Finite Group Databasebeta

← #225249

Subgroups — #225249

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626C211yesmaximal, characteristic
2#22600C211yesmaximal, characteristic
3#20816C211yesmaximal, characteristic
4#9596C211yesmaximal, characteristic, complemented
5#9596C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#9535C211yesmaximal, characteristic, complemented
9#9534C211yesmaximal, characteristic, complemented
10#8710C211yesmaximal, characteristic
11#8709C211yesmaximal, characteristic
12#8596C211yesmaximal, characteristic
13#8286C211yesmaximal, characteristic, complemented
14#8285C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic
16C2×#433—21no—
17C22×C4×C8C2211yescharacteristic
18#2285C2211yescharacteristic
19#2285C2211yescharacteristic
20#1939C2211yescharacteristic
21#1939C2211yescharacteristic
22#1939C2211yescharacteristic
23#1939C2211yescharacteristic
24#1917—21no—
25#1917—21no—
26#1916—21no—
27#1916—21no—
28#1844C411yescharacteristic
29#1844C411yescharacteristic
30#1834C411yescharacteristic
31#1834C411yescharacteristic
32#1818—21no—
33#1807C411yescharacteristic
34#1807C411yescharacteristic
35#1807C411yescharacteristic
36#1807C411yescharacteristic
37#1798—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#322C2211yescharacteristic
54C2×#322C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8C2×C411yescharacteristic
68C23×C8C2×C411yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#433—44no—
73C2×#106C2×C411yescharacteristic
74C2×#106C2×C411yescharacteristic
75C2×#106C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2×C411yescharacteristic
101C2×C4×C8C2×C411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382—21no—
113#382—21no—
114#380—21no—
115#380—21no—
116#380C2×C411yescharacteristic
117#380C2×C411yescharacteristic
118#380C2×C411yescharacteristic
119#380C2×C411yescharacteristic
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#336—21no—
135#336—21no—
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#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#335—22no—
149#322—22no—
150#322—22no—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—41no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174#117—42no—
175#117—42no—
176#117—81no—
177#117—81no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
195C2×C22⋊C4—21no—
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×C42C22×C411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#106—22no—
211#106—22no—
212#106—22no—
213#106—22no—
214#106—22no—
215#106—22no—
216#106—41no—
217#106—41no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—42no—
233C22⋊C8—42no—
234C22⋊C8—42no—
235#97—42no—
236#97—81no—
237C4×C8—22no—
238C4×C8—22no—
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—21no—
249C4×C8—42no—
250C4×C8—81no—
251C4×C8#4112yes—
252C4×C8#4112yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24C4×D411yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4#13712yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
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—44no—
312C2×C8—44no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
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#11712yes—
332C2×C8#11712yes—
333C2×C8#13112yes—
334C2×C8#13112yes—
335C2×C8#13112yes—
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—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—81no—
363C22⋊C4—81no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42C2×#4111yescharacteristic
379C42C2×#4111yescharacteristic
380C422−1+411yescharacteristic
381C422−1+411yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#44012yes—
389C23C2×C4×D411yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
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—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#44012yes—
434C2×C4#44412yes—
435C2×C4C2×#11711yescharacteristic
436C2×C4#51911yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4#55211yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4C2×#13111yescharacteristic
442C2×C4C2×#13111yescharacteristic
443C2×C4C2×#13111yescharacteristic
444C2×C4#56811yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C8—22no—
450C8—22no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C8—81no—
462C8#43314yescomplemented
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22#234411yescharacteristic
471C22#293111yescharacteristic
472C22#293511yescharacteristic
473C22C2×#44011yescharacteristic
474C22#294911yescharacteristic
475C22C2×#44411yescharacteristic
476C22#295711yescharacteristic
477C4—21no—
478C4—21no—
479C4—22no—
480C4—22no—
481C4—41no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4C2×#43311yescharacteristic
491C4#292411yescharacteristic
492C4#298511yescharacteristic
493C4#298611yescharacteristic
494C4#334411yescharacteristic
495C4#334411yescharacteristic
496C4#334511yescharacteristic
497C4#334511yescharacteristic
498C2—42no—
499C2#1796011yescharacteristic
500C2#1797111yescharacteristic
501C2#1807111yescharacteristic
502C2#1808911yescharacteristic
503C2#2260011yescharacteristic
504C2#2260711yescharacteristic
505C2#2261511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×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. #9596C2semidirect
6. #9539C2semidirect
7. #9538C2semidirect
8. #9535C2semidirect
9. #9534C2semidirect
13. #8286C2semidirect
14. #8285C2semidirect
462. C8#433semidirect

Maximal subgroups

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

subgroupindex
#226262
#226002
#208162
#95962
#95392
#95382
#95352
#95342
#87102
#87092
#85962
#82862
#82852
#81442