Finite Group Databasebeta

← #229047

Subgroups — #229047

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775C211yesmaximal, characteristic
2#23153C211yesmaximal, characteristic, complemented
3#22701C211yesmaximal, characteristic
4#9877C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9600C211yesmaximal, characteristic, complemented
7#9593C211yesmaximal, characteristic, complemented
8#9539C211yesmaximal, characteristic, complemented
9#9538C211yesmaximal, characteristic, complemented
10#8847C211yesmaximal, characteristic
11#8845C211yesmaximal, characteristic
12#8262C211yesmaximal, characteristic, complemented
13#8206C211yesmaximal, characteristic
14#8204C211yesmaximal, characteristic
15#8181C211yesmaximal, characteristic
16C2×#446—21no—
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2293C2211yescharacteristic
21#2100C411yescharacteristic
22#2094C411yescharacteristic
23#2088C411yescharacteristic
24#2088C411yescharacteristic
25#2065C411yescharacteristic
26#2065C411yescharacteristic
27#2058C411yescharacteristic
28#2023C411yescharacteristic
29#1984—21no—
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1917—21no—
35#1917—21no—
36#1851—21no—
37#1850—21no—
38#1831C2211yescharacteristic
39#1798—21no—
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1729—21no—
50C4×#97C2211yescharacteristic
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—
68C23×C8—21no—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#446—44no—
74#436—22no—
75#436—41no—
76C8×D4—22no—
77C8×D4—41no—
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#390C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#389—21no—
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#388—21no—
120#388—21no—
121#384C2×C411yescharacteristic
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
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#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—41no—
150#334—41no—
151D4⋊C8—22no—
152D4⋊C8Q814yescomplemented
153#322—41no—
154C82—41no—
155C22×Q8—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—81no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4C2×Q812yes—
185C4×D4#4112yes—
186#117—81no—
187#117—81no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#106—22no—
218#106—41no—
219#106—41no—
220#106—41no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8C2×Q812yes—
237#102—22no—
238#102—41no—
239#102—41no—
240#102—41no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243#97—24no—
244#97—42no—
245#97—42no—
246#97—42no—
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—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—42no—
259C4×C8C2×Q812yes—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267C24—21no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C4×Q812yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297M16—84no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
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—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—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—84no—
339C2×C8#13112yes—
340C2×C8AGL(1,ℤ/8)12yes—
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—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4C4×Q812yes—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—42no—
365C22⋊C4—81no—
366C22⋊C4—81no—
367C22⋊C4—81no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—22no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C42H(5,𝔽2)11yescharacteristic
387C422−1+411yescharacteristic
388C23—22no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C2×C4×D411yescharacteristic
396Q8—84no—
397D4—22no—
398D4—22no—
399D4—41no—
400D4—41no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
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—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
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#44312yes—
440C2×C4C2×C4×Q811yescharacteristic
441C2×C4#52011yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4D4×Q811yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4#55411yescharacteristic
446C2×C4C2×#13111yescharacteristic
447C2×C4#56811yescharacteristic
448C2×C4#56811yescharacteristic
449C2×C4#56811yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57711yescharacteristic
454C8—24no—
455C8—22no—
456C8—22no—
457C8—42no—
458C8—42no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—81no—
466C8—81no—
467C8—81no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#236011yescharacteristic
476C22#293011yescharacteristic
477C22#293511yescharacteristic
478C22C4×AGL(1,ℤ/8)11yescharacteristic
479C22#294911yescharacteristic
480C22C2×#44311yescharacteristic
481C22#295611yescharacteristic
482C4—22no—
483C4—21no—
484C4—21no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—42no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4#296111yescharacteristic
496C4#296211yescharacteristic
497C4#299211yescharacteristic
498C4#299511yescharacteristic
499C4#330611yescharacteristic
500C4#331111yescharacteristic
501C4#338411yescharacteristic
502C4#338811yescharacteristic
503C2—82no—
504C2#1794211yescharacteristic
505C2#1795111yescharacteristic
506C2#1807811yescharacteristic
507C2#1808511yescharacteristic
508C2#2296111yescharacteristic
509C2#2297511yescharacteristic
510C2#2298211yescharacteristic
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
2. #23153C2semidirect
4. #9877C2semidirect
5. #9603C2semidirect
6. #9600C2semidirect
7. #9593C2semidirect
8. #9539C2semidirect
9. #9538C2semidirect
12. #8262C2semidirect
152. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#237752
#231532
#227012
#98772
#96032
#96002
#95932
#95392
#95382
#88472
#88452
#82622
#82062
#82042
#81812