Finite Group Databasebeta

← #218280

Subgroups — #218280

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22299C211yesmaximal, characteristic, complemented
2#22008C211yesmaximal, characteristic
3C4×#478C211yesmaximal, characteristic
4#9866C211yesmaximal, characteristic, complemented
5#9601C211yesmaximal, characteristic, complemented
6#9600C211yesmaximal, characteristic, complemented
7#9598C211yesmaximal, characteristic, complemented
8#9560C211yesmaximal, characteristic
9#9558C211yesmaximal, characteristic
10#9541C211yesmaximal, characteristic
11#9541C211yesmaximal, characteristic
12#9536C211yesmaximal, characteristic, complemented
13#9534C211yesmaximal, characteristic, complemented
14#8563C211yesmaximal, characteristic
15#8265C211yesmaximal, characteristic, complemented
16C2×#478C411yescharacteristic
17C4×#125C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1989C411yescharacteristic
20#1954C2211yescharacteristic
21#1954C2211yescharacteristic
22#1949C411yescharacteristic
23#1939—21no—
24#1939—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1923—21no—
30#1923—21no—
31#1921C411yescharacteristic
32#1919C411yescharacteristic
33#1916—21no—
34#1916—21no—
35#1864C411yescharacteristic
36#1862C411yescharacteristic
37#1842C2211yescharacteristic
38#1817—21no—
39#1804C411yescharacteristic
40#1797—21no—
41C4×#107C2211yescharacteristic
42C4×#106C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
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⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#389C2×C411yescharacteristic
108#389C2×C411yescharacteristic
109#387—21no—
110#387—21no—
111#382C2×C411yescharacteristic
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C43C2311yescharacteristic
123#340C2×C411yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336C2×C411yescharacteristic
140#336C2×C411yescharacteristic
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—41no—
146#335—41no—
147D4⋊C8—41no—
148D4⋊C8—41no—
149#322—41no—
150#322—41no—
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#125M1614yes—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
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—22no—
209#106—22no—
210#106—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—42no—
222C4⋊C8—81no—
223C4⋊C8—81no—
224C4⋊C8—81no—
225C4⋊C8—81no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228#102—22no—
229#102—22no—
230#102—22no—
231#102—22no—
232C22⋊C8—81no—
233C22⋊C8—81no—
234C22⋊C8—81no—
235C22⋊C8—81no—
236C4×C8—21no—
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—81no—
249C4×C8—81no—
250C4×C8—81no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4#13112yes—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
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—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
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×C8—41no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—24no—
339C4⋊C4—24no—
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—81no—
349C4⋊C4—81no—
350C4⋊C4C2×M1612yes—
351C4⋊C4C2×M1612yes—
352C4⋊C4#13112yes—
353C22⋊C4—24no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—42no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—81no—
372C42—81no—
373C42C2×M1612yes—
374C42#13112yes—
375C42H(5,𝔽2)11yescharacteristic
376C422−1+411yescharacteristic
377C422−1+411yescharacteristic
378C422−1+411yescharacteristic
379C23—22no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23C2×C4×D411yescharacteristic
387D4—22no—
388D4—22no—
389D4—41no—
390D4—41no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
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#43512yes—
431C2×C4#43512yes—
432C2×C4#51911yescharacteristic
433C2×C4#52011yescharacteristic
434C2×C4#53611yescharacteristic
435C2×C4#53711yescharacteristic
436C2×C4#54111yescharacteristic
437C2×C4#54211yescharacteristic
438C2×C4C22×M1611yescharacteristic
439C2×C4C2×#13111yescharacteristic
440C2×C4C2×#13111yescharacteristic
441C2×C4C2×#13111yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57811yescharacteristic
446C8—42no—
447C8—41no—
448C8—41no—
449C8—41no—
450C8—41no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22#232311yescharacteristic
468C22C2×#43511yescharacteristic
469C22#292911yescharacteristic
470C22#294611yescharacteristic
471C22#295111yescharacteristic
472C22#295611yescharacteristic
473C22#295711yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—42no—
480C4—42no—
481C4—41no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4#298411yescharacteristic
488C4#298611yescharacteristic
489C4#299111yescharacteristic
490C4#299411yescharacteristic
491C4#324111yescharacteristic
492C4#324311yescharacteristic
493C4#325811yescharacteristic
494C4#325911yescharacteristic
495C2—82no—
496C2#1787511yescharacteristic
497C2#1788111yescharacteristic
498C2#1791811yescharacteristic
499C2#1792611yescharacteristic
500C2#2210011yescharacteristic
501C2#2212611yescharacteristic
502C2#2213211yescharacteristic
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. #22299C2semidirect
4. #9866C2semidirect
5. #9601C2semidirect
6. #9600C2semidirect
7. #9598C2semidirect
12. #9536C2semidirect
13. #9534C2semidirect
15. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#222992
#220082
C4×#4782
#98662
#96012
#96002
#95982
#95602
#95582
#95412
#95362
#95342
#85632
#82652