Finite Group Databasebeta

← #219248

Subgroups — #219248

Order 512.

605 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23819C211yesmaximal, characteristic
2#22301C211yesmaximal, characteristic, complemented
3#22262C211yesmaximal, characteristic
4#9871C211yesmaximal, characteristic, complemented
5#9638C211yesmaximal, characteristic
6#9546C211yesmaximal, characteristic, complemented
7#9545C211yesmaximal, characteristic, complemented
8#9542C211yesmaximal, characteristic, complemented
9#9541C211yesmaximal, characteristic, complemented
10#8847C211yesmaximal, characteristic, complemented
11#8847C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8812C211yesmaximal, characteristic, complemented
15#8133C211yesmaximal, characteristic, complemented
16C2×#436C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C4×#127C2211yescharacteristic
19C4×#125C2211yescharacteristic
20#2100C411yescharacteristic
21#2100C411yescharacteristic
22#2073C411yescharacteristic
23#2073C411yescharacteristic
24#2041C411yescharacteristic
25#2041C411yescharacteristic
26#2025C411yescharacteristic
27#2025C411yescharacteristic
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1940—21no—
31#1939—21no—
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1917—21no—
35#1916—21no—
36#1870—21no—
37#1870—21no—
38#1869—21no—
39#1869—21no—
40#1842C2211yescharacteristic
41#1831C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#127C2×C411yescharacteristic
70C2×#125C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75#436—41no—
76#436—41no—
77C8×D4—41no—
78C8×D4—41no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#102C2×C411yescharacteristic
88C2×#102C2×C411yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#390—21no—
112#390—21no—
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#387—21no—
118#387—21no—
119#384C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C43C2311yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—41no—
149#335—41no—
150#334—41no—
151#334—41no—
152D4⋊C8—41no—
153D4⋊C8—41no—
154D4⋊C8—41no—
155D4⋊C8—41no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C2×M16—42no—
163C2×M16—81no—
164C2×M16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—81no—
173C22×C8—81no—
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182#127—22no—
183#127—41no—
184#125—22no—
185#125—41no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—81no—
195C4×D4—81no—
196C4×D4—81no—
197C4×D4—81no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237#102—22no—
238#102—22no—
239#102—22no—
240#102—22no—
241#102—41no—
242#102—41no—
243#102—41no—
244#102—41no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247C22⋊C8—81no—
248C22⋊C8—81no—
249C22⋊C8—81no—
250C22⋊C8—81no—
251#97—81no—
252C4×C8—24no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8—81no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273C24—21no—
274C24—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4#13112yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318M16—84no—
319C2×C8—22no—
320C2×C8—22no—
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—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—84no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4#13112yes—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—81no—
385C22⋊C4—81no—
386C22⋊C4—81no—
387C22⋊C4—81no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42H(5,𝔽2)11yescharacteristic
403C42H(5,𝔽2)11yescharacteristic
404C422−1+411yescharacteristic
405C422−1+411yescharacteristic
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23C2×C4×D411yescharacteristic
421D4—22no—
422D4—22no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—84no—
430D4—84no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#44312yes—
470C2×C4#44412yes—
471C2×C4#52011yescharacteristic
472C2×C4#52011yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#54211yescharacteristic
476C2×C4#54211yescharacteristic
477C2×C4C2×#13111yescharacteristic
478C2×C4C2×#13111yescharacteristic
479C2×C4#56811yescharacteristic
480C2×C4#56811yescharacteristic
481C2×C4#57611yescharacteristic
482C2×C4#57611yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57711yescharacteristic
485C8—42no—
486C8—42no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22#232611yescharacteristic
514C22#293311yescharacteristic
515C22#293311yescharacteristic
516C22#294611yescharacteristic
517C22#294611yescharacteristic
518C22C2×#44311yescharacteristic
519C22C2×#44411yescharacteristic
520C4—22no—
521C4—22no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4#299211yescharacteristic
533C4#299311yescharacteristic
534C4#299411yescharacteristic
535C4#299511yescharacteristic
536C4#322011yescharacteristic
537C4#322211yescharacteristic
538C4#324111yescharacteristic
539C4#324111yescharacteristic
540C2—82no—
541C2—161no—
542C2#1801411yescharacteristic
543C2#1801811yescharacteristic
544C2#1802011yescharacteristic
545C2#1802211yescharacteristic
546C2#2221711yescharacteristic
547C2#2223211yescharacteristic
548C2#2223411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22301C2semidirect
4. #9871C2semidirect
6. #9546C2semidirect
7. #9545C2semidirect
8. #9542C2semidirect
9. #9541C2semidirect
10. #8847C2semidirect
11. #8847C2semidirect
12. #8845C2semidirect
13. #8845C2semidirect
14. #8812C2semidirect
15. #8133C2semidirect

Maximal subgroups

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

subgroupindex
#238192
#223012
#222622
#98712
#96382
#95462
#95452
#95422
#95412
#88472
#88452
#88122
#81332