Finite Group Databasebeta

← #503828

Subgroups — #503828

Order 512.

636 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2235C211yesmaximal, characteristic, complemented
2C22×#367C211yesmaximal, characteristic, complemented
3#24781C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13494C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13132C211yesmaximal, characteristic, complemented
10#13111C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic, complemented
12#12993C211yesmaximal, characteristic, complemented
13#12976C211yesmaximal, characteristic
14#12920C211yesmaximal, characteristic, complemented
15#12920C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23#2417—21no—
24D4×C4⋊C4—21no—
25#2235—24no—
26C2×#367—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×#367C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#2068—21no—
41#2036—21no—
42#1969C2211yescharacteristic
43#1947—21no—
44#1942C2211yescharacteristic
45#1872—21no—
46#1852—21no—
47#1852—21no—
48#1845C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×D4—41no—
67C2×C4×D4D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16—22no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#467—22no—
79#467—22no—
80#467—22no—
81#467—41no—
82C8×D4—22no—
83C8×D4—22no—
84C2×#107—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—41no—
92C2×#107—41no—
93C2×#107—41no—
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C22⋊C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#398—41no—
109#396—41no—
110#391—21no—
111#389—21no—
112#389—41no—
113#386—41no—
114#385—41no—
115#385—41no—
116#384—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—41no—
121C4×C4⋊C4—21no—
122#367—22no—
123#367—22no—
124#367—22no—
125#367—22no—
126#367—41no—
127#367D414yescomplemented
128#358—22no—
129#358—22no—
130#358—22no—
131#358—41no—
132C22⋊C16—22no—
133C22⋊C16—22no—
134C22⋊C16—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×D4D811yescharacteristic
142C23×C4—41no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8#4112yes—
167C22×C8#4112yes—
168C22×C8C2411yescharacteristic
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C22⋊Q8—81no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—84no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4D811yescharacteristic
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4D811yescharacteristic
204C2×C22⋊C4D811yescharacteristic
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C16—28no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16C2×D412yes—
219C2×C16#4112yes—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—42no—
233#107—42no—
234#107—42no—
235#107—81no—
236#107C2×D412yes—
237#107C2×D412yes—
238C4⋊C8—22no—
239#103—42no—
240#103—42no—
241#103—81no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24C2×D811yescharacteristic
254C24C2×D811yescharacteristic
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4SD3214yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—161no—
289C22×C4—161no—
290C22×C4—161no—
291C22×C4#12312yes—
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4#13712yes—
297C22×C4C22×D411yescharacteristic
298Q16—41no—
299Q16—41no—
300Q16—41no—
301Q16—41no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8#12312yes—
319C2×C8C22×D411yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8C2×#4111yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8H(5,𝔽2)11yescharacteristic
324C2×C82−1+411yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—161no—
347C4⋊C4—161no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—161no—
355C42—21no—
356C42—21no—
357C42—82no—
358C16—22no—
359C16—22no—
360C16—41no—
361C16—41no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23#48012yes—
373C23C2×SD3212yes—
374C23C2×SD3212yes—
375C23#51012yes—
376C23C22×D811yescharacteristic
377Q8—81no—
378Q8—81no—
379Q8—81no—
380Q8—81no—
381D4—28no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
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—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—164no—
425C2×C4—164no—
426C2×C4—164no—
427C2×C4#48012yes—
428C2×C4C2×SD3212yes—
429C2×C4C2×#12311yescharacteristic
430C2×C4D4211yescharacteristic
431C2×C4#54711yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57611yescharacteristic
434C2×C4#57811yescharacteristic
435C8—22no—
436C8—41no—
437C8—41no—
438C8#52511yescharacteristic
439C8#53911yescharacteristic
440C8#54811yescharacteristic
441C8#55311yescharacteristic
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22#223514yescomplemented
456C22C2×#48011yescharacteristic
457C22D4×D811yescharacteristic
458C22#333411yescharacteristic
459C22C22×SD3211yescharacteristic
460C22C2×#51011yescharacteristic
461C22#341611yescharacteristic
462C22#341611yescharacteristic
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—82no—
468C4—82no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4#244311yescharacteristic
476C4#309811yescharacteristic
477C4#327011yescharacteristic
478C4#334611yescharacteristic
479C2—24no—
480C2—42no—
481C2#2612611yescharacteristic
482C2C2×#223511yescharacteristic
483C2#3365311yescharacteristic
484C2#3383211yescharacteristic
485C2D4×SD3211yescharacteristic
486C2#3390211yescharacteristic
487C2#3391111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2235C2semidirect
2. C22×#367C2semidirect
5. #13531C2semidirect
6. #13506C2semidirect
7. #13494C2semidirect
8. #13382C2semidirect
9. #13132C2semidirect
10. #13111C2semidirect
11. #13060C2semidirect
12. #12993C2semidirect
14. #12920C2semidirect
15. #12920C2semidirect
127. #367D4semidirect
455. C22#2235semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
C22×#3672
#247812
D4×#1072
#135312
#135062
#134942
#133822
#131322
#131112
#130602
#129932
#129762
#129202