Finite Group Databasebeta

← #226839

Subgroups — #226839

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22799C211yesmaximal, characteristic, complemented
2#22795C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#9665C212yesmaximal
5#9546C212yesmaximal, complemented
6#9542C212yesmaximal, complemented
7#8805C211yesmaximal, characteristic, complemented
8#8563C211yesmaximal, characteristic, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8288C212yesmaximal, complemented
11#8261C211yesmaximal, characteristic, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×#432—21no—
16C2×#431—21no—
17C42×D4C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1954C2212yes—
20#1954C2212yes—
21#1924—22no—
22#1923—22no—
23#1842C2211yescharacteristic
24#1842C2211yescharacteristic
25#1797—21no—
26#1797—21no—
27C4×#107C411yescharacteristic
28C4×#107C411yescharacteristic
29C4×#107C2211yescharacteristic
30#1770C2212yes—
31C8×C4⋊C4C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34C4×#102C411yescharacteristic
35C4×#102C411yescharacteristic
36C4×#102C411yescharacteristic
37C4×#102C411yescharacteristic
38C4×#102C2211yescharacteristic
39C4×#102C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×#334C2212yes—
42C2×#334C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×#322C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1273C2212yes—
49C22×D8C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C4×D8—24no—
59C4×D8—24no—
60C4×D8—22no—
61C4×D8—22no—
62#432—44no—
63#431—44no—
64C2×#107C2×C411yescharacteristic
65C2×#107C2×C411yescharacteristic
66C2×#107C2×C411yescharacteristic
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#102C2×C411yescharacteristic
77C2×#102C2×C411yescharacteristic
78C2×#102C2×C411yescharacteristic
79C2×#102C2×C411yescharacteristic
80C2×#102C2×C411yescharacteristic
81C2×#102C2×C411yescharacteristic
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—22no—
87C2×C4×C8—22no—
88C2×C4×C8C2×C411yescharacteristic
89C2×C4×C8C2×C411yescharacteristic
90C2×C4×C8C2×C411yescharacteristic
91C2×C4×C8C2×C411yescharacteristic
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96C4×C4⋊C4C2×C411yescharacteristic
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C43C2311yescharacteristic
109#340C2×C411yescharacteristic
110#340C2×C411yescharacteristic
111#340C2×C411yescharacteristic
112#340C2×C411yescharacteristic
113#340C2×C411yescharacteristic
114#340C2×C411yescharacteristic
115#336—22no—
116#336—22no—
117#336—21no—
118#336D412yes—
119#334—24no—
120#334—42no—
121D4⋊C8—42no—
122D4⋊C8—42no—
123#322—24no—
124C22×D4C22×C411yescharacteristic
125C22×D4C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C22×C411yescharacteristic
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4M1614yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
189#107—22no—
190#107—22no—
191#107—22no—
192C4⋊C8—24no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8—81no—
202C4⋊C8—81no—
203#102—22no—
204#102—22no—
205#102—22no—
206#102—22no—
207#102—22no—
208#102M1614yes—
209C22⋊C8—44no—
210C22⋊C8—44no—
211C22⋊C8—82no—
212C22⋊C8—82no—
213#97—44no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8—21no—
222C4×C8—22no—
223C4×C8—41no—
224C4×C8—44no—
225C4×C8#4112yes—
226C4×C8#4112yes—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233C24—21no—
234C24—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C2×M1612yes—
246C2×D4C2×M1612yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4111yescharacteristic
284D8—28no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—24no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8#11712yes—
310C2×C8#11712yes—
311C2×C8C2×M1612yes—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4C2×M1612yes—
323C4⋊C4C2×M1612yes—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—22no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42C2×M1612yes—
347C42#13112yes—
348C42AGL(1,ℤ/8)12yes—
349C42AGL(1,ℤ/8)12yes—
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C42H(5,𝔽2)11yescharacteristic
353C422−1+411yescharacteristic
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23C2×C4×D411yescharacteristic
369D4—22no—
370D4—22no—
371D4—22no—
372D4—22no—
373D4—41no—
374D4—41no—
375D4—41no—
376D4—41no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—48no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—84no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4#43512yes—
417C2×C4#43512yes—
418C2×C4#44212yes—
419C2×C4#44212yes—
420C2×C4#44312yes—
421C2×C4C2×#11711yescharacteristic
422C2×C4#52011yescharacteristic
423C2×C4#54711yescharacteristic
424C2×C4#55011yescharacteristic
425C2×C4#55511yescharacteristic
426C2×C4#55611yescharacteristic
427C2×C4C22×M1611yescharacteristic
428C2×C4C22×M1611yescharacteristic
429C2×C4C2×#13111yescharacteristic
430C2×C4#56811yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C8—22no—
436C8—22no—
437C8—22no—
438C8—44no—
439C8—42no—
440C8—42no—
441C8—41no—
442C8—82no—
443C8—82no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22#235411yescharacteristic
459C22C2×#43511yescharacteristic
460C22D4×M1611yescharacteristic
461C22C2×#44211yescharacteristic
462C22#295211yescharacteristic
463C22C2×#44311yescharacteristic
464C22#295711yescharacteristic
465C4—21no—
466C4—21no—
467C4—22no—
468C4—22no—
469C4—42no—
470C4—41no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4#158614yes—
477C4#292011yescharacteristic
478C4#292511yescharacteristic
479C4#298911yescharacteristic
480C4#299011yescharacteristic
481C4#335411yescharacteristic
482C4#335411yescharacteristic
483C4#339711yescharacteristic
484C4#339711yescharacteristic
485C2—82no—
486C2—82no—
487C2C2×#158611yescharacteristic
488C2#1788211yescharacteristic
489C2#1804411yescharacteristic
490C2#1805711yescharacteristic
491C2#2279711yescharacteristic
492C2#2280211yescharacteristic
493C2#2281311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22799C2semidirect
2. #22795C2semidirect
5. #9546C2semidirect
6. #9542C2semidirect
7. #8805C2semidirect
8. #8563C2semidirect
9. #8563C2semidirect
10. #8288C2semidirect
11. #8261C2semidirect

Maximal subgroups

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

subgroupindex
#227992
#227952
C42×D82
#96652
#95462
#95422
#88052
#85632
#82882
#82612