Finite Group Databasebeta

← #498135

Subgroups — #498135

Order 512.

814 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2186C211yesmaximal, characteristic
2C2×C4×SD32C211yesmaximal, characteristic, complemented
3C2×#2138C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic, complemented
5#22451C211yesmaximal, characteristic, complemented
6#13235C211yesmaximal, characteristic, complemented
7#13147C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic, complemented
10#13013C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14C4×#358C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C22×D16C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×#427C2211yescharacteristic
25#2363—21no—
26#2341—21no—
27#2186C418yescomplemented
28C4×SD32—24no—
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31#2138—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1942C2211yescharacteristic
47#1926—21no—
48#1880—21no—
49#1880—21no—
50#1875—21no—
51#1864—21no—
52#1808C2211yescharacteristic
53#1788—21no—
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×#41—21no—
63C2×#135—22no—
64C2×#135—22no—
65C2×#135—22no—
66C2×#135—41no—
67C2×#135—41no—
68C2×#135C2×C414yescomplemented
69C22×Q16C2311yescharacteristic
70C22×SD16—21no—
71C22×D8C2311yescharacteristic
72C22×D8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×#121—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×D4—21no—
78C2×#117—21no—
79C2×SD32—22no—
80C2×SD32—22no—
81C2×SD32C2×C414yes—
82C2×D16—41no—
83C2×D16—41no—
84C2×D16C2×C414yes—
85C22×C16—22no—
86C22×C16—22no—
87C22×C16—41no—
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#466—22no—
93#466C2×C414yes—
94C4×Q16—22no—
95C4×Q16—22no—
96C4×Q16—41no—
97C4×Q16—41no—
98C4×D8—22no—
99C4×D8—41no—
100#427—22no—
101#427—41no—
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#106—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#390—21no—
119#390—21no—
120#390—21no—
121#390—41no—
122#390—41no—
123#390—41no—
124#389—21no—
125#388—41no—
126#386—41no—
127#380—21no—
128#380—41no—
129C4×C4⋊C4—21no—
130C4×C22⋊C4—41no—
131C4×C22⋊C4—41no—
132C4×C22⋊C4D411yescharacteristic
133#367—22no—
134#366—41no—
135#358—22no—
136#358C2×C414yescomplemented
137#358D414yescomplemented
138#357—22no—
139#357—22no—
140#357—22no—
141#357—41no—
142#357C2×C414yescomplemented
143C22⋊C16C2×C414yescomplemented
144C4×C16—22no—
145C4×C16—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C2×#41—24no—
152C2×#41—42no—
153C2×#41—42no—
154C2×#41—42no—
155C2×#41—81no—
156C2×#41—81no—
157C22×Q8—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—41no—
161C22×D4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165#135—28no—
166#135—44no—
167#135—44no—
168#135—44no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16C22×C412yes—
173C2×Q16C2×D412yes—
174C2×Q16#4112yes—
175C2×SD16—24no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8C22×C412yes—
189C2×D8C22×C412yes—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8C22×C412yes—
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8#4112yes—
199C22×C8C2411yescharacteristic
200#121—24no—
201#121—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—81no—
205C4×Q8—81no—
206C4×D4—42no—
207C4×D4—81no—
208#117—42no—
209#117—81no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4D811yescharacteristic
229C2×C22⋊C4D811yescharacteristic
230C2×C42—41no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236SD32—28no—
237D16—44no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—21no—
247C2×C16—44no—
248C2×C16C22×C412yes—
249C2×C16C22×C412yes—
250C2×C16C2×D412yes—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—21no—
258#107—21no—
259#107C22×C412yes—
260#107C2×D412yes—
261#106—42no—
262#103—42no—
263#103—42no—
264#102—24no—
265#102—42no—
266#102—81no—
267#102—81no—
268#102—81no—
269C22⋊C8—22no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95D811yescharacteristic
278#95D811yescharacteristic
279C24—41no—
280C24—41no—
281C24C2×D811yescharacteristic
282#41—48no—
283#41—84no—
284#41—84no—
285#41—84no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—161no—
309C2×D4—161no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—161no—
326C22×C4C4×D412yes—
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×D811yescharacteristic
332C22×C4#13512yes—
333C22×C4C22×D411yescharacteristic
334Q16—22no—
335Q16—22no—
336Q16—22no—
337Q16C4×D414yes—
338SD16—48no—
339D8—22no—
340D8—22no—
341D8—22no—
342D8—22no—
343D8—41no—
344D8—41no—
345D8—41no—
346D8—41no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8C4×D412yes—
360C2×C8C23×C411yescharacteristic
361C2×C8C22×D411yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4—161no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—161no—
388C22⋊C4—161no—
389C22⋊C4—161no—
390C22⋊C4—161no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—82no—
396C42—82no—
397C42—82no—
398C42—161no—
399C16—22no—
400C16—22no—
401C16—22no—
402C16—41no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—42no—
410C23—42no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23C4×D812yes—
424C23C22×D811yescharacteristic
425Q8—24no—
426Q8—42no—
427Q8—42no—
428Q8—42no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—88no—
434D4—81no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—164no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—88no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—164no—
475C2×C4C4×D812yes—
476C2×C4#50812yes—
477C2×C4#50812yes—
478C2×C4C2×C4×D411yescharacteristic
479C2×C4D4211yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4#57611yescharacteristic
483C2×C4#57711yescharacteristic
484C8—22no—
485C8—41no—
486C8—41no—
487C8C2×C4×D411yescharacteristic
488C8#51811yescharacteristic
489C8#54611yescharacteristic
490C8#55111yescharacteristic
491C22—22no—
492C22—22no—
493C22—22no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—161no—
505C22—161no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22—161no—
512C22C2×C4×D811yescharacteristic
513C22D4×D811yescharacteristic
514C22#329411yescharacteristic
515C22C2×#50811yescharacteristic
516C22C2×#50811yescharacteristic
517C22#341511yescharacteristic
518C22#341611yescharacteristic
519C4—22no—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—44no—
525C4—82no—
526C4—82no—
527C4—161no—
528C4—161no—
529C4—161no—
530C4#235811yescharacteristic
531C4#294611yescharacteristic
532C4#328211yescharacteristic
533C4#329711yescharacteristic
534C2—42no—
535C2—162no—
536C2—162no—
537C2#2287011yescharacteristic
538C2C4×#50811yescharacteristic
539C2#3345411yescharacteristic
540C2#3384811yescharacteristic
541C2#3385311yescharacteristic
542C2#3386811yescharacteristic
543C2#3387211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2186C2
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. C2×C4×SD32C2semidirect
3. C2×#2138C2semidirect
4. #23027C2semidirect
5. #22451C2semidirect
6. #13235C2semidirect
7. #13147C2semidirect
8. #13131C2semidirect
9. #13059C2semidirect
10. #13013C2semidirect
11. #13012C2semidirect
12. #12936C2semidirect
13. #12921C2semidirect
14. C4×#358C2semidirect
15. C16×C22⋊C4C2semidirect
27. #2186C4semidirect
68. C2×#135C2×C4semidirect
136. #358C2×C4semidirect
137. #358D4semidirect
142. #357C2×C4semidirect
143. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21862
C2×C4×SD322
C2×#21382
#230272
#224512
#132352
#131472
#131312
#130592
#130132
#130122
#129362
#129212
C4×#3582
C16×C22⋊C42