Finite Group Databasebeta

← #500561

Subgroups — #500561

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic
2C2×#2227C211yesmaximal, characteristic, complemented
3C2×#2207C211yesmaximal, characteristic, complemented
4C4×#467C211yesmaximal, characteristic, complemented
5#13380C212yesmaximal, complemented
6#13305C212yesmaximal, complemented
7#13276C211yesmaximal, characteristic, complemented
8#13252C211yesmaximal, characteristic, complemented
9#13014C212yesmaximal, complemented
10#12935C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#532—21no—
13C22×SD32C2212yes—
14#3169—24no—
15#3169—24no—
16C2×#500C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#486—21no—
19C2×#467C2211yescharacteristic
20C2×#466—21no—
21C2×#466C2212yes—
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8—21no—
24C2×C4×D8C2212yes—
25C22×C4×C8C2211yescharacteristic
26C4×C22⋊Q8—21no—
27#2227—24no—
28#2207—24no—
29C2×#366C2212yes—
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2212yes—
33C2×#357C2212yes—
34C2×#357C2212yes—
35C2×#357C2212yes—
36C2×C22⋊C16C2212yes—
37#1969C2211yescharacteristic
38#1942C2211yescharacteristic
39#1933—21no—
40#1892—21no—
41#1808C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#103—21no—
44#1380C2212yes—
45#1380C2212yes—
46#1374C2212yes—
47C22×#41—41no—
48C22×Q16C2311yescharacteristic
49C22×D8—21no—
50C22×D8C2312yes—
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53#532—28no—
54#532—44no—
55C2×C4⋊Q8—21no—
56C2×#127—21no—
57C2×#124—41no—
58C2×C22⋊Q8—21no—
59C2×#121—22no—
60C2×#121—41no—
61C2×C4×Q8—21no—
62C2×C4×D4—22no—
63C2×C4×D4—41no—
64C22×C42D411yescharacteristic
65C2×SD32—24no—
66C2×SD32—24no—
67C2×SD32—42no—
68C22×C16C2312yes—
69C22×C16C2312yes—
70#500—22no—
71#500D414yes—
72#493—22no—
73#493D414yes—
74#486—28no—
75#467—22no—
76#467—22no—
77#466—28no—
78#466—24no—
79#466—24no—
80C4×Q16—22no—
81C4×Q16—22no—
82C4×D8—28no—
83C4×D8—24no—
84C4×D8—24no—
85C2×#107—21no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2312yes—
89C2×#107C2311yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#102—22no—
94C2×#102—22no—
95C2×C4×C8—22no—
96C2×C4×C8—22no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—21no—
102#389—21no—
103#387—41no—
104#385—41no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—41no—
108C4×C22⋊C4—41no—
109#366—24no—
110C4⋊C16—22no—
111C4⋊C16—22no—
112#358—24no—
113#357—24no—
114#357—24no—
115#357—42no—
116C22⋊C16—42no—
117#340—21no—
118#340—21no—
119#340—21no—
120C2×#41—48no—
121C2×#41—84no—
122C2×#41—84no—
123C22×Q8—21no—
124C22×Q8—41no—
125C22×D4—22no—
126C22×D4—42no—
127C22×D4—41no—
128C22×D4—41no—
129C23×C4—42no—
130C23×C4D811yescharacteristic
131C23×C4D811yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16#4112yescomplemented
139C2×D8—24no—
140C2×D8—22no—
141C2×D8—22no—
142C2×D8—22no—
143C2×D8—22no—
144C2×D8—22no—
145C2×D8—42no—
146C2×D8—42no—
147C2×D8C2×D414yes—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161C4⋊Q8—24no—
162C4⋊Q8—42no—
163#127—24no—
164#127—42no—
165#124—48no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168#121—28no—
169#121—44no—
170#121—48no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×D4—28no—
174C4×D4—44no—
175C4×D4—48no—
176C2×C4⋊C4—22no—
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×C22⋊C4—41no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—41no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42D811yescharacteristic
193C2×C42D811yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196SD32—48no—
197C2×C16—22no—
198C2×C16—22no—
199C2×C16—22no—
200C2×C16—22no—
201C2×C16—22no—
202C2×C16—22no—
203#107—24no—
204#107—22no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107C2×D414yes—
211#107#4112yescomplemented
212#103—24no—
213#103—42no—
214#103—42no—
215#102—28no—
216#102—28no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—22no—
221C4×C8C2×D412yes—
222C4×C8C2×D412yes—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228C24—42no—
229C24C2×D811yescharacteristic
230#41—832no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—44no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×D4—24no—
241C2×D4—48no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—82no—
254C2×D4—82no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—48no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C22×D411yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285D8—24no—
286D8—24no—
287D8—48no—
288D8—42no—
289D8—42no—
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—44no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8#12012yes—
304C2×C8C22×D411yescharacteristic
305C2×C8C22×D411yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8C2×#4111yescharacteristic
308C2×C8H(5,𝔽2)11yescharacteristic
309C2×C82−1+411yescharacteristic
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—44no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—82no—
326C42—21no—
327C42—21no—
328C42—22no—
329C42—22no—
330C42—82no—
331C42—82no—
332C42#12012yes—
333C42AGL(1,ℤ/8)12yes—
334C16—42no—
335C16—42no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—44no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23#50912yes—
350C23C22×D811yescharacteristic
351Q8—42no—
352Q8—42no—
353Q8—88no—
354Q8—81no—
355Q8—81no—
356D4—44no—
357D4—44no—
358D4—88no—
359D4—816no—
360D4—82no—
361D4—88no—
362D4—82no—
363C2×C4—28no—
364C2×C4—21no—
365C2×C4—21no—
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—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—816no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
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#44712yes—
396C2×C4#44712yes—
397C2×C4#50812yes—
398C2×C4#50912yes—
399C2×C4#51012yes—
400C2×C4C2×#12011yescharacteristic
401C2×C4#52511yescharacteristic
402C2×C4#53711yescharacteristic
403C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
404C2×C4#57511yescharacteristic
405C2×C4#57511yescharacteristic
406C8—22no—
407C8—41no—
408C8—41no—
409C8#54611yescharacteristic
410C8#54711yescharacteristic
411C8#54811yescharacteristic
412C8#55011yescharacteristic
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22—162no—
424C22—162no—
425C22—162no—
426C22—162no—
427C22C2×#44711yescharacteristic
428C22#310011yescharacteristic
429C22#324511yescharacteristic
430C22C2×#50811yescharacteristic
431C22C2×#50911yescharacteristic
432C22C2×#50911yescharacteristic
433C22C2×#51011yescharacteristic
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—82no—
439C4—82no—
440C4—84no—
441C4—161no—
442C4—161no—
443C4#218914yes—
444C4#240711yescharacteristic
445C4#300511yescharacteristic
446C4#311511yescharacteristic
447C4#325111yescharacteristic
448C2—42no—
449C2—164no—
450C2#2415011yescharacteristic
451C2C2×#218911yescharacteristic
452C2#3351711yescharacteristic
453C2#3366611yescharacteristic
454C2#3367211yescharacteristic
455C2#3381311yescharacteristic
456C2#3382011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3169C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2227C2semidirect
3. C2×#2207C2semidirect
4. C4×#467C2semidirect
5. #13380C2semidirect
6. #13305C2semidirect
7. #13276C2semidirect
8. #13252C2semidirect
9. #13014C2semidirect
10. #12935C2semidirect
11. #12783C2semidirect
138. C2×Q16#41semidirect
211. #107#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#22272
C2×#22072
C4×#4672
#133802
#133052
#132762
#132522
#130142
#129352
#127832