Finite Group Databasebeta

← #503839

Subgroups — #503839

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2237C211yesmaximal, characteristic, complemented
2C2×#2158C211yesmaximal, characteristic, complemented
3#24779C211yesmaximal, characteristic
4#22658C211yesmaximal, characteristic
5#13530C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13493C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13132C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic, complemented
12#12994C211yesmaximal, characteristic, complemented
13#12976C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic, complemented
15#12921C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22C2×#427C2211yescharacteristic
23#2415—21no—
24#2354—21no—
25#2237—24no—
26#2158—24no—
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#2066—21no—
41#2035—21no—
42#1969C2211yescharacteristic
43#1948—21no—
44#1942C2211yescharacteristic
45#1873—21no—
46#1854—21no—
47#1854—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×D4D411yescharacteristic
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C16—22no—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#467—22no—
78#467—22no—
79#467—22no—
80#467—41no—
81C8×D4—22no—
82C8×D4—22no—
83#427—22no—
84#427—22no—
85#427—22no—
86#427—41no—
87#427—41no—
88#427—41no—
89C2×#107—21no—
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#397—41no—
108#394—41no—
109#391—21no—
110#390—41no—
111#389—21no—
112#389—41no—
113#388—41no—
114#388—41no—
115#388—41no—
116#384—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—41no—
122#367—22no—
123#367—22no—
124#367—22no—
125#366—22no—
126#366—41no—
127#366D414yescomplemented
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×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8#4112yes—
164C22×C8#4112yes—
165C22×C8C2411yescharacteristic
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—81no—
170C4×D4—22no—
171C4×D4—22no—
172#117—42no—
173#117—42no—
174#117—42no—
175#117—81no—
176#117—81no—
177#117—81no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4D811yescharacteristic
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4D811yescharacteristic
200C2×C22⋊C4D811yescharacteristic
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C16—28no—
207C2×C16—21no—
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×C16C2×D412yes—
218C2×C16#4112yes—
219#107—21no—
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—81no—
232#107C2×D412yes—
233#107C2×D412yes—
234#106—42no—
235#106—42no—
236#106—42no—
237C4⋊C8—22no—
238#103—42no—
239#103—42no—
240#103—81no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C4×C8—21no—
244C4×C8—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24C2×D811yescharacteristic
253C24C2×D811yescharacteristic
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C22×C4—21no—
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—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4#12312yes—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4#13712yes—
287C22×C4C22×D411yescharacteristic
288Q16—41no—
289Q16—41no—
290Q16—41no—
291Q16—41no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8#12312yes—
309C2×C8C22×D411yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8H(5,𝔽2)11yescharacteristic
314C2×C82−1+411yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—161no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—161no—
344C22⋊C4—161no—
345C42—21no—
346C42—21no—
347C42—82no—
348C42—82no—
349C42—82no—
350C42—82no—
351C16—22no—
352C16—22no—
353C16—41no—
354C16—41no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23#48012yes—
369C23C22×D811yescharacteristic
370Q8—81no—
371Q8—81no—
372Q8—81no—
373Q8—81no—
374D4—44no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
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#48012yes—
418C2×C4#50812yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54711yescharacteristic
422C2×C4C2×#13711yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57811yescharacteristic
425C8—22no—
426C8—41no—
427C8—41no—
428C8#52511yescharacteristic
429C8#53911yescharacteristic
430C8#54811yescharacteristic
431C8#55311yescharacteristic
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22C2×#48011yescharacteristic
447C22D4×D811yescharacteristic
448C22#333411yescharacteristic
449C22C2×#50811yescharacteristic
450C22#341411yescharacteristic
451C22#341611yescharacteristic
452C22#341711yescharacteristic
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—82no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4—161no—
465C4#244311yescharacteristic
466C4#309811yescharacteristic
467C4#327011yescharacteristic
468C4#334611yescharacteristic
469C2—42no—
470C2—81no—
471C2#2612611yescharacteristic
472C2#3364511yescharacteristic
473C2#3365111yescharacteristic
474C2#3383711yescharacteristic
475C2#3384911yescharacteristic
476C2#3390711yescharacteristic
477C2#3391611yescharacteristic
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×#2237C2semidirect
2. C2×#2158C2semidirect
5. #13530C2semidirect
6. #13506C2semidirect
7. #13493C2semidirect
8. #13382C2semidirect
9. #13132C2semidirect
10. #13109C2semidirect
11. #13060C2semidirect
12. #12994C2semidirect
14. #12921C2semidirect
15. #12921C2semidirect
127. #366D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C2×#21582
#247792
#226582
#135302
#135062
#134932
#133822
#131322
#131092
#130602
#129942
#129762
#129212