Finite Group Databasebeta

← #183985

Subgroups — #183985

Order 512.

728 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21279C211yesmaximal, characteristic
2#20571C211yesmaximal, characteristic, complemented
3#20562C211yesmaximal, characteristic
4C2×#1590C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#8287C211yesmaximal, characteristic, complemented
7C8×#107C211yesmaximal, characteristic, complemented
8#8275C211yesmaximal, characteristic, complemented
9#8213C211yesmaximal, characteristic, complemented
10#8212C211yesmaximal, characteristic
11#8203C211yesmaximal, characteristic, complemented
12#8202C211yesmaximal, characteristic
13#8183C211yesmaximal, characteristic
14#8182C211yesmaximal, characteristic, complemented
15#7844C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21#2295C2211yescharacteristic
22#2293C2211yescharacteristic
23#1973C412yes—
24#1944C412yes—
25#1936C412yes—
26#1935C412yes—
27C4×#107C2211yescharacteristic
28#1771—21no—
29#1770C2211yescharacteristic
30#1770C2211yescharacteristic
31C8×C4⋊C4—21no—
32C8×C4⋊C4C2211yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34#1768—21no—
35C4×C4⋊C8C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37C4×#103C2211yescharacteristic
38C4×#102C2211yescharacteristic
39#1753—21no—
40C8×C22⋊C4—21no—
41#1750—21no—
42#1746—21no—
43#1744C2211yescharacteristic
44#1744C2211yescharacteristic
45#1729—21no—
46C4×#97C2211yescharacteristic
47#1590C418yescomplemented
48SD16⋊C8C418yescomplemented
49C2×#335C2211yescharacteristic
50C2×#334C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊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×SD16C2×C411yescharacteristic
66C22×M16—21no—
67C23×C8—21no—
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71#446—22no—
72#446C2×C414yes—
73C8×Q8—22no—
74C8×Q8C2×C414yes—
75C4×SD16C2×C414yes—
76C4×SD16C2×C414yes—
77#436—22no—
78#436C2×C414yes—
79C8×D4—22no—
80C8×D4C2×C414yes—
81C2×#107C2×C412yes—
82C2×#106C2×C411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#103C2×C412yes—
90C2×#103C2×C411yescharacteristic
91C2×#102C2×C412yes—
92C2×#102C2×C411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97C2×C412yes—
99C2×#97C2×C412yes—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391C2×C412yes—
112#390C2×C412yes—
113#389C2×C412yes—
114#389C2×C412yes—
115#388—22no—
116#387—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121#376—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—
139Q8⋊C8C2×C414yes—
140Q8⋊C8C2×C414yes—
141D4⋊C8C2×C414yes—
142D4⋊C8C2×C414yes—
143#322—22no—
144#322—22no—
145#322C2×C414yes—
146C82C2×C414yes—
147C22×Q8C22×C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C2×SD16—24no—
151C2×SD16C4214yes—
152C2×M16—24no—
153C2×M16—81no—
154C2×M16—81no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—24no—
162C22×C8—81no—
163C22×C8—81no—
164C22×C8C22×C412yes—
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8C22×C412yes—
174C4×Q8C22×C412yes—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4C22×C412yes—
178C4×D4C22×C412yes—
179#117—81no—
180#117—81no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C22×C412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4112yes—
198C2×C42C2411yescharacteristic
199#107—24no—
200#106C4214yes—
201C4⋊C8—24no—
202C4⋊C8—24no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8C22×C412yes—
208C4⋊C8C22×C412yes—
209C4⋊C8C22×C412yes—
210C4⋊C8C22×C412yes—
211#103—42no—
212#103C4214yes—
213#102—42no—
214#102C4214yes—
215C22⋊C8—24no—
216C22⋊C8—24no—
217#97—24no—
218#97—24no—
219#97—42no—
220#97—81no—
221#97—81no—
222C4×C8—24no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—81no—
228C4×C8—81no—
229C4×C8—81no—
230C4×C8—81no—
231C4×C8C22×C412yes—
232C4×C8C22×C412yes—
233C4×C8C22×C412yes—
234C4×C8C22×C412yes—
235C4×C8C22×C412yes—
236C4×C8C22×C412yes—
237C4×C8#4112yes—
238C4×C8#4112yes—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245C24—21no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8C2×C4212yes—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4C2×C4212yes—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270SD16—28no—
271M16—84no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—84no—
301C2×C8C2×C4212yes—
302C2×C8C2×C4212yes—
303C2×C8C4×D412yes—
304C2×C8C4×D412yes—
305C2×C8C4×D412yes—
306C2×C8C4×D412yes—
307C2×C8#13114yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—44no—
317C4⋊C4—82no—
318C4⋊C4C2×C4212yes—
319C4⋊C4C2×C4212yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—42no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42C23×C411yescharacteristic
338C42C23×C411yescharacteristic
339C42C2×#4111yescharacteristic
340C42C2×#4111yescharacteristic
341C23—22no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23C2×C4×D411yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—42no—
350D4—22no—
351D4—22no—
352D4—42no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4C22×C4211yescharacteristic
378C2×C4C2×C4×D411yescharacteristic
379C2×C4C2×C4×D411yescharacteristic
380C2×C4C4×#4111yescharacteristic
381C2×C4C4×#4111yescharacteristic
382C2×C4C4×#4111yescharacteristic
383C2×C4C2×#13112yes—
384C2×C4#56812yes—
385C2×C4#57611yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57711yescharacteristic
388C2×C4#57811yescharacteristic
389C8—22no—
390C8—22no—
391C8—22no—
392C8—22no—
393C8—22no—
394C8—42no—
395C8—42no—
396C8—42no—
397C8—42no—
398C8—42no—
399C8—42no—
400C8—42no—
401C8#40514yes—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22C42×D411yescharacteristic
408C22#293512yes—
409C22#294611yescharacteristic
410C22#294711yescharacteristic
411C22#295512yes—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—82no—
421C4C2×#40511yescharacteristic
422C4#287511yescharacteristic
423C4#294611yescharacteristic
424C4#294711yescharacteristic
425C4#294911yescharacteristic
426C4#295111yescharacteristic
427C4#296912yes—
428C2—82no—
429C2#1808012yes—
430C2#1808312yes—
431C2#2049511yescharacteristic
432C2#2050411yescharacteristic
433C2#2051011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×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. #20571C2semidirect
6. #8287C2semidirect
7. C8×#107C2semidirect
8. #8275C2semidirect
9. #8213C2semidirect
11. #8203C2semidirect
14. #8182C2semidirect
15. #7844C2semidirect
47. #1590C4semidirect
48. SD16⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#212792
#205712
#205622
C2×#15902
C2×SD16⋊C82
#82872
C8×#1072
#82752
#82132
#82122
#82032
#82022
#81832
#81822
#78442