Finite Group Databasebeta

← #505671

Subgroups — #505671

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C211yesmaximal, characteristic
2C2×#2232C211yesmaximal, characteristic
3#21524C211yesmaximal, characteristic
4#20803C211yesmaximal, characteristic, complemented
5#13507C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13482C211yesmaximal, characteristic, complemented
8#13480C211yesmaximal, characteristic, complemented
9#13178C211yesmaximal, characteristic, complemented
10#13176C211yesmaximal, characteristic, complemented
11#12868C211yesmaximal, characteristic, complemented
12#12867C211yesmaximal, characteristic, complemented
13#12813C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12784C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#428—21no—
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2324—21no—
23#2317—21no—
24#2234—24no—
25#2232—24no—
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38#1970C2211yescharacteristic
39#1942C2211yescharacteristic
40#1936—21no—
41#1895—21no—
42#1847—21no—
43#1847—21no—
44#1846C2211yescharacteristic
45#1846C2211yescharacteristic
46#1845C2211yescharacteristic
47#1845C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1762—21no—
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#466—22no—
74#466D414yes—
75C4×D8—22no—
76C4×D8—22no—
77#428—44no—
78#427—22no—
79#427—22no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98#405—22no—
99#405—22no—
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#390—21no—
105#390—41no—
106#389—21no—
107#389—41no—
108#389—41no—
109#388—41no—
110#386—41no—
111#385—41no—
112#385—41no—
113#385—41no—
114#384—21no—
115#384—21no—
116#382—21no—
117#382—21no—
118#380—21no—
119C4×C4⋊C4—21no—
120#367—22no—
121#366—22no—
122#366—22no—
123#366—22no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135C22×D4—21no—
136C22×D4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8C2×D412yes—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D412yes—
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8#4112yes—
158C22×C8C2411yescharacteristic
159#121—24no—
160#121—42no—
161C4×D4—42no—
162C4×D4—42no—
163#117—22no—
164#117—22no—
165#117—42no—
166#117—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D811yescharacteristic
183C2×C4⋊C4D811yescharacteristic
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4D811yescharacteristic
191C2×C22⋊C4D811yescharacteristic
192C2×C42—41no—
193C2×C42—41no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—42no—
222#107—42no—
223#107—42no—
224#107C2×D412yes—
225#106—42no—
226#106—42no—
227#106—42no—
228#102—24no—
229#102—42no—
230#102—42no—
231#97—22no—
232#97—22no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8#4112yes—
236C4×C8#4112yes—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244C24—41no—
245C24C2×D811yescharacteristic
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—161no—
254C2×D4—161no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—161no—
269C22×C4—161no—
270C22×C4—161no—
271C22×C4—161no—
272C22×C4#12112yes—
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4AGL(1,ℤ/8)12yes—
279C22×C4C22×D411yescharacteristic
280D8—22no—
281D8—22no—
282D8—41no—
283D8—41no—
284M16—28no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8#12112yes—
296C2×C8#12312yes—
297C2×C8#13712yes—
298C2×C8C22×D411yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C82−1+411yescharacteristic
303C2×C82−1+411yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—82no—
321C4⋊C4—161no—
322C4⋊C4—161no—
323C4⋊C4—161no—
324C4⋊C4—161no—
325C4⋊C4—161no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—44no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—161no—
332C22⋊C4—161no—
333C22⋊C4—161no—
334C42—21no—
335C42—21no—
336C42—82no—
337C42—161no—
338C42#12312yes—
339C42#13712yes—
340C16—41no—
341C16—41no—
342C16—41no—
343C16—41no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#45912yes—
358C23C22×D811yescharacteristic
359D4—42no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—164no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—164no—
402C2×C4—164no—
403C2×C4#45912yes—
404C2×C4#48012yes—
405C2×C4#48012yes—
406C2×C4C2×#12111yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4#53611yescharacteristic
409C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4C2×#13711yescharacteristic
412C8—22no—
413C8—22no—
414C8—22no—
415C8#54711yescharacteristic
416C8#54711yescharacteristic
417C8#55211yescharacteristic
418C8#55211yescharacteristic
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22C2×#45911yescharacteristic
434C22C2×#48011yescharacteristic
435C22#322611yescharacteristic
436C22#341511yescharacteristic
437C22#341511yescharacteristic
438C22#341611yescharacteristic
439C22#341711yescharacteristic
440C4—22no—
441C4—22no—
442C4—22no—
443C4—41no—
444C4—41no—
445C4—82no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4#248611yescharacteristic
453C4#303911yescharacteristic
454C4#309211yescharacteristic
455C4#323811yescharacteristic
456C2—42no—
457C2—162no—
458C2#2764411yescharacteristic
459C2#3358511yescharacteristic
460C2#3359011yescharacteristic
461C2#3365011yescharacteristic
462C2#3365211yescharacteristic
463C2#3380711yescharacteristic
464C2#3380911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20803C2semidirect
5. #13507C2semidirect
6. #13505C2semidirect
7. #13482C2semidirect
8. #13480C2semidirect
9. #13178C2semidirect
10. #13176C2semidirect
11. #12868C2semidirect
12. #12867C2semidirect
13. #12813C2semidirect
14. #12811C2semidirect
15. #12784C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C2×#22322
#215242
#208032
#135072
#135052
#134822
#134802
#131782
#131762
#128682
#128672
#128132
#128112
#127842