Finite Group Databasebeta

← #495873

Subgroups — #495873

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×C4×M32C211yesmaximal, characteristic, complemented
4C4×#467C211yesmaximal, characteristic, complemented
5C4×#466C211yesmaximal, characteristic
6#13237C211yesmaximal, characteristic, complemented
7#13165C211yesmaximal, characteristic, complemented
8#13148C211yesmaximal, characteristic, complemented
9#13147C211yesmaximal, characteristic, complemented
10#13062C211yesmaximal, characteristic, complemented
11#13061C211yesmaximal, characteristic
12#12836C211yesmaximal, characteristic, complemented
13#12674C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×C22⋊Q8—21no—
25C4×#121—21no—
26#2220C418yescomplemented
27#2175—24no—
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C4×M32—24no—
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1934—21no—
47#1933—21no—
48#1893—21no—
49#1892—21no—
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C4×#103—21no—
55C4×#102—21no—
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×C4×Q8—21no—
69C2×C4×D4—21no—
70C2×C4×D4—41no—
71C22×C42D411yescharacteristic
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32C2×C414yescomplemented
75C2×M32—22no—
76C2×M32—22no—
77C2×M32—22no—
78C2×M32—22no—
79C2×M32—22no—
80C2×M32C2×C414yescomplemented
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85#467—22no—
86#467C2×C414yescomplemented
87#466—22no—
88#466C2×C414yes—
89C4×Q16—22no—
90C4×Q16—22no—
91C4×D8—22no—
92C4×D8—22no—
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8D411yescharacteristic
108C2×C4×C8D411yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#388—41no—
116#387—41no—
117#386—41no—
118#385—41no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—41no—
124C4×C22⋊C4—41no—
125C4×C22⋊C4—41no—
126#367—22no—
127#366—22no—
128#366C2×C414yescomplemented
129#358—22no—
130#358—22no—
131#358C2×C414yescomplemented
132#357—22no—
133#357—22no—
134#357C2×C414yescomplemented
135#346—22no—
136#346D414yescomplemented
137C4×C16—22no—
138C4×C16—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—41no—
149C23×C4D811yescharacteristic
150C23×C4D811yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16C22×C412yes—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8C22×C412yes—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C22×C412yes—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8#4112yes—
176C22×C8C2411yescharacteristic
177C22⋊Q8—24no—
178C22⋊Q8—42no—
179#121—24no—
180#121—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—84no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42D811yescharacteristic
209C2×C42D811yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212SD32—28no—
213M32—28no—
214M32—28no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16C22×C412yes—
222C2×C16C22×C412yes—
223C2×C16C2×D412yes—
224C2×C16C2×D412yes—
225C2×C16#4112yes—
226C2×C16#4112yes—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107C22×C412yes—
238#107C22×C412yes—
239#103—24no—
240#103—42no—
241#103—42no—
242#102—24no—
243#102—42no—
244#102—42no—
245C4×C8—22no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—22no—
249C4×C8C2×D412yes—
250C4×C8#4112yes—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—41no—
260C24C2×D811yescharacteristic
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
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—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4C4×D412yes—
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4#13512yes—
303C22×C4C22×D411yescharacteristic
304Q16—22no—
305Q16—22no—
306Q16—41no—
307Q16—41no—
308D8—22no—
309D8—22no—
310D8—41no—
311D8—41no—
312C2×C8—28no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8C4×D412yes—
324C2×C8#12112yes—
325C2×C8#13512yes—
326C2×C8C23×C411yescharacteristic
327C2×C8C22×D411yescharacteristic
328C2×C8C2×#4111yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C2×C8C2×#4111yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—22no—
359C42—82no—
360C42—82no—
361C42—82no—
362C42#12112yes—
363C42#13512yes—
364C16—22no—
365C16—22no—
366C16—22no—
367C16C4×D414yescomplemented
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23C4×D812yes—
382C23C22×D811yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—81no—
386Q8—81no—
387D4—42no—
388D4—42no—
389D4—88no—
390D4—81no—
391D4—81no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—81no—
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—164no—
430C2×C4C4×D812yes—
431C2×C4#46612yes—
432C2×C4#46612yes—
433C2×C4#50912yes—
434C2×C4#51012yes—
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C8—22no—
442C8—22no—
443C8—22no—
444C8C2×C4×D411yescharacteristic
445C8C4×#4111yescharacteristic
446C8#52511yescharacteristic
447C8#52911yescharacteristic
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22C2×C4×D811yescharacteristic
463C22C2×#46611yescharacteristic
464C22#316911yescharacteristic
465C22C2×#50911yescharacteristic
466C22C2×#51011yescharacteristic
467C22#341411yescharacteristic
468C22#341411yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4C4×#12111yescharacteristic
482C4C4×#13511yescharacteristic
483C4#305011yescharacteristic
484C4#316811yescharacteristic
485C2—42no—
486C2—162no—
487C2C4×#46611yescharacteristic
488C2C4×#50911yescharacteristic
489C2C4×#51011yescharacteristic
490C2#3360511yescharacteristic
491C2#3360611yescharacteristic
492C2#3375011yescharacteristic
493C2#3375111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×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×#2175C2semidirect
3. C2×C4×M32C2semidirect
4. C4×#467C2semidirect
6. #13237C2semidirect
7. #13165C2semidirect
8. #13148C2semidirect
9. #13147C2semidirect
10. #13062C2semidirect
12. #12836C2semidirect
13. #12674C2semidirect
14. #12638C2semidirect
15. #12637C2semidirect
26. #2220C4semidirect
74. C2×SD32C2×C4semidirect
80. C2×M32C2×C4semidirect
86. #467C2×C4semidirect
128. #366C2×C4semidirect
131. #358C2×C4semidirect
134. #357C2×C4semidirect
136. #346D4semidirect
367. C16C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#21752
C2×C4×M322
C4×#4672
C4×#4662
#132372
#131652
#131482
#131472
#130622
#130612
#128362
#126742
#126382
#126372