Finite Group Databasebeta

← #496447

Subgroups — #496447

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic
2C2×#2164C211yesmaximal, characteristic
3#20821C211yesmaximal, characteristic
4#13284C211yesmaximal, characteristic, complemented
5#13283C211yesmaximal, characteristic, complemented
6#13282C211yesmaximal, characteristic, complemented
7#13281C211yesmaximal, characteristic, complemented
8#13277C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13276C211yesmaximal, characteristic, complemented
11#13275C211yesmaximal, characteristic, complemented
12#12868C211yesmaximal, characteristic
13#12868C211yesmaximal, characteristic
14#12867C211yesmaximal, characteristic
15#12866C211yesmaximal, characteristic
16C2×#544—21no—
17#3164—24no—
18#3164C418yescomplemented
19C2×#501—21no—
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#499C2211yescharacteristic
24C2×#499C2211yescharacteristic
25C2×#428—21no—
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2314—21no—
30#2164—24no—
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1846C2211yescharacteristic
38#1845C2211yescharacteristic
39#1836—21no—
40#1809—21no—
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1776—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1380C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64#544—28no—
65#544—44no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—41no—
69C2×#125—21no—
70C2×#125—21no—
71C2×#125—41no—
72C2×C22⋊Q8—41no—
73C2×C22⋊Q8—41no—
74C2×#117—21no—
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#501—28no—
82#501—28no—
83#500—22no—
84#500—22no—
85#500C2×C414yes—
86#500C2×C414yes—
87#499—22no—
88#499—22no—
89#499C2×C414yes—
90#499C2×C414yes—
91#428—28no—
92#427—22no—
93#427C2×C414yes—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#107—22no—
98C2×#107—21no—
99C2×#107C2×C414yes—
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#107C2311yescharacteristic
108C2×#106—21no—
109C2×#106—21no—
110C2×#106—21no—
111C2×#106—21no—
112#405C2×C414yes—
113#405D414yes—
114C2×#97D411yescharacteristic
115C2×#97D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#388—41no—
119#388—41no—
120#387—41no—
121#387—41no—
122#384—21no—
123#382—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4⋊C16—22no—
129C4⋊C16—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#346—22no—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138C22×Q8—41no—
139C22×Q8—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C2×M16—22no—
143C2×M16—22no—
144C2×M16C22⋊C414yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8C22×C412yes—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D412yes—
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—48no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160#125—24no—
161#125—24no—
162#125—48no—
163#125—42no—
164#125—42no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167#117—24no—
168#117—42no—
169#117C22⋊C414yes—
170#117SD1614yes—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4D811yescharacteristic
191C2×C4⋊C4D811yescharacteristic
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4D811yescharacteristic
196C2×C22⋊C4D811yescharacteristic
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214#107—24no—
215#107—24no—
216#107—24no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—42no—
230#107C22×C412yes—
231#107C22×C412yes—
232#107C22×C412yes—
233#107C22×C412yes—
234#106—24no—
235#106—24no—
236#106—24no—
237#106—42no—
238#97C22⋊C414yes—
239#97C22⋊C414yes—
240C4×C8C22×C412yes—
241C4×C8C22×C412yes—
242C4×C8C2×D412yes—
243C4×C8C2×D412yes—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250C24C2×D811yescharacteristic
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—161no—
254C2×Q8—161no—
255C2×Q8—161no—
256C2×Q8—161no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—161no—
271C22×C4—161no—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×SD1612yes—
279C22×C4C22×D411yescharacteristic
280M16—28no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8C2×C22⋊C412yes—
290C2×C8C2×C22⋊C412yes—
291C2×C8C2×C22⋊C412yes—
292C2×C8#13712yes—
293C2×C8#13712yes—
294C2×C8C23×C411yescharacteristic
295C2×C8C22×D411yescharacteristic
296C2×C82−1+411yescharacteristic
297C2×C82−1+411yescharacteristic
298C2×C82−1+411yescharacteristic
299C2×C82−1+411yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—161no—
324C4⋊C4—161no—
325C4⋊C4#10214yes—
326C4⋊C4#10214yes—
327C22⋊C4—44no—
328C22⋊C4—161no—
329C22⋊C4—161no—
330C22⋊C4#10214yes—
331C22⋊C4#10214yes—
332C42—44no—
333C42—161no—
334C42—161no—
335C42C2×C22⋊C412yes—
336C42C2×C22⋊C412yes—
337C42C2×SD1612yes—
338C42C2×SD1612yes—
339C16—41no—
340C16—41no—
341C16—41no—
342C16—41no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23C2×#10212yes—
350C23C22×D811yescharacteristic
351Q8—164no—
352Q8—164no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
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—164no—
392C2×C4C2×#10212yes—
393C2×C4C2×#10212yes—
394C2×C4C2×#10212yes—
395C2×C4C2×#10212yes—
396C2×C4C2×#10212yes—
397C2×C4C22×C22⋊C411yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4C22×SD1611yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C8—22no—
404C8—22no—
405C8#41914yes—
406C8#51911yescharacteristic
407C8#51911yescharacteristic
408C8#53611yescharacteristic
409C8#53611yescharacteristic
410C22—22no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22C22×#10211yescharacteristic
418C22#322611yescharacteristic
419C22#322611yescharacteristic
420C22#341511yescharacteristic
421C22#341611yescharacteristic
422C22#341611yescharacteristic
423C22#341711yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#231011yescharacteristic
438C4C2×#41911yescharacteristic
439C4#322911yescharacteristic
440C4#322911yescharacteristic
441C2—42no—
442C2#2183311yescharacteristic
443C2#3338111yescharacteristic
444C2#3338211yescharacteristic
445C2#3380711yescharacteristic
446C2#3380811yescharacteristic
447C2#3380911yescharacteristic
448C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×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. #13284C2semidirect
5. #13283C2semidirect
6. #13282C2semidirect
7. #13281C2semidirect
8. #13277C2semidirect
9. #13277C2semidirect
10. #13276C2semidirect
11. #13275C2semidirect
18. #3164C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#21642
#208212
#132842
#132832
#132822
#132812
#132772
#132762
#132752
#128682
#128672
#128662