Finite Group Databasebeta

← #495824

Subgroups — #495824

Order 512.

746 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2207C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×C4×SD32C211yesmaximal, characteristic, complemented
4#21697C211yesmaximal, characteristic
5#20507C211yesmaximal, characteristic, complemented
6#13252C211yesmaximal, characteristic
7#13178C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13013C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic, complemented
12#12814C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#442—21no—
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25#2328—21no—
26#2276—21no—
27#2207C418yescomplemented
28#2175—24no—
29C4×SD32—24no—
30C2×#367C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×C4×C16C2211yescharacteristic
44#1972—21no—
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1961—21no—
48#1846C2211yescharacteristic
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×D8—21no—
64C22×D8C2311yescharacteristic
65C22×D8C2311yescharacteristic
66C2×#127—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×SD32—22no—
75C2×SD32—22no—
76C2×SD32C2×C414yescomplemented
77C2×SD32C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#493—22no—
83#493C2×C414yes—
84#442—44no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×Q16C2×C414yescomplemented
90C4×Q16D414yescomplemented
91C4×D8—22no—
92C4×D8—22no—
93C4×D8—22no—
94C4×D8—22no—
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#106—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#390—21no—
115#390—21no—
116#386—41no—
117#386—41no—
118#386—41no—
119#386—41no—
120#382—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#367—22no—
130#366—22no—
131C4⋊C16—22no—
132C4⋊C16C2×C414yescomplemented
133#358—22no—
134#358—22no—
135#357—22no—
136#357—22no—
137#357—22no—
138#357—22no—
139#357C2×C414yescomplemented
140#357C2×C414yescomplemented
141#346—22no—
142C4×C16—22no—
143#340—21no—
144#340—21no—
145#340—21no—
146C22×Q8—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—41no—
150C22×D4—41no—
151C23×C4—41no—
152C23×C4—41no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16C22×C412yes—
157C2×Q16C2×D412yes—
158C2×Q16#4112yes—
159C2×D8—24no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—42no—
171C2×D8—42no—
172C2×D8C22×C412yes—
173C2×D8C22×C412yes—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2411yescharacteristic
178#127—24no—
179#127—42no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
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×C22⋊C4—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
213SD32—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#107C22×C412yes—
235#107C2×D412yes—
236#107#4112yes—
237#106—42no—
238#103—24no—
239#103—24no—
240#103—24no—
241#102—42no—
242#102—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—
264C24—41no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—44no—
284C2×D4—44no—
285C2×D4—161no—
286C2×D4—161no—
287C2×D4—161no—
288C2×D4—161no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—161no—
298C22×C4—161no—
299C22×C4—161no—
300C22×C4—161no—
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D811yescharacteristic
305C22×C4C2×D811yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4C22×D411yescharacteristic
308Q16—22no—
309Q16—22no—
310Q16—22no—
311Q16C4×D414yes—
312D8—22no—
313D8—22no—
314D8—22no—
315D8—22no—
316D8—48no—
317D8—41no—
318D8—41no—
319D8—41no—
320D8—41no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8C4×D412yes—
329C2×C8#12112yes—
330C2×C8C23×C411yescharacteristic
331C2×C8C22×D411yescharacteristic
332C2×C8C2×#4111yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C8C2×#4111yescharacteristic
335C2×C8C2×#4111yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C22⋊C4—161no—
354C22⋊C4—161no—
355C42—21no—
356C42—21no—
357C42—44no—
358C42—44no—
359C42—44no—
360C42—161no—
361C42—161no—
362C42—161no—
363C42C4×D412yes—
364C42#12112yes—
365C42#13512yes—
366C42AGL(1,ℤ/8)12yes—
367C16—22no—
368C16—22no—
369C16—41no—
370C16—41no—
371C23—42no—
372C23—42no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23C22×D811yescharacteristic
386Q8—24no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—81no—
395D4—81no—
396D4—81no—
397D4—81no—
398D4—164no—
399D4—164no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
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—81no—
431C2×C4—81no—
432C2×C4—164no—
433C2×C4—164no—
434C2×C4C4×D812yes—
435C2×C4C4×D812yes—
436C2×C4#45912yes—
437C2×C4#45912yes—
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×#12111yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4#57511yescharacteristic
444C8—22no—
445C8—22no—
446C8—41no—
447C8C2×C4×D411yescharacteristic
448C8C4×#4111yescharacteristic
449C8#52911yescharacteristic
450C8#53211yescharacteristic
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22C2×C4×D811yescharacteristic
466C22C2×#45911yescharacteristic
467C22#310011yescharacteristic
468C22#341511yescharacteristic
469C22#341611yescharacteristic
470C22#341611yescharacteristic
471C22#341611yescharacteristic
472C4—22no—
473C4—22no—
474C4—22no—
475C4—22no—
476C4—41no—
477C4—41no—
478C4—44no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—161no—
483C4—161no—
484C4C4×#12111yescharacteristic
485C4C4×AGL(1,ℤ/8)11yescharacteristic
486C4#303511yescharacteristic
487C4#310611yescharacteristic
488C2—162no—
489C2—162no—
490C2C4×#45911yescharacteristic
491C2#3345411yescharacteristic
492C2#3345711yescharacteristic
493C2#3358711yescharacteristic
494C2#3358911yescharacteristic
495C2#3367911yescharacteristic
496C2#3368611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
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
2. C2×#2175C2semidirect
3. C2×C4×SD32C2semidirect
5. #20507C2semidirect
7. #13178C2semidirect
8. #13131C2semidirect
9. #13130C2semidirect
10. #13013C2semidirect
11. #13013C2semidirect
12. #12814C2semidirect
13. #12654C2semidirect
14. #12637C2semidirect
15. C4×#357C2semidirect
27. #2207C4semidirect
76. C2×SD32C2×C4semidirect
77. C2×SD32C2×C4semidirect
89. C4×Q16C2×C4semidirect
90. C4×Q16D4semidirect
132. C4⋊C16C2×C4semidirect
139. #357C2×C4semidirect
140. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22072
C2×#21752
C2×C4×SD322
#216972
#205072
#132522
#131782
#131312
#131302
#130132
#128142
#126542
#126372
C4×#3572