Finite Group Databasebeta

← #190355

Subgroups — #190355

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8779C214yesmaximal, complemented
5#8777C214yesmaximal, complemented
6#8048C212yesmaximal, complemented
7#7941C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10C2×#410—21no—
11#2357C2212yes—
12#2356C412yescomplemented
13#2355—21no—
14#2354—21no—
15#2353C412yescomplemented
16#2353C2212yes—
17#2352—21no—
18#2352—21no—
19#2298—21no—
20#2298—21no—
21#2297—21no—
22#2285C411yescharacteristic, complemented
23#2283C411yescharacteristic, complemented
24#2283C2211yescharacteristic
25C4×#117—21no—
26C4×#117C411yescharacteristic, complemented
27C4×#117C2212yes—
28C4×#117C2211yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34#1740—21no—
35#1467—21no—
36#1466—21no—
37#1465C2212yes—
38#1465C2212yes—
39#1458—21no—
40#1458—21no—
41#1297C2218yes—
42#1284C2218yes—
43C22×#41D411yescharacteristic
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117—22no—
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C2×#117D411yescharacteristic
63C22×C4⋊C4—21no—
64C22×C42—21no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2311yescharacteristic
68#432—22no—
69#432—22no—
70#431—22no—
71#431—22no—
72#423—22no—
73#410—28no—
74C2×#99—22no—
75C2×#99—22no—
76#407C2314yes—
77C4×M16—22no—
78#391—22no—
79#390—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#386—22no—
89#386—22no—
90#384—21no—
91#383C2×C412yes—
92#383C2×C412yes—
93#382—21no—
94#382—21no—
95#382C2×C412yes—
96#381C2×C412yes—
97#381C2312yes—
98#380—21no—
99#380—21no—
100#380—21no—
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4C2314yes—
125C4×C22⋊C4C2312yes—
126#376—21no—
127#376—21no—
128#376—22no—
129#376C2×C412yes—
130#376C2×C412yes—
131C2×#95C2×C412yes—
132C43—21no—
133C43—21no—
134C43—22no—
135#342—22no—
136#342—22no—
137#324—22no—
138#324—22no—
139#323—22no—
140#323—22no—
141C2×#41—24no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—42no—
145C22×Q8C22⋊C411yescharacteristic
146C22×D4C22⋊C411yescharacteristic
147C22×D4C2×D411yescharacteristic
148C22×D4C2×D411yescharacteristic
149C23×C4—22no—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×M16—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—28no—
183#117—44no—
184#117—44no—
185#117—44no—
186#117—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C411yescharacteristic
209C2×C22⋊C4C22⋊C411yescharacteristic
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C22⋊C4C2×D411yescharacteristic
212C2×C22⋊C4C2×D411yescharacteristic
213C2×C22⋊C4C2×D411yescharacteristic
214C2×C42—21no—
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—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—44no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C4⋊C8—42no—
250C4⋊C8—42no—
251#99—28no—
252#99—28no—
253C22⋊C8—24no—
254C22⋊C8—24no—
255#97—42no—
256C4×C8—42no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—42no—
269#95C22×C412yes—
270#95C22×C412yes—
271#95C22×C412yes—
272#95C22×C412yes—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275C24C22×D411yescharacteristic
276#41—48no—
277#41—48no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—81no—
318C22×C4—82no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4#10414yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4#12312yes—
333C22×C4#12312yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4112yes—
340M16—84no—
341C2×C8—44no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—28no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—44no—
382C42—44no—
383C42—44no—
384C42—48no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—84no—
390C42—82no—
391C42—82no—
392C42—84no—
393C42#11711yescharacteristic
394C42#11711yescharacteristic
395C42#11712yes—
396C42#11712yes—
397C42#11711yescharacteristic
398C42#11711yescharacteristic
399C42#12312yes—
400C42#12312yes—
401C42#12312yes—
402C42#12312yes—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23C2×#10412yes—
417C23C22×C22⋊C411yescharacteristic
418C23C2×#12312yes—
419Q8—44no—
420D4—44no—
421D4—44no—
422D4—44no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—88no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4#38111yescharacteristic
477C2×C4#38112yes—
478C2×C4#38112yes—
479C2×C4#38111yescharacteristic
480C2×C4#38111yescharacteristic
481C2×C4#38111yescharacteristic
482C2×C4C2×#10412yes—
483C2×C4C2×#10412yes—
484C2×C4#42112yes—
485C2×C4C2×#11711yescharacteristic
486C2×C4C2×#11711yescharacteristic
487C2×C4C2×#12312yes—
488C8—84no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22C2×#38111yescharacteristic
507C22C22×#10411yescharacteristic
508C22C2×#42111yescharacteristic
509C4—22no—
510C4—22no—
511C4—21no—
512C4—22no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—88no—
526C4#182612yes—
527C4#182612yes—
528C4#182712yes—
529C2—21no—
530C2—21no—
531C2—42no—
532C2—42no—
533C2—81no—
534C2C2×#182611yescharacteristic
535C2C2×#182711yescharacteristic
536C2#2071011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8779C2semidirect
5. #8777C2semidirect
6. #8048C2semidirect
7. #7941C2semidirect
12. #2356C4semidirect
15. #2353C4semidirect
22. #2285C4semidirect
23. #2283C4semidirect
26. C4×#117C4semidirect
29. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#341212
C4×#4102
#173612
#87792
#87772
#80482
#79412