal subgroups

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

subgroupindex
#219732
#213612
#205352
#95402
#95392
#95382
#95372
#95362
#95352
#95342
#95332
#78982
C2×#12792
#78792
#78492