Finite Group Databasebeta

← #224552

Subgroups — #224552

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34954C211yesmaximal, characteristic
2#22821C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9675C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic
6#9645C212yesmaximal, complemented
7#9638C211yesmaximal, characteristic
8#8569C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8183C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12#2593—22no—
13#2582C411yescharacteristic
14#2571C411yescharacteristic
15#2548C411yescharacteristic
16#2548C411yescharacteristic
17#2513—21no—
18#2513C412yes—
19#2448—21no—
20#2448C412yes—
21#2395—22no—
22C4×#127C2211yescharacteristic
23C4×#124—21no—
24C4×#124C2212yes—
25#2293—21no—
26#2293C2212yes—
27C42×Q8C2211yescharacteristic
28C42×Q8C2211yescharacteristic
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⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2312yes—
65C2×#117—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80#400—22no—
81#399—22no—
82#398—22no—
83#397—22no—
84#397—22no—
85#397—22no—
86#394—22no—
87#394—22no—
88#391—21no—
89#391C2×C412yes—
90#390—21no—
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2×C412yes—
99#389—21no—
100#389—21no—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#384C2×C411yescharacteristic
112#382—21no—
113#382C2×C412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C43—21no—
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—22no—
141Q8⋊C8—22no—
142Q8⋊C8—42no—
143D4⋊C8—24no—
144D4⋊C8—24no—
145D4⋊C8—42no—
146D4⋊C8—42no—
147C22×Q8C22⋊C411yescharacteristic
148C22×Q8C22×C411yescharacteristic
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C23×C4—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159#127—22no—
160#127—22no—
161#124—24no—
162#124—42no—
163#124—44no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—41no—
176C4×Q8—42no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—81no—
184C4×D4—81no—
185#117—82no—
186#117—82no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—22no—
238C4⋊C8—21no—
239C4⋊C8—42no—
240C4⋊C8—44no—
241C4⋊C8—42no—
242C4⋊C8—81no—
243C4⋊C8#4112yes—
244C4×C8—22no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—22no—
248C4×C8—21no—
249C4×C8—42no—
250C4×C8—42no—
251C4×C8—42no—
252C4×C8—82no—
253C4×C8#4112yes—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266C24—22no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8#13112yes—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—81no—
287C2×D4—81no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—81no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C42—21no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—21no—
372C42—22no—
373C42—21no—
374C42—22no—
375C42—41no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—42no—
384C42—42no—
385C42—81no—
386C42#13112yes—
387C42#13112yes—
388C42#13512yes—
389C42#13512yes—
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C42H(5,𝔽2)11yescharacteristic
393C422−1+411yescharacteristic
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23C22×C22⋊C411yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406Q8—44no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—84no—
412D4—84no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4#40812yes—
454C2×C4#40812yes—
455C2×C4#41612yes—
456C2×C4#41612yes—
457C2×C4#42112yes—
458C2×C4C4×#4111yescharacteristic
459C2×C4#52011yescharacteristic
460C2×C4#54811yescharacteristic
461C2×C4#54811yescharacteristic
462C2×C4#55011yescharacteristic
463C2×C4#55011yescharacteristic
464C2×C4C2×#13111yescharacteristic
465C2×C4C2×#13111yescharacteristic
466C2×C4C2×#13111yescharacteristic
467C2×C4#56811yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C8—42no—
473C8—42no—
474C8—42no—
475C8—82no—
476C8—81no—
477C8—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22#234111yescharacteristic
486C22C2×#40811yescharacteristic
487C22#288011yescharacteristic
488C22C2×#41611yescharacteristic
489C22#289711yescharacteristic
490C22C2×#42111yescharacteristic
491C22#290111yescharacteristic
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—41no—
502C4—82no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4#148214yes—
507C4#296611yescharacteristic
508C4#296911yescharacteristic
509C4#299311yescharacteristic
510C4#299411yescharacteristic
511C4#336111yescharacteristic
512C4#336511yescharacteristic
513C4#339411yescharacteristic
514C4#339611yescharacteristic
515C2—162no—
516C2C2×#148211yescharacteristic
517C2#1746811yescharacteristic
518C2#1751011yescharacteristic
519C2#1752611yescharacteristic
520C2#2253111yescharacteristic
521C2#2254311yescharacteristic
522C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#127C2×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
4. #9675C2semidirect
6. #9645C2semidirect
8. #8569C2semidirect
9. #8564C2semidirect
10. #8183C2semidirect
11. C4×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#349542
#228212
#213392
#96752
#96742
#96452
#96382
#85692
#85642
#81832
C4×Q8⋊C82