Finite Group Databasebeta

← #203308

Subgroups — #203308

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22019C211yesmaximal, characteristic, complemented
2#21114C211yesmaximal, characteristic
3#20567C211yesmaximal, characteristic
4#9544C211yesmaximal, characteristic, complemented
5#9543C211yesmaximal, characteristic, complemented
6#9536C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8949C211yesmaximal, characteristic
9#8949C211yesmaximal, characteristic
10#8849C211yesmaximal, characteristic
11#8847C211yesmaximal, characteristic
12#8279C211yesmaximal, characteristic, complemented
13#8176C211yesmaximal, characteristic, complemented
14C4×D4⋊C8C211yesmaximal, characteristic
15#8133C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19#2295C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1984C2211yescharacteristic
22#1984C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1928C411yescharacteristic
26#1928C411yescharacteristic
27#1919C411yescharacteristic
28#1919C411yescharacteristic
29#1916—21no—
30#1916—21no—
31#1877C411yescharacteristic
32#1877C411yescharacteristic
33#1862C411yescharacteristic
34#1862C411yescharacteristic
35#1840—21no—
36#1832—21no—
37C4×#107C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40#1767C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1753—21no—
45#1753—21no—
46C4×C22⋊C8—21no—
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
49#1728—21no—
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75C4×SD16—22no—
76C4×SD16—22no—
77#436—22no—
78#436—22no—
79#436—22no—
80#436—22no—
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#387—21no—
115#387—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125C43C2311yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—22no—
151#334—22no—
152Q8⋊C8—22no—
153Q8⋊C8—22no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156C22×Q8C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C2×SD16—22no—
162C2×SD16—41no—
163C2×SD16—41no—
164C2×M16—24no—
165C2×M16—24no—
166C2×M16—81no—
167C2×M16—81no—
168C2×M16—81no—
169C2×M16—81no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4#4112yes—
195C4×D4#4112yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—81no—
214C2×C42—81no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—22no—
223#107—22no—
224#106—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—81no—
232C4⋊C8—81no—
233C4⋊C8#4112yes—
234C4⋊C8#4112yes—
235#103—22no—
236#103—22no—
237#103—22no—
238#102—22no—
239#102—41no—
240#102—41no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243#97—81no—
244C4×C8—24no—
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—21no—
257C4×C8—42no—
258C4×C8—81no—
259C4×C8—81no—
260C4×C8—81no—
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—
271C24—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4#11712yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310SD16—44no—
311M16—84no—
312M16—84no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
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—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—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
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—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4#11712yes—
369C4⋊C4#13112yes—
370C4⋊C4#13112yes—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—22no—
384C42—22no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—81no—
392C42—81no—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C23—22no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23C2×C4×D411yescharacteristic
405Q8—41no—
406Q8—41no—
407Q8—41no—
408Q8—41no—
409D4—22no—
410D4—22no—
411D4—41no—
412D4—41no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4#44312yes—
453C2×C4#44412yes—
454C2×C4C2×#11711yescharacteristic
455C2×C4C4×#4111yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52511yescharacteristic
458C2×C4#52911yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4C2×#13111yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4#56811yescharacteristic
463C2×C4#56811yescharacteristic
464C2×C4#57611yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—42no—
469C8—42no—
470C8—41no—
471C8—42no—
472C8—41no—
473C8—41no—
474C8—42no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—81no—
481C8—81no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22#229011yescharacteristic
490C22#293311yescharacteristic
491C22#293311yescharacteristic
492C22#294611yescharacteristic
493C22#294711yescharacteristic
494C22C2×#44311yescharacteristic
495C22C2×#44411yescharacteristic
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—42no—
501C4—41no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4#292111yescharacteristic
510C4#292211yescharacteristic
511C4#296911yescharacteristic
512C4#296911yescharacteristic
513C4#311811yescharacteristic
514C4#312011yescharacteristic
515C4#312311yescharacteristic
516C4#312411yescharacteristic
517C2—82no—
518C2#1801411yescharacteristic
519C2#1801911yescharacteristic
520C2#1802011yescharacteristic
521C2#1802411yescharacteristic
522C2#2110611yescharacteristic
523C2#2112511yescharacteristic
524C2#2113711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×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
1. #22019C2semidirect
4. #9544C2semidirect
5. #9543C2semidirect
6. #9536C2semidirect
7. #9533C2semidirect
12. #8279C2semidirect
13. #8176C2semidirect
15. #8133C2semidirect

Maximal subgroups

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

subgroupindex
#220192
#211142
#205672
#95442
#95432
#95362
#95332
#89492
#88492
#88472
#82792
#81762
C4×D4⋊C82
#81332