Finite Group Databasebeta

← #230485

Subgroups — #230485

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23257C211yesmaximal, characteristic, complemented
2#22839C211yesmaximal, characteristic
3C4×#478C211yesmaximal, characteristic
4#9871C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9556C211yesmaximal, characteristic
7#9556C211yesmaximal, characteristic
8#9546C211yesmaximal, characteristic
9#9545C211yesmaximal, characteristic
10#9539C211yesmaximal, characteristic, complemented
11#9537C211yesmaximal, characteristic, complemented
12#8287C211yesmaximal, characteristic, complemented
13#8286C211yesmaximal, characteristic, complemented
14#8261C211yesmaximal, characteristic, complemented
15C4×D4⋊C8C211yesmaximal, characteristic
16C2×#478C411yescharacteristic
17C4×#125C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1989C411yescharacteristic
20#1954C2211yescharacteristic
21#1954C2211yescharacteristic
22#1949C411yescharacteristic
23#1940—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1924—21no—
27#1924—21no—
28#1921C411yescharacteristic
29#1919C411yescharacteristic
30#1917—21no—
31#1916—21no—
32#1864C411yescharacteristic
33#1862C411yescharacteristic
34#1831C2211yescharacteristic
35#1817—21no—
36#1804C411yescharacteristic
37C4×#107C2211yescharacteristic
38C4×#106C2211yescharacteristic
39#1771—21no—
40#1770C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1749—21no—
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×#125C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C22×C42—21no—
72#478—24no—
73C2×#107C2×C411yescharacteristic
74C2×#107C2×C411yescharacteristic
75C2×#106C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2×C411yescharacteristic
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
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×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#389C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#387—21no—
111#387—21no—
112#382C2×C411yescharacteristic
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C43C2311yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
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#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#334—22no—
147D4⋊C8—22no—
148D4⋊C8—22no—
149#322—22no—
150#322—22no—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C22×C411yescharacteristic
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169#125—22no—
170#125—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—41no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—81no—
199C2×C42—81no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42C2411yescharacteristic
207#107—22no—
208#107—41no—
209#106—22no—
210#106—41no—
211C4⋊C8—22no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8—42no—
222C4⋊C8—81no—
223C4⋊C8M1614yes—
224C4⋊C8#4112yes—
225#102—22no—
226#102—22no—
227#102—22no—
228#102—22no—
229C22⋊C8—42no—
230C22⋊C8—42no—
231C22⋊C8—81no—
232C22⋊C8—81no—
233#97—42no—
234#97—81no—
235#97—81no—
236#97—81no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—42no—
249C4×C8—42no—
250C4×C8—81no—
251C4×C8#4112yes—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4#13112yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8C2×M1612yes—
335C2×C8C2×M1612yes—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C22⋊C4—42no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—22no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—81no—
373C42—81no—
374C42C2×M1612yes—
375C42#13112yes—
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C422−1+411yescharacteristic
379C422−1+411yescharacteristic
380C23—22no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C2×C4×D411yescharacteristic
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4#43612yes—
431C2×C4#43612yes—
432C2×C4#44412yes—
433C2×C4C4×#4111yescharacteristic
434C2×C4#51911yescharacteristic
435C2×C4#54711yescharacteristic
436C2×C4#54811yescharacteristic
437C2×C4#55211yescharacteristic
438C2×C4#55611yescharacteristic
439C2×C4C22×M1611yescharacteristic
440C2×C4C2×#13111yescharacteristic
441C2×C4C2×#13111yescharacteristic
442C2×C4#56811yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C8—22no—
448C8—22no—
449C8—42no—
450C8—42no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—41no—
457C8—41no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22#236211yescharacteristic
469C22C2×#43611yescharacteristic
470C22#293311yescharacteristic
471C22#294311yescharacteristic
472C22#294611yescharacteristic
473C22C2×#44411yescharacteristic
474C22#295611yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4#296611yescharacteristic
489C4#297011yescharacteristic
490C4#298411yescharacteristic
491C4#298511yescharacteristic
492C4#335111yescharacteristic
493C4#335511yescharacteristic
494C4#337011yescharacteristic
495C4#337111yescharacteristic
496C2—82no—
497C2#1789011yescharacteristic
498C2#1789311yescharacteristic
499C2#1800211yescharacteristic
500C2#1802011yescharacteristic
501C2#2305511yescharacteristic
502C2#2307111yescharacteristic
503C2#2308111yescharacteristic
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 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23257C2semidirect
4. #9871C2semidirect
5. #9603C2semidirect
10. #9539C2semidirect
11. #9537C2semidirect
12. #8287C2semidirect
13. #8286C2semidirect
14. #8261C2semidirect

Maximal subgroups

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

subgroupindex
#232572
#228392
C4×#4782
#98712
#96032
#95562
#95462
#95452
#95392
#95372
#82872
#82862
#82612
C4×D4⋊C82