Finite Group Databasebeta

← #212377

Subgroups — #212377

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22311C211yesmaximal, characteristic, complemented
2C4×#474C211yesmaximal, characteristic
3#20568C211yesmaximal, characteristic
4#9864C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic
9#8847C211yesmaximal, characteristic
10#8565C211yesmaximal, characteristic
11#8287C211yesmaximal, characteristic, complemented
12#8282C211yesmaximal, characteristic, complemented
13#8264C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic
15#8206C211yesmaximal, characteristic
16C2×#474C411yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1988C411yescharacteristic
22#1985—21no—
23#1985—21no—
24#1947C411yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1918C411yescharacteristic
28#1918C411yescharacteristic
29#1917—21no—
30#1917—21no—
31#1862C411yescharacteristic
32#1862C411yescharacteristic
33#1831C2211yescharacteristic
34#1802C411yescharacteristic
35C4×#107C2211yescharacteristic
36C4×#107C2211yescharacteristic
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40#1770C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1753—21no—
47#1753—21no—
48#1749—21no—
49#1744C2211yescharacteristic
50#1728—21no—
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75#474—24no—
76#436—22no—
77#436—22no—
78#436—22no—
79#436Q814yescomplemented
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8C2×C411yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#389—21no—
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C43C2311yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
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#336—21no—
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—22no—
150#334—22no—
151D4⋊C8—22no—
152D4⋊C8—22no—
153#322—22no—
154C82—22no—
155C22×Q8—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—81no—
163C2×M16—81no—
164C2×M16—81no—
165C2×M16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4×Q8—81no—
183C4×Q8—81no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4C2×Q812yes—
191C4×D4#4112yes—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—81no—
213C2×C42—81no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#107—22no—
222#107—22no—
223#107—41no—
224#107—41no—
225C4⋊C8—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8M1614yescomplemented
236C4⋊C8C2×Q812yes—
237C4⋊C8#4112yes—
238#102—22no—
239#102—22no—
240#102—41no—
241#102—41no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244#97—81no—
245#97—81no—
246#97—81no—
247#97—81no—
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—21no—
258C4×C8—81no—
259C4×C8C2×Q812yes—
260C4×C8#4112yes—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268C24—21no—
269C2×Q8—42no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C4×Q812yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
304M16—84no—
305M16—84no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—21no—
319C2×C8—21no—
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—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8C22⋊Q812yes—
345C2×C8C22⋊Q812yes—
346C2×C8C2×M1612yes—
347C2×C8C2×M1612yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4C4×Q812yes—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—81no—
386C42—81no—
387C42C2×M1612yes—
388C42#13112yes—
389C42C22×Q811yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23C2×C4×D411yescharacteristic
401Q8—84no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4#43612yes—
446C2×C4#43612yes—
447C2×C4C2×C4×Q811yescharacteristic
448C2×C4C4×#4111yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4#52511yescharacteristic
451C2×C4#52911yescharacteristic
452C2×C4#53311yescharacteristic
453C2×C4C22×M1611yescharacteristic
454C2×C4C2×#13111yescharacteristic
455C2×C4#56811yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4#57611yescharacteristic
458C2×C4#57611yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57811yescharacteristic
461C8—22no—
462C8—22no—
463C8—22no—
464C8—22no—
465C8—42no—
466C8—42no—
467C8—41no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—81no—
473C8—81no—
474C8—81no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22C4×C22⋊Q811yescharacteristic
483C22C2×#43611yescharacteristic
484C22#293011yescharacteristic
485C22#294711yescharacteristic
486C22#294811yescharacteristic
487C22#295611yescharacteristic
488C22#295711yescharacteristic
489C4—21no—
490C4—21no—
491C4—22no—
492C4—22no—
493C4—41no—
494C4—41no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4#296211yescharacteristic
503C4Q8×M1611yescharacteristic
504C4#296911yescharacteristic
505C4#297011yescharacteristic
506C4#308111yescharacteristic
507C4#308311yescharacteristic
508C4#311811yescharacteristic
509C4#312411yescharacteristic
510C2—82no—
511C2#1789411yescharacteristic
512C2#1789511yescharacteristic
513C2#1794911yescharacteristic
514C2#1795011yescharacteristic
515C2#2159811yescharacteristic
516C2#2162411yescharacteristic
517C2#2162911yescharacteristic
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. #22311C2semidirect
4. #9864C2semidirect
5. #9603C2semidirect
6. #9540C2semidirect
7. #9537C2semidirect
11. #8287C2semidirect
12. #8282C2semidirect
13. #8264C2semidirect
79. #436Q8semidirect
235. C4⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#223112
C4×#4742
#205682
#98642
#96032
#95402
#95372
#88492
#88472
#85652
#82872
#82822
#82642
#82082
#82062