Finite Group Databasebeta

← #495821

Subgroups — #495821

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2211C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic
3C2×C4×Q32C211yesmaximal, characteristic, complemented
4#21696C211yesmaximal, characteristic
5#20507C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic
7#13176C211yesmaximal, characteristic, complemented
8#13133C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic
10#13015C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic
12#12812C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×#440—21no—
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25#2326—21no—
26#2276—21no—
27#2211C418yescomplemented
28#2175—24no—
29C4×Q32—24no—
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×C4×C16C2211yescharacteristic
44#1973—21no—
45#1970C2211yescharacteristic
46#1969C2211yescharacteristic
47#1959—21no—
48#1845C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1763—21no—
53#1763—21no—
54#1763—21no—
55#1726C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×SD16—21no—
65C22×D8C2311yescharacteristic
66C2×#124—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×Q32—22no—
73C2×Q32—22no—
74C2×Q32C2×C414yescomplemented
75C2×SD32—22no—
76C2×SD32—22no—
77C2×SD32C2×C414yes—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#495—22no—
83#495C2×C414yes—
84#440—44no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×Q16—22no—
90C4×Q16—22no—
91C4×Q16C2×C414yescomplemented
92C4×Q16D414yes—
93C4×D8—22no—
94C4×D8—22no—
95C2×#107—21no—
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#390—21no—
116#388—41no—
117#388—41no—
118#385—41no—
119#385—41no—
120#384—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—41no—
125C4×C4⋊C4—41no—
126C4×C4⋊C4—41no—
127#376—41no—
128#376D411yescharacteristic
129#366—22no—
130#366—22no—
131C4⋊C16—22no—
132C4⋊C16C2×C414yescomplemented
133#358—22no—
134#358—22no—
135#358—22no—
136#358—22no—
137#358C2×C414yescomplemented
138#357—22no—
139#357—22no—
140#357C2×C414yes—
141#346—22no—
142C4×C16—22no—
143#340—21no—
144#340—21no—
145#340—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—21no—
149C23×C4—41no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16C22×C412yes—
159C2×Q16C22×C412yes—
160C2×Q16C2×D412yes—
161C2×Q16#4112yes—
162C2×SD16—24no—
163C2×SD16—42no—
164C2×SD16—42no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8C22×C412yes—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2411yescharacteristic
175#124—24no—
176#124—42no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×D4—42no—
186C4×D4—42no—
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×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⋊C4—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213Q32—28no—
214SD32—28no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16C22×C412yes—
226C2×C16C22×C412yes—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—42no—
235#107C22×C412yes—
236#107C2×D412yes—
237#107#4112yes—
238#103—24no—
239#103—24no—
240#103—24no—
241#103—42no—
242#103—42no—
243#102—42no—
244#102—42no—
245#97—22no—
246#97—22no—
247#97—41no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C22×C412yes—
251C4×C8C2×D412yes—
252C4×C8#4112yes—
253C4×C8#4112yes—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95D811yescharacteristic
260#95D811yescharacteristic
261#95D811yescharacteristic
262#95D811yescharacteristic
263C24—41no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—161no—
291C22×C4—161no—
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C22×D411yescharacteristic
299Q16—22no—
300Q16—22no—
301Q16—22no—
302Q16—22no—
303Q16—22no—
304Q16—41no—
305Q16—41no—
306Q16C4×D414yes—
307SD16—48no—
308D8—22no—
309D8—22no—
310D8—41no—
311D8—41no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8C4×D412yes—
320C2×C8#12112yes—
321C2×C8C23×C411yescharacteristic
322C2×C8C22×D411yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—44no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—161no—
347C22⋊C4—161no—
348C42—21no—
349C42—21no—
350C42—44no—
351C42—44no—
352C42—44no—
353C42—82no—
354C42—161no—
355C42—161no—
356C42—161no—
357C42C4×D412yes—
358C42#12112yes—
359C42#13512yes—
360C42AGL(1,ℤ/8)12yes—
361C16—22no—
362C16—22no—
363C16—41no—
364C16—41no—
365C23—42no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23C22×D811yescharacteristic
373Q8—24no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
381D4—42no—
382D4—42no—
383D4—81no—
384D4—81no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
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—164no—
421C2×C4C4×D812yes—
422C2×C4C4×D812yes—
423C2×C4#45912yes—
424C2×C4#45912yes—
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×#12111yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
430C2×C4#57511yescharacteristic
431C8—22no—
432C8—22no—
433C8—41no—
434C8C2×C4×D411yescharacteristic
435C8C4×#4111yescharacteristic
436C8#52911yescharacteristic
437C8#53211yescharacteristic
438C22—82no—
439C22—82no—
440C22—82no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22C2×C4×D811yescharacteristic
446C22C2×#45911yescharacteristic
447C22#310011yescharacteristic
448C22#341511yescharacteristic
449C22#341611yescharacteristic
450C22#341711yescharacteristic
451C22#341711yescharacteristic
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—44no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4—161no—
464C4—161no—
465C4C4×#12111yescharacteristic
466C4C4×AGL(1,ℤ/8)11yescharacteristic
467C4#303511yescharacteristic
468C4#310611yescharacteristic
469C2—162no—
470C2C4×#45911yescharacteristic
471C2#3345511yescharacteristic
472C2#3345911yescharacteristic
473C2#3358811yescharacteristic
474C2#3358911yescharacteristic
475C2#3368111yescharacteristic
476C2#3368411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×C4×Q32C2semidirect
5. #20507C2semidirect
7. #13176C2semidirect
8. #13133C2semidirect
10. #13015C2semidirect
12. #12812C2semidirect
13. #12654C2semidirect
14. #12638C2semidirect
27. #2211C4semidirect
74. C2×Q32C2×C4semidirect
91. C4×Q16C2×C4semidirect
132. C4⋊C16C2×C4semidirect
137. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22112
C2×#21752
C2×C4×Q322
#216962
#205072
#132542
#131762
#131332
#131302
#130152
#130132
#128122
#126542
#126382
C4×#3572