Finite Group Databasebeta

← #225241

Subgroups — #225241

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22632C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic
3C4×#428C211yesmaximal, characteristic
4#9596C211yesmaximal, characteristic, complemented
5#9593C211yesmaximal, characteristic, complemented
6#9536C211yesmaximal, characteristic, complemented
7#9535C211yesmaximal, characteristic, complemented
8#9534C211yesmaximal, characteristic, complemented
9#9533C211yesmaximal, characteristic, complemented
10#8708C211yesmaximal, characteristic
11#8707C211yesmaximal, characteristic
12#8597C211yesmaximal, characteristic
13#8284C211yesmaximal, characteristic, complemented
14C8×#107C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic
16C2×#431—21no—
17C2×#428C411yescharacteristic
18C2×#428C411yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C4×#117C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1939C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1923—21no—
27#1923—21no—
28#1916—21no—
29#1916—21no—
30#1831—21no—
31#1817—21no—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34#1776C411yescharacteristic
35#1776C411yescharacteristic
36#1776C411yescharacteristic
37#1776C411yescharacteristic
38C4×#107C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×C22⋊C8C2211yescharacteristic
47C4×C22⋊C8C2211yescharacteristic
48#1744—21no—
49#1736C411yescharacteristic
50#1736C411yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#322C2211yescharacteristic
54C2×C82C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16C2×C411yescharacteristic
68C22×M16C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#431—44no—
76#428—24no—
77#428C818yescomplemented
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C4×M16—22no—
100C4×M16—22no—
101C2×#97C2×C411yescharacteristic
102C2×#97C2×C411yescharacteristic
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125#376—21no—
126C2×#95—21no—
127C43—21no—
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—22no—
150#334—22no—
151#322—22no—
152C82—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—41no—
163C2×M16C2×C814yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178#117—22no—
179#117—22no—
180#117—41no—
181#117C2×C814yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×C814yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—22no—
219#107C2×C814yes—
220#107C2×C814yes—
221#107M1614yes—
222#106—22no—
223#106—22no—
224#106C2×C814yes—
225#106C2×C814yes—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—81no—
237C4⋊C8—81no—
238C22⋊C8—22no—
239C22⋊C8—41no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—81no—
243#97—22no—
244#97—22no—
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—42no—
256C4×C8—42no—
257C4×C8—42no—
258C4×C8—42no—
259C4×C8#4112yes—
260C4×C8#4112yes—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24C4×D411yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C22×C812yes—
302C22×C4#13112yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306M16—28no—
307M16—44no—
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—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—41no—
328C2×C8—41no—
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×C8#11712yes—
345C2×C8#11712yes—
346C2×C8C22×C812yes—
347C2×C8C22×C812yes—
348C2×C8C2×M1612yes—
349C2×C8AGL(1,ℤ/8)12yes—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—24no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4C22×C812yes—
363C4⋊C4C22×C812yes—
364C4⋊C4C22×C812yes—
365C4⋊C4C22×C812yes—
366C4⋊C4C2×M1612yes—
367C4⋊C4C2×M1612yes—
368C22⋊C4—24no—
369C22⋊C4—42no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—24no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—81no—
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C422−1+411yescharacteristic
393C422−1+411yescharacteristic
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23C8×D412yes—
401C23C2×C4×D411yescharacteristic
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—48no—
423C2×C4—41no—
424C2×C4—41no—
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—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—84no—
448C2×C4C8×D412yes—
449C2×C4C2×#11711yescharacteristic
450C2×C4#51911yescharacteristic
451C2×C4#54711yescharacteristic
452C2×C4#54711yescharacteristic
453C2×C4#55211yescharacteristic
454C2×C4#55211yescharacteristic
455C2×C4C23×C811yescharacteristic
456C2×C4C22×M1611yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4#56811yescharacteristic
459C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57811yescharacteristic
463C8—22no—
464C8—22no—
465C8—22no—
466C8—42no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—81no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8#43114yescomplemented
477C22—22no—
478C22—22no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22#234411yescharacteristic
485C22C2×C8×D411yescharacteristic
486C22D4×M1611yescharacteristic
487C22C4×AGL(1,ℤ/8)11yescharacteristic
488C22#294711yescharacteristic
489C22#295511yescharacteristic
490C22#295611yescharacteristic
491C4—22no—
492C4—21no—
493C4—21no—
494C4—41no—
495C4—41no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—81no—
503C4—81no—
504C4C2×#43111yescharacteristic
505C4#292411yescharacteristic
506C4#298211yescharacteristic
507C4#298411yescharacteristic
508C4#334311yescharacteristic
509C4#334811yescharacteristic
510C4#335211yescharacteristic
511C4#335411yescharacteristic
512C2—42no—
513C2C8×AGL(1,ℤ/8)11yescharacteristic
514C2#1785311yescharacteristic
515C2#1804611yescharacteristic
516C2#1805311yescharacteristic
517C2#2259911yescharacteristic
518C2#2260811yescharacteristic
519C2#2262411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9596C2semidirect
5. #9593C2semidirect
6. #9536C2semidirect
7. #9535C2semidirect
8. #9534C2semidirect
9. #9533C2semidirect
13. #8284C2semidirect
14. C8×#107C2semidirect
77. #428C8semidirect
476. C8#431semidirect

Maximal subgroups

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

subgroupindex
#226322
#225992
C4×#4282
#95962
#95932
#95362
#95352
#95342
#95332
#87082
#87072
#85972
#82842
C8×#1072
#81462