Finite Group Databasebeta

← #224550

Subgroups — #224550

Order 512.

806 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#34945C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9675C212yesmaximal
5#9673C211yesmaximal, characteristic, complemented
6#9645C212yesmaximal
7#9637C211yesmaximal, characteristic, complemented
8#8570C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic
10#8182C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic
12#2573—22no—
13#2571C411yescharacteristic
14#2568C411yescharacteristic
15#2547C411yescharacteristic
16#2538C411yescharacteristic
17#2514C412yes—
18#2494—21no—
19#2447C412yes—
20#2421—21no—
21#2401—22no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#124C2212yes—
24C4×C22⋊Q8—21no—
25#2295C2212yes—
26#2283—21no—
27C42×Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1916—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36C8×C4⋊C4—21no—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2312yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C22×C42—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#401—22no—
79#400—22no—
80#400—22no—
81#400—22no—
82#397—22no—
83#397—22no—
84#397—22no—
85#396—22no—
86#391—21no—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2×C412yes—
91#390C2×C412yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389C2×C412yes—
96#389C2×C412yes—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#383—22no—
105#382—21no—
106#382—21no—
107#382C2×C412yes—
108#381—22no—
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129#376—22no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140Q8⋊C8—24no—
141Q8⋊C8—24no—
142Q8⋊C8—42no—
143Q8⋊C8—42no—
144D4⋊C8—22no—
145D4⋊C8—22no—
146D4⋊C8—42no—
147C22×Q8C22⋊C411yescharacteristic
148C22×Q8C22×C412yes—
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161#124—24no—
162#124—42no—
163C22⋊Q8—44no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—42no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C412yes—
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×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C412yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—21no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—42no—
230C4⋊C8—44no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8#4112yes—
234C4×C8—21no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—42no—
240C4×C8—42no—
241C4×C8—42no—
242C4×C8—82no—
243C4×C8#4112yes—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—21no—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4#13112yes—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—81no—
345C4⋊C4—82no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—42no—
353C22⋊C4—82no—
354C42—21no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—21no—
364C42—21no—
365C42—22no—
366C42—22no—
367C42—41no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—41no—
373C42—41no—
374C42—42no—
375C42—42no—
376C42#13112yes—
377C42#13112yes—
378C42#13512yes—
379C42#13512yes—
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C42H(5,𝔽2)11yescharacteristic
383C422−1+411yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23C22×C22⋊C411yescharacteristic
392Q8—44no—
393Q8—42no—
394Q8—42no—
395Q8—42no—
396Q8—42no—
397D4—22no—
398D4—22no—
399D4—41no—
400D4—41no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—84no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—84no—
443C2×C4#40812yes—
444C2×C4#40812yes—
445C2×C4#41612yes—
446C2×C4#41612yes—
447C2×C4#42112yes—
448C2×C4C4×#4111yescharacteristic
449C2×C4#52011yescharacteristic
450C2×C4#54811yescharacteristic
451C2×C4#54811yescharacteristic
452C2×C4#55011yescharacteristic
453C2×C4#55011yescharacteristic
454C2×C4C2×#13111yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4C2×#13111yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4C2×#13511yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57811yescharacteristic
462C8—42no—
463C8—42no—
464C8—42no—
465C8—82no—
466C8—81no—
467C8—82no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#234111yescharacteristic
476C22C2×#40811yescharacteristic
477C22#288011yescharacteristic
478C22C2×#41611yescharacteristic
479C22#289811yescharacteristic
480C22C2×#42111yescharacteristic
481C22#290111yescharacteristic
482C4—21no—
483C4—21no—
484C4—22no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—42no—
491C4—82no—
492C4—81no—
493C4—82no—
494C4—81no—
495C4—81no—
496C4#148214yes—
497C4#296611yescharacteristic
498C4#296911yescharacteristic
499C4#299311yescharacteristic
500C4#299411yescharacteristic
501C4#336011yescharacteristic
502C4#336411yescharacteristic
503C4#339311yescharacteristic
504C4#339411yescharacteristic
505C2—82no—
506C2C2×#148211yescharacteristic
507C2#1746911yescharacteristic
508C2#1750811yescharacteristic
509C2#1752311yescharacteristic
510C2#2253111yescharacteristic
511C2#2254111yescharacteristic
512C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22821C2semidirect
3. #21339C2semidirect
5. #9673C2semidirect
7. #9637C2semidirect
8. #8570C2semidirect
10. #8182C2semidirect

Maximal subgroups

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

subgroupindex
#349452
#228212
#213392
#96752
#96732
#96452
#96372
#85702
#85632
#81822
C4×D4⋊C82