Finite Group Databasebeta

← #184287

Subgroups — #184287

Order 512.

768 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
1#21277C211yesmaximal, characteristic
2#20562C211yesmaximal, characteristic, complemented
3#20562C211yesmaximal, characteristic, complemented
4C2×#1586C211yesmaximal, characteristic
5C2×#1581C211yesmaximal, characteristic
6#8290C211yesmaximal, characteristic, complemented
7#8289C211yesmaximal, characteristic, complemented
8#8275C211yesmaximal, characteristic, complemented
9#8208C211yesmaximal, characteristic, complemented
10#8206C211yesmaximal, characteristic, complemented
11#8200C211yesmaximal, characteristic, complemented
12C8×#102C211yesmaximal, characteristic, complemented
13#8183C211yesmaximal, characteristic, complemented
14#8183C211yesmaximal, characteristic, complemented
15#7844C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21#2293C2211yescharacteristic
22#2293C2211yescharacteristic
23#1972C412yes—
24#1943C412yes—
25#1936C412yes—
26#1936C412yes—
27C4×#106C2211yescharacteristic
28#1770C2211yescharacteristic
29#1770C2211yescharacteristic
30C8×C4⋊C4C2211yescharacteristic
31C8×C4⋊C4C2211yescharacteristic
32C4×C4⋊C8C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34C4×#102C2211yescharacteristic
35C4×#102C2211yescharacteristic
36#1753—21no—
37#1753—21no—
38C8×C22⋊C4—21no—
39C8×C22⋊C4—21no—
40#1750—21no—
41#1750—21no—
42#1744C2211yescharacteristic
43#1744C2211yescharacteristic
44#1729—21no—
45#1729—21no—
46C4×#97C2211yescharacteristic
47#1586C418yescomplemented
48#1581C418yescomplemented
49C2×#335C2211yescharacteristic
50C2×#334C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65C22×D8C2×C411yescharacteristic
66C22×M16—21no—
67C22×M16—21no—
68C23×C8—21no—
69C23×C8—21no—
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74C4×D8C2×C414yes—
75C4×D8C2×C414yes—
76#436—22no—
77#436—22no—
78#436C2×C414yes—
79#436C2×C414yes—
80C8×D4—22no—
81C8×D4—22no—
82C8×D4C2×C414yes—
83C8×D4C2×C414yes—
84C2×#107C2×C411yescharacteristic
85C2×#106C2×C412yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102C2×C412yes—
91C2×#102C2×C412yes—
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97C2×C412yes—
101C2×#97C2×C412yes—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#390C2×C412yes—
113#390C2×C412yes—
114#389C2×C412yes—
115#389C2×C412yes—
116#388—22no—
117#388—22no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122C43C2311yescharacteristic
123#340C2×C412yes—
124#340C2×C412yes—
125#340C2×C412yes—
126#336—21no—
127#336—21no—
128#336—21no—
129#336—21no—
130#336—21no—
131#336—21no—
132#336—22no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135#336D411yescharacteristic
136#336D411yescharacteristic
137#335C2×C414yes—
138#334C2×C414yes—
139D4⋊C8C2×C414yes—
140D4⋊C8C2×C414yes—
141D4⋊C8C2×C414yes—
142D4⋊C8C2×C414yes—
143#322—22no—
144#322C2×C414yes—
145#322C2×C414yes—
146C82—22no—
147C22×D4C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C2×D8—42no—
152C2×D8C4214yes—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16—81no—
156C2×M16—81no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—24no—
167C22×C8—81no—
168C22×C8—81no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4C22×C412yes—
183C4×D4C22×C412yes—
184C4×D4C22×C412yes—
185C4×D4C22×C412yes—
186#117—81no—
187#117—81no—
188#117—81no—
189#117—81no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209#107C4214yes—
210#106—42no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8C22×C412yes—
216C4⋊C8C22×C412yes—
217C4⋊C8C22×C412yes—
218C4⋊C8C22×C412yes—
219#102—24no—
220#102—24no—
221#102C4214yes—
222#102C4214yes—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227#97—24no—
228#97—42no—
229#97—81no—
230#97—81no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—81no—
236C4×C8—81no—
237C4×C8—81no—
238C4×C8—81no—
239C4×C8C22×C412yes—
240C4×C8C22×C412yes—
241C4×C8C22×C412yes—
242C4×C8C22×C412yes—
243C4×C8C22×C412yes—
244C4×C8C22×C412yes—
245C4×C8#4112yes—
246C4×C8#4112yes—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251C24—21no—
252C24—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4C2×C4212yes—
260C2×D4C2×C4212yes—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282D8—44no—
283M16—84no—
284M16—84no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—84no—
314C2×C8—84no—
315C2×C8C2×C4212yes—
316C2×C8C2×C4212yes—
317C2×C8C4×D412yes—
318C2×C8C4×D412yes—
319C2×C8C4×D412yes—
320C2×C8C4×D412yes—
321C2×C8#13114yes—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4C2×C4212yes—
331C4⋊C4C2×C4212yes—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—42no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42C23×C411yescharacteristic
352C42C23×C411yescharacteristic
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C23—22no—
356C23—22no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23C2×C4×D411yescharacteristic
366D4—22no—
367D4—22no—
368D4—22no—
369D4—22no—
370D4—42no—
371D4—42no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4C22×C4211yescharacteristic
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C4×#4111yescharacteristic
403C2×C4C4×#4111yescharacteristic
404C2×C4C4×#4111yescharacteristic
405C2×C4C2×#13112yes—
406C2×C4#56812yes—
407C2×C4#57611yescharacteristic
408C2×C4#57611yescharacteristic
409C2×C4#57711yescharacteristic
410C2×C4#57711yescharacteristic
411C8—22no—
412C8—22no—
413C8—22no—
414C8—22no—
415C8—22no—
416C8—22no—
417C8—24no—
418C8—42no—
419C8—42no—
420C8—42no—
421C8—42no—
422C8—42no—
423C8—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22C42×D411yescharacteristic
435C22#293512yes—
436C22#294611yescharacteristic
437C22#294611yescharacteristic
438C22#295512yes—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—82no—
446C4—82no—
447C4#287311yescharacteristic
448C4C4×#13111yescharacteristic
449C4#294611yescharacteristic
450C4#294611yescharacteristic
451C4#294811yescharacteristic
452C4#295211yescharacteristic
453C4#296812yes—
454C2—82no—
455C2—82no—
456C2#1808012yes—
457C2#1808212yes—
458C2#2049211yescharacteristic
459C2#2050311yescharacteristic
460C2#2051011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20562C2semidirect
3. #20562C2semidirect
6. #8290C2semidirect
7. #8289C2semidirect
8. #8275C2semidirect
9. #8208C2semidirect
10. #8206C2semidirect
11. #8200C2semidirect
12. C8×#102C2semidirect
13. #8183C2semidirect
14. #8183C2semidirect
15. #7844C2semidirect
47. #1586C4semidirect
48. #1581C4semidirect

Maximal subgroups

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

subgroupindex
#212772
#205622
C2×#15862
C2×#15812
#82902
#82892
#82752
#82082
#82062
#82002
C8×#1022
#81832
#78442