Finite Group Databasebeta

← #184223

Subgroups — #184223

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21277C211yesmaximal, characteristic
2#20565C211yesmaximal, characteristic, complemented
3#20565C211yesmaximal, characteristic, complemented
4C2×#1586C211yesmaximal, characteristic
5C2×#1581C211yesmaximal, characteristic
6#8290C211yesmaximal, characteristic, complemented
7#8289C211yesmaximal, characteristic, complemented
8#8276C211yesmaximal, characteristic, complemented
9#8212C211yesmaximal, characteristic, complemented
10#8210C211yesmaximal, characteristic, complemented
11#8204C211yesmaximal, characteristic, complemented
12#8202C211yesmaximal, characteristic, complemented
13#8177C211yesmaximal, characteristic, complemented
14#8177C211yesmaximal, characteristic, complemented
15#7842C211yesmaximal, 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
32#1767C2211yescharacteristic
33#1767C2211yescharacteristic
34C4×#102C2211yescharacteristic
35C4×#102C2211yescharacteristic
36#1753—21no—
37#1753—21no—
38#1752—21no—
39#1752—21no—
40#1750—21no—
41#1750—21no—
42#1744C2211yescharacteristic
43#1729—21no—
44#1729—21no—
45C4×#97C2211yescharacteristic
46C42×C8C2211yescharacteristic
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×#322C2211yescharacteristic
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×#97—21no—
101C2×#97—21no—
102C2×#97C2×C412yes—
103C2×#97C2×C412yes—
104C2×C4×C8—22no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390C2×C412yes—
114#390C2×C412yes—
115#389C2×C412yes—
116#389C2×C412yes—
117#388—22no—
118#388—22no—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C43C2311yescharacteristic
124#340C2×C412yes—
125#340C2×C412yes—
126#340C2×C412yes—
127#336—21no—
128#336—21no—
129#336—21no—
130#336—21no—
131#336—21no—
132#336—21no—
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#322—22no—
145#322C2×C414yes—
146#322C2×C414yes—
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×C412yes—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
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—42no—
220#102—42no—
221#102C4214yes—
222#102C4214yes—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227#97—42no—
228#97—42no—
229#97—81no—
230#97—81no—
231#97—81no—
232#97—81no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—44no—
238C4×C8—81no—
239C4×C8—81no—
240C4×C8C22×C412yes—
241C4×C8C22×C412yes—
242C4×C8C22×C412yes—
243C4×C8C22×C412yes—
244C4×C8C22×C412yes—
245C4×C8C22×C412yes—
246C4×C8#4112yes—
247C4×C8#4112yes—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252C24—21no—
253C24—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4C2×C4212yes—
261C2×D4C2×C4212yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
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×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283D8—44no—
284M16—84no—
285M16—84no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—84no—
316C2×C8—84no—
317C2×C8C2×C4212yes—
318C2×C8C2×C4212yes—
319C2×C8C4×D412yes—
320C2×C8C4×D412yes—
321C2×C8C4×D412yes—
322C2×C8C4×D412yes—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4C2×C4212yes—
332C4⋊C4C2×C4212yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—22no—
342C42—22no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—42no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42C23×C411yescharacteristic
353C42C23×C411yescharacteristic
354C42C2×#4111yescharacteristic
355C42C2×#4111yescharacteristic
356C23—22no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23C2×C4×D411yescharacteristic
367D4—22no—
368D4—22no—
369D4—22no—
370D4—22no—
371D4—42no—
372D4—42no—
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—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
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×C4—42no—
400C2×C4C22×C4211yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C2×C4×D411yescharacteristic
403C2×C4C4×#4111yescharacteristic
404C2×C4C4×#4111yescharacteristic
405C2×C4C4×#4111yescharacteristic
406C2×C4#56812yes—
407C2×C4#56812yes—
408C2×C4#57611yescharacteristic
409C2×C4#57611yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57711yescharacteristic
412C8—24no—
413C8—22no—
414C8—22no—
415C8—22no—
416C8—22no—
417C8—22no—
418C8—22no—
419C8—42no—
420C8—42no—
421C8—42no—
422C8—42no—
423C8—42no—
424C8—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22C42×D411yescharacteristic
436C22#293212yes—
437C22#294611yescharacteristic
438C22#294611yescharacteristic
439C22#295612yes—
440C4—21no—
441C4—21no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—82no—
447C4—82no—
448C4#287311yescharacteristic
449C4#287511yescharacteristic
450C4#294611yescharacteristic
451C4#294611yescharacteristic
452C4#294811yescharacteristic
453C4#295211yescharacteristic
454C4#296712yes—
455C2—82no—
456C2—82no—
457C2#1798712yes—
458C2#1798712yes—
459C2#2048811yescharacteristic
460C2#2050311yescharacteristic
461C2#2051211yescharacteristic
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. #20565C2semidirect
3. #20565C2semidirect
6. #8290C2semidirect
7. #8289C2semidirect
8. #8276C2semidirect
9. #8212C2semidirect
10. #8210C2semidirect
11. #8204C2semidirect
12. #8202C2semidirect
13. #8177C2semidirect
14. #8177C2semidirect
15. #7842C2semidirect
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
#205652
C2×#15862
C2×#15812
#82902
#82892
#82762
#82122
#82102
#82042
#82022
#81772
#78422