Finite Group Databasebeta

← #509903

Subgroups — #509903

Order 512.

643 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic, complemented
4#25414C211yesmaximal, characteristic
5#13506C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13407C211yesmaximal, characteristic, complemented
8#13381C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic, complemented
10#13363C211yesmaximal, characteristic, complemented
11#13202C211yesmaximal, characteristic, complemented
12#13166C211yesmaximal, characteristic, complemented
13#13120C211yesmaximal, characteristic, complemented
14#12991C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2428—21no—
24#2233—24no—
25#2226—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42#2083—21no—
43#2057—21no—
44#2041—21no—
45#1999—21no—
46#1970C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1879—21no—
51#1852—21no—
52#1845C2211yescharacteristic
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8—21no—
62C22×D8C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66C2×#127—21no—
67C2×C22⋊Q8—41no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—41no—
71C2×C4×D4—21no—
72C22×C4⋊C4—21no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#500—22no—
78#500—41no—
79#493—22no—
80#493—41no—
81#466—22no—
82#466—22no—
83#466—41no—
84#466—41no—
85C4×D8—22no—
86C4×D8—41no—
87C2×#107—22no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—41no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C22⋊C8D411yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#398—41no—
106#395—41no—
107#394—41no—
108#392—41no—
109#390—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#386—41no—
114#385—41no—
115#384—21no—
116#380—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—41no—
121#367D414yescomplemented
122C4⋊C16—22no—
123#357—22no—
124#357—22no—
125#357—22no—
126#357—22no—
127#357—22no—
128#357—41no—
129#357—41no—
130#357—41no—
131C22⋊C16—22no—
132C22⋊C16—41no—
133C4×C16—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—41no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—41no—
146C22×D4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4C2×D411yescharacteristic
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—42no—
162C2×D8—42no—
163C2×D8—81no—
164C2×D8#4112yes—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2411yescharacteristic
175C4⋊Q8—42no—
176C4⋊Q8—81no—
177#127—42no—
178#127—81no—
179C22⋊Q8—84no—
180#121—42no—
181#121—42no—
182#121—81no—
183#121—81no—
184#121—84no—
185C4×D4—42no—
186C4×D4—81no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—81no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4D811yescharacteristic
209C2×C22⋊C4D811yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
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×C16C2×D412yes—
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—21no—
233#107—21no—
234#107—42no—
235#107—42no—
236#107—81no—
237#107C2×D412yes—
238#107C2×D412yes—
239#102—42no—
240#102—42no—
241#102—42no—
242#102—81no—
243C22⋊C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95D811yescharacteristic
253#95D811yescharacteristic
254C24—41no—
255C24—41no—
256C24C2×D811yescharacteristic
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8—161no—
260C2×D4—22no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—161no—
281C2×D4—161no—
282C2×D4—161no—
283C2×D4—161no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—161no—
301C22×C4—161no—
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D811yescharacteristic
305C22×C4C2×D811yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4C22×D411yescharacteristic
308D8—22no—
309D8—22no—
310D8—41no—
311D8—41no—
312D8—41no—
313D8—41no—
314D8—41no—
315D8—41no—
316D8—84no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8#13712yes—
330C2×C8C22×D411yescharacteristic
331C2×C8C2×#4111yescharacteristic
332C2×C8H(5,𝔽2)11yescharacteristic
333C2×C8H(5,𝔽2)11yescharacteristic
334C2×C82−1+411yescharacteristic
335C2×C82−1+411yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—161no—
355C22⋊C4—22no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—161no—
362C22⋊C4SD3214yes—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C16—22no—
368C16—41no—
369C16—41no—
370C16—41no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23C2×SD3212yes—
392C23C22×D811yescharacteristic
393Q8—164no—
394D4—42no—
395D4—42no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401D4—81no—
402D4—164no—
403D4—164no—
404D4—164no—
405D4—164no—
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—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
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—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—164no—
441C2×C4—164no—
442C2×C4C2×SD3212yes—
443C2×C4C2×SD3212yes—
444C2×C4#50912yes—
445C2×C4#53611yescharacteristic
446C2×C4D4211yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C2×C4#57611yescharacteristic
450C2×C4#57711yescharacteristic
451C8—22no—
452C8—41no—
453C8—41no—
454C8#53711yescharacteristic
455C8#53911yescharacteristic
456C8#54811yescharacteristic
457C8#56211yescharacteristic
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22—161no—
479C22#322611yescharacteristic
480C22D4×D811yescharacteristic
481C22#329411yescharacteristic
482C22C22×SD3211yescharacteristic
483C22C2×#50911yescharacteristic
484C22#341511yescharacteristic
485C22#341611yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—82no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4#268111yescharacteristic
498C4#324111yescharacteristic
499C4#327811yescharacteristic
500C4#331411yescharacteristic
501C2—42no—
502C2—162no—
503C2—321no—
504C2#3111611yescharacteristic
505C2#3379311yescharacteristic
506C2#3380711yescharacteristic
507C2#3384011yescharacteristic
508C2D4×SD3211yescharacteristic
509C2#3386211yescharacteristic
510C2#3387311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#25414C2
LCSG ▷ C22×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
1. C2×#2233C2semidirect
2. C2×#2226C2semidirect
3. #26065C2semidirect
5. #13506C2semidirect
6. #13476C2semidirect
7. #13407C2semidirect
8. #13381C2semidirect
9. #13381C2semidirect
10. #13363C2semidirect
11. #13202C2semidirect
12. #13166C2semidirect
13. #13120C2semidirect
14. #12991C2semidirect
15. #12937C2semidirect
121. #367D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
C2×#22262
#260652
#254142
#135062
#134762
#134072
#133812
#133632
#132022
#131662
#131202
#129912
#129372