Finite Group Databasebeta

← #499497

Subgroups — #499497

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2176C211yesmaximal, characteristic, complemented
2C2×#2164C211yesmaximal, characteristic
3#21525C211yesmaximal, characteristic
4#21525C211yesmaximal, characteristic
5#13516C211yesmaximal, characteristic, complemented
6#13494C211yesmaximal, characteristic
7#13477C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13450C211yesmaximal, characteristic, complemented
10#13383C211yesmaximal, characteristic
11#13382C211yesmaximal, characteristic
12#13306C211yesmaximal, characteristic
13#12674C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C4×#358C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2323—21no—
23#2323—21no—
24#2176—24no—
25#2164—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1935—21no—
46#1935—21no—
47#1894—21no—
48#1894—21no—
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1763—21no—
54#1763—21no—
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×M16D411yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117D411yescharacteristic
70C2×Q32—22no—
71C2×Q32—22no—
72C2×Q32—22no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#467—22no—
78#467—22no—
79#467—22no—
80#467—22no—
81C4×Q16—22no—
82C4×Q16—22no—
83C4×Q16—22no—
84C4×Q16—22no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97#405—22no—
98#405—22no—
99C2×#97D411yescharacteristic
100C2×#97D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#391—21no—
105#389—21no—
106#389—21no—
107#389—41no—
108#389—41no—
109#388—41no—
110#388—41no—
111#388—41no—
112#388—41no—
113#387—41no—
114#387—41no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119#367—22no—
120#367—22no—
121#366—22no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#358—22no—
128#358—22no—
129#358—22no—
130C22⋊C16—22no—
131C22⋊C16—22no—
132#346D414yes—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4C2×D411yescharacteristic
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16#4112yes—
154C2×Q16#4112yes—
155C2×M16—22no—
156C2×M16—41no—
157C2×M16—41no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2411yescharacteristic
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176#117—22no—
177#117—22no—
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—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
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⋊C4—41no—
196C2×C22⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204Q32—28no—
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—
214C2×C16—21no—
215C2×C16C2×D412yes—
216C2×C16C2×D412yes—
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#4112yes—
228#107#4112yes—
229#103—24no—
230#103—24no—
231#103—42no—
232#103—42no—
233#103—42no—
234#103—42no—
235#97—22no—
236#97—22no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8C2×D412yes—
240C4×C8#4112yes—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24C2×D811yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C22×D411yescharacteristic
284Q16—22no—
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—41no—
289Q16—41no—
290Q16—41no—
291Q16—41no—
292M16—44no—
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—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8#12112yes—
306C2×C8C22×D411yescharacteristic
307C2×C8C2×#4111yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—161no—
331C4⋊C4—161no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—161no—
337C22⋊C4—161no—
338C42—21no—
339C42—21no—
340C42—82no—
341C42—82no—
342C42—161no—
343C42—161no—
344C42#12112yes—
345C42#13512yes—
346C16—22no—
347C16—22no—
348C16—41no—
349C16—41no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C22×D811yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
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—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#46612yes—
408C2×C4#46612yes—
409C2×C4C2×#12111yescharacteristic
410C2×C4#52511yescharacteristic
411C2×C4#52511yescharacteristic
412C2×C4C2×#13511yescharacteristic
413C2×C4#57511yescharacteristic
414C2×C4#57511yescharacteristic
415C8—22no—
416C8—41no—
417C8—41no—
418C8#52511yescharacteristic
419C8#52911yescharacteristic
420C8#52911yescharacteristic
421C8#53211yescharacteristic
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22C2×#46611yescharacteristic
430C22#310011yescharacteristic
431C22#310011yescharacteristic
432C22#341611yescharacteristic
433C22#341611yescharacteristic
434C22#341711yescharacteristic
435C22#341711yescharacteristic
436C4—22no—
437C4—22no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4—161no—
449C4#239211yescharacteristic
450C4#305411yescharacteristic
451C4#310211yescharacteristic
452C4#310211yescharacteristic
453C2—81no—
454C2#2369611yescharacteristic
455C2#3361311yescharacteristic
456C2#3361311yescharacteristic
457C2#3368011yescharacteristic
458C2#3368111yescharacteristic
459C2#3368511yescharacteristic
460C2#3368711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
1. C2×#2176C2semidirect
5. #13516C2semidirect
7. #13477C2semidirect
8. #13475C2semidirect
9. #13450C2semidirect
13. #12674C2semidirect
14. C4×#358C2semidirect
15. C4×#358C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21762
C2×#21642
#215252
#135162
#134942
#134772
#134752
#134502
#133832
#133822
#133062
#126742
C4×#3582