Finite Group Databasebeta

← #512169

Subgroups — #512169

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×#2211C211yesmaximal, characteristic
3#33240C211yesmaximal, characteristic
4C42×Q16C211yesmaximal, characteristic, complemented
5#13277C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic
7#13133C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic
10#13130C211yesmaximal, characteristic
11#13015C211yesmaximal, characteristic, complemented
12#13015C211yesmaximal, characteristic, complemented
13#13013C211yesmaximal, characteristic
14#13013C211yesmaximal, characteristic
15#12879C211yesmaximal, characteristic, complemented
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#488—21no—
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24#2861—21no—
25C42×Q8—21no—
26#2229—24no—
27#2211—24no—
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1968—21no—
45#1968—21no—
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#103—21no—
50C4×#103—21no—
51C4×#103—21no—
52C42×C8C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×SD16—21no—
64C22×D8C2311yescharacteristic
65C2×#125—21no—
66C2×#124—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×Q32—22no—
71C2×Q32—22no—
72C2×Q32—22no—
73C2×SD32—22no—
74C2×SD32—22no—
75C2×SD32—22no—
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#499—22no—
81#499—22no—
82#495—22no—
83#495—22no—
84#488—44no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×Q16—22no—
90C4×Q16D414yes—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#106—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C4×C8D411yescharacteristic
105C2×C4×C8D411yescharacteristic
106C2×C4×C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#388—41no—
115#388—41no—
116#388—41no—
117#388—41no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—41no—
122C4×C4⋊C4—41no—
123C4×C4⋊C4—41no—
124C43—41no—
125C43D411yescharacteristic
126#367—22no—
127#366—22no—
128C4⋊C16—22no—
129C4⋊C16—22no—
130#358—22no—
131#358—22no—
132#358—22no—
133#358—22no—
134#358—22no—
135#357—22no—
136#357—22no—
137#357—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×D4—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×Q16C2×D412yes—
154C2×Q16#4112yes—
155C2×Q16#4112yes—
156C2×SD16—42no—
157C2×SD16—42no—
158C2×SD16—42no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#125—42no—
170#125—42no—
171#124—42no—
172#124—42no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
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×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42D811yescharacteristic
202C2×C42D811yescharacteristic
203C2×C42D811yescharacteristic
204C2×C42D811yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208Q32—44no—
209SD32—44no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107C2×D412yes—
230#107#4112yes—
231#107#4112yes—
232#106—42no—
233#103—24no—
234#103—24no—
235#103—24no—
236#103—42no—
237#103—42no—
238#102—42no—
239#102—42no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—22no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8C2×D412yes—
248C4×C8#4112yes—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—41no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—22no—
292Q16—41no—
293Q16—41no—
294Q16—41no—
295Q16—41no—
296Q16#12414yes—
297SD16—84no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8#12112yes—
311C2×C8C22×D411yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42—41no—
345C42—41no—
346C42—44no—
347C42—44no—
348C42—44no—
349C42—82no—
350C42—82no—
351C42—82no—
352C42#12112yes—
353C42AGL(1,ℤ/8)12yes—
354C16—41no—
355C16—41no—
356C16—41no—
357C16—41no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23C22×D811yescharacteristic
366Q8—24no—
367Q8—42no—
368Q8—42no—
369Q8—42no—
370Q8—81no—
371Q8—81no—
372Q8—81no—
373Q8—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
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—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4#45912yes—
414C2×C4#45912yes—
415C2×C4#50812yes—
416C2×C4#51012yes—
417C2×C4C2×#12111yescharacteristic
418C2×C4#52511yescharacteristic
419C2×C4#52511yescharacteristic
420C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
421C2×C4#57511yescharacteristic
422C2×C4#57511yescharacteristic
423C8—22no—
424C8—41no—
425C8—41no—
426C8C2×#12411yescharacteristic
427C8#52911yescharacteristic
428C8#52911yescharacteristic
429C8#53211yescharacteristic
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22C2×#45911yescharacteristic
438C22#310011yescharacteristic
439C22#310011yescharacteristic
440C22C2×#50811yescharacteristic
441C22C2×#51011yescharacteristic
442C22#341411yescharacteristic
443C22#341411yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—44no—
449C4—41no—
450C4—41no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4#285011yescharacteristic
458C4#304011yescharacteristic
459C4#311511yescharacteristic
460C4#311511yescharacteristic
461C2—321no—
462C2#3302411yescharacteristic
463C2#3357911yescharacteristic
464C2#3358111yescharacteristic
465C2#3366211yescharacteristic
466C2#3366811yescharacteristic
467C2#3367111yescharacteristic
468C2#3367711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2229C2semidirect
4. C42×Q16C2semidirect
5. #13277C2semidirect
7. #13133C2semidirect
8. #13132C2semidirect
11. #13015C2semidirect
12. #13015C2semidirect
15. #12879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×#22112
#332402
C42×Q162
#132772
#132542
#131332
#131322
#131312
#131302
#130152
#130132
#128792