Finite Group Databasebeta

← #498159

Subgroups — #498159

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2188C211yesmaximal, characteristic
2C2×#2176C211yesmaximal, characteristic, complemented
3C2×#2141C211yesmaximal, characteristic
4#23026C211yesmaximal, characteristic
5C22⋊C4×Q16C211yesmaximal, characteristic
6#13236C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic
10#13015C211yesmaximal, characteristic, complemented
11#13015C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic
14C4×#358C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C22×Q32C2211yescharacteristic
18C22×Q32C2211yescharacteristic
19C22×M32C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×#427C2211yescharacteristic
25#2362—21no—
26Q8×C22⋊C4—21no—
27#2188C418yescomplemented
28#2176—24no—
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31#2141—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1927—21no—
48#1878—21no—
49#1878—21no—
50#1873—21no—
51#1865—21no—
52#1808C2211yescharacteristic
53#1787—21no—
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C23×Q8—21no—
63C22×Q16—22no—
64C22×Q16—22no—
65C22×Q16—21no—
66C22×Q16—22no—
67C22×Q16—41no—
68C22×Q16—41no—
69C22×Q16C2×C414yes—
70C22×Q16C2311yescharacteristic
71C22×Q16C2311yescharacteristic
72C22×Q16C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×C22⋊Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×#117—21no—
79C2×Q32—22no—
80C2×Q32—22no—
81C2×Q32—41no—
82C2×Q32—41no—
83C2×Q32C2×C414yes—
84C2×Q32C2×C414yes—
85C2×M32—22no—
86C2×M32—22no—
87C2×M32—41no—
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#467—22no—
93#467C2×C414yes—
94C4×Q16—22no—
95C4×Q16—22no—
96C4×Q16—22no—
97C4×Q16—41no—
98C4×Q16—41no—
99C4×Q16—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×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#103—21no—
114C2×#103—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#391—21no—
121#391—41no—
122#391—41no—
123#389—21no—
124#388—41no—
125#388—41no—
126#387—41no—
127#380—21no—
128#380—41no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—41no—
131C4×C22⋊C4—41no—
132C4×C22⋊C4D411yescharacteristic
133#367—22no—
134#366—41no—
135#358—22no—
136#358—22no—
137#358—22no—
138#358—22no—
139#358—41no—
140#358C2×C414yes—
141#358C2×C414yes—
142#358D414yes—
143C22⋊C16C2×C414yes—
144#346—22no—
145C4×C16—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C22×Q8—24no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—41no—
158C22×Q8—42no—
159C22×Q8—81no—
160C22×Q8—81no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C2×Q16—28no—
164C2×Q16—24no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×Q16—21no—
178C2×Q16—44no—
179C2×Q16—44no—
180C2×Q16—44no—
181C2×Q16—42no—
182C2×Q16—42no—
183C2×Q16C22×C412yes—
184C2×Q16C22×C412yes—
185C2×Q16C22×C412yes—
186C2×Q16C2×D412yes—
187C2×Q16#4112yes—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8C22×C412yes—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8#4112yes—
197C22×C8C2411yescharacteristic
198C22⋊Q8—24no—
199C22⋊Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—81no—
204C4×Q8—81no—
205C4×Q8—81no—
206#117—42no—
207#117—81no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
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⋊C4D811yescharacteristic
222C2×C22⋊C4D811yescharacteristic
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230Q32—28no—
231Q32—44no—
232M32—44no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16C22×C412yes—
243C2×C16C22×C412yes—
244C2×C16C2×D412yes—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107C22×C412yes—
254#107C2×D412yes—
255#106—42no—
256#103—24no—
257#103—42no—
258#103—42no—
259#103—42no—
260#103—81no—
261#103—81no—
262#103—81no—
263C22⋊C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95D811yescharacteristic
278#95D811yescharacteristic
279C24C2×D811yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—48no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4C4×D412yes—
321C22×C4C2×D811yescharacteristic
322C22×C4C2×D811yescharacteristic
323C22×C4C2×D811yescharacteristic
324C22×C4C2×D811yescharacteristic
325C22×C4C2×D811yescharacteristic
326C22×C4#13512yes—
327C22×C4C22×D411yescharacteristic
328Q16—22no—
329Q16—22no—
330Q16—22no—
331Q16—22no—
332Q16—22no—
333Q16—22no—
334Q16—22no—
335Q16—48no—
336Q16—41no—
337Q16—41no—
338Q16—41no—
339Q16—41no—
340Q16C4×D414yes—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8C4×D412yes—
354C2×C8C23×C411yescharacteristic
355C2×C8C22×D411yescharacteristic
356C2×C8C2×#4111yescharacteristic
357C2×C8C2×#4111yescharacteristic
358C2×C8H(5,𝔽2)11yescharacteristic
359C2×C8H(5,𝔽2)11yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C22⋊C4—22no—
376C22⋊C4—41no—
377C22⋊C4—44no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—82no—
385C42—82no—
386C42—82no—
387C42—82no—
388C42—82no—
389C16—22no—
390C16—22no—
391C16—22no—
392C16—41no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23C4×D812yes—
400C23C22×D811yescharacteristic
401Q8—24no—
402Q8—42no—
403Q8—42no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408Q8—42no—
409Q8—88no—
410Q8—81no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—88no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4C4×D812yes—
457C2×C4#51012yes—
458C2×C4C2×C4×D411yescharacteristic
459C2×C4D4211yescharacteristic
460C2×C4#54611yescharacteristic
461C2×C4C2×#13511yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57711yescharacteristic
464C8—22no—
465C8—41no—
466C8—41no—
467C8C2×C4×D411yescharacteristic
468C8#51811yescharacteristic
469C8#54611yescharacteristic
470C8#55111yescharacteristic
471C22—22no—
472C22—22no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22C2×C4×D811yescharacteristic
479C22D4×D811yescharacteristic
480C22#329411yescharacteristic
481C22C2×#51011yescharacteristic
482C22#341411yescharacteristic
483C22#341711yescharacteristic
484C22#341711yescharacteristic
485C4—22no—
486C4—22no—
487C4—41no—
488C4—41no—
489C4—44no—
490C4—41no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4—161no—
496C4—161no—
497C4—161no—
498C4#235811yescharacteristic
499C4#294611yescharacteristic
500C4#328211yescharacteristic
501C4#329711yescharacteristic
502C2—42no—
503C2#2287011yescharacteristic
504C2C4×#51011yescharacteristic
505C2#3345811yescharacteristic
506C2#3384211yescharacteristic
507C2#3384311yescharacteristic
508C2#3387911yescharacteristic
509C2#3388011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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×#2176C2semidirect
6. #13236C2semidirect
7. #13150C2semidirect
8. #13132C2semidirect
10. #13015C2semidirect
11. #13015C2semidirect
12. #12936C2semidirect
14. C4×#358C2semidirect
27. #2188C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21882
C2×#21762
C2×#21412
#230262
C22⋊C4×Q162
#132362
#131502
#131322
#130622
#130152
#129362
#129212
C4×#3582
#126122