Finite Group Databasebeta

← #497729

Subgroups — #497729

Order 512.

732 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2139C211yesmaximal, characteristic
2#22611C211yesmaximal, characteristic, complemented
3#22448C211yesmaximal, characteristic
4#13150C211yesmaximal, characteristic, complemented
5#13148C211yesmaximal, characteristic, complemented
6#13133C211yesmaximal, characteristic, complemented
7#13132C211yesmaximal, characteristic, complemented
8#12937C211yesmaximal, characteristic, complemented
9#12935C211yesmaximal, characteristic, complemented
10#12867C211yesmaximal, characteristic, complemented
11#12836C211yesmaximal, characteristic, complemented
12#12733C211yesmaximal, characteristic
13#12637C211yesmaximal, characteristic, complemented
14C4×#357C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2344—21no—
22#2339—21no—
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25#2139C418yescomplemented
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1969C2211yescharacteristic
38#1969C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1918—21no—
42#1918—21no—
43#1877—21no—
44#1877—21no—
45#1823—21no—
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1788—21no—
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C8×C22⋊C4C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×#41—21no—
61C2×#135—22no—
62C2×#135—22no—
63C2×#135—22no—
64C2×#135—41no—
65C2×#135—41no—
66C2×#135C2×C414yes—
67C22×Q16C2311yescharacteristic
68C22×SD16—21no—
69C22×D8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×#117—41no—
74C22×C4⋊C4—21no—
75C2×M32—41no—
76C2×M32—41no—
77C2×M32C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C4×D8—22no—
83C4×D8—22no—
84C4×D8—41no—
85C4×D8—41no—
86C2×#107—21no—
87C2×#107—22no—
88C2×#107C2×C414yescomplemented
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#391—21no—
103#391—21no—
104#389—21no—
105#389—21no—
106#387—41no—
107#387—41no—
108#386—41no—
109#386—41no—
110#381—41no—
111#381—41no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4D411yescharacteristic
117C2×#95—41no—
118#367—41no—
119#366—41no—
120#358—41no—
121#358—41no—
122#358C2×C414yescomplemented
123#358C2×C414yescomplemented
124#357—22no—
125#357—22no—
126#357C2×C414yes—
127#357C2×C414yes—
128C22⋊C16—22no—
129#346—41no—
130C4×C16—41no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—81no—
144C2×#41—81no—
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153#135—28no—
154#135—44no—
155#135—44no—
156#135—44no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16C22×C412yes—
163C2×SD16—24no—
164C2×SD16—42no—
165C2×SD16—42no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8C22×C412yes—
170C2×D8#4112yes—
171C2×D8#4112yes—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8C22×C412yes—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8#4112yes—
181C22×C8C2411yescharacteristic
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—81no—
185C4×D4—81no—
186#117—84no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4D811yescharacteristic
203C2×C22⋊C4D811yescharacteristic
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208M32—44no—
209C2×C16—21no—
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×C16C22×C412yes—
220C2×C16C22×C412yes—
221#107—24no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107C22×C412yes—
233#107C22×C412yes—
234#103—81no—
235#103—81no—
236#102—42no—
237#102—42no—
238C22⋊C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95D811yescharacteristic
254#95D811yescharacteristic
255C24—41no—
256C24C2×D811yescharacteristic
257#41—48no—
258#41—84no—
259#41—84no—
260#41—84no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—161no—
298C22×C4—161no—
299C22×C4C4×D412yes—
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D811yescharacteristic
305C22×C4#13512yes—
306C22×C4C22×D411yescharacteristic
307Q16—22no—
308Q16—22no—
309Q16—41no—
310Q16—41no—
311SD16—48no—
312D8—22no—
313D8—22no—
314D8—22no—
315D8#11714yescomplemented
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8C4×D412yes—
329C2×C8C23×C411yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C2×C8C2×#4111yescharacteristic
332C2×C8C2×#4111yescharacteristic
333C2×C8H(5,𝔽2)11yescharacteristic
334C2×C8H(5,𝔽2)11yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—44no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—161no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—82no—
361C42—82no—
362C16—22no—
363C16—22no—
364C16—41no—
365C16—41no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23C4×D812yes—
380C23C22×D811yescharacteristic
381Q8—42no—
382Q8—42no—
383Q8—81no—
384Q8—81no—
385D4—24no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—88no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—88no—
411C2×C4—88no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—164no—
431C2×C4—164no—
432C2×C4C4×D812yes—
433C2×C4#50912yes—
434C2×C4C2×C4×D411yescharacteristic
435C2×C4#54611yescharacteristic
436C2×C4#54611yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C8—22no—
441C8—41no—
442C8—41no—
443C8C2×#11711yescharacteristic
444C8#51811yescharacteristic
445C8#55111yescharacteristic
446C8#55111yescharacteristic
447C22—22no—
448C22—22no—
449C22—22no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—44no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22C2×C4×D811yescharacteristic
462C22#329411yescharacteristic
463C22#329411yescharacteristic
464C22C2×#50911yescharacteristic
465C22#341411yescharacteristic
466C22#341611yescharacteristic
467C22#341711yescharacteristic
468C4—22no—
469C4—22no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4#234211yescharacteristic
481C4#294911yescharacteristic
482C4#329611yescharacteristic
483C4#329611yescharacteristic
484C2—42no—
485C2—84no—
486C2#2257011yescharacteristic
487C2C4×#50911yescharacteristic
488C2#3345511yescharacteristic
489C2#3387311yescharacteristic
490C2#3388211yescharacteristic
491C2#3389011yescharacteristic
492C2#3389711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22611C2semidirect
4. #13150C2semidirect
5. #13148C2semidirect
6. #13133C2semidirect
7. #13132C2semidirect
8. #12937C2semidirect
9. #12935C2semidirect
10. #12867C2semidirect
11. #12836C2semidirect
13. #12637C2semidirect
14. C4×#357C2semidirect
15. #12612C2semidirect
25. #2139C4semidirect
77. C2×M32C2×C4semidirect
88. C2×#107C2×C4semidirect
122. #358C2×C4semidirect
123. #358C2×C4semidirect
315. D8#117semidirect

Maximal subgroups

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

subgroupindex
C2×#21392
#226112
#224482
#131502
#131482
#131332
#131322
#129372
#129352
#128672
#128362
#127332
#126372
C4×#3572
#126122