Finite Group Databasebeta

← #501926

Subgroups — #501926

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28227C211yesmaximal, characteristic
2#22656C211yesmaximal, characteristic, complemented
3#13493C211yesmaximal, characteristic, complemented
4#13492C211yesmaximal, characteristic, complemented
5#13380C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic, complemented
7#13190C211yesmaximal, characteristic, complemented
8#13176C211yesmaximal, characteristic, complemented
9#12992C211yesmaximal, characteristic, complemented
10#12991C211yesmaximal, characteristic, complemented
11#12890C211yesmaximal, characteristic, complemented
12#12868C211yesmaximal, characteristic, complemented
13#12867C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2537—21no—
23#2352—21no—
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#2033—21no—
37#2033—21no—
38#1949—21no—
39#1854—21no—
40#1853—21no—
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1802—21no—
48C4×#107C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—41no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#466—22no—
74#466—22no—
75#466—41no—
76#466—41no—
77C4×D8—22no—
78C4×D8—22no—
79C8×D4—22no—
80C8×D4—22no—
81#427—22no—
82#427—41no—
83C2×#107—22no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—41no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#106—21no—
96C2×C4⋊C8D411yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#394—41no—
104#394—41no—
105#388—41no—
106#387—41no—
107#386—41no—
108#384—21no—
109#382—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—41no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4—41no—
117#366—41no—
118#366—41no—
119C4⋊C16—22no—
120#357—22no—
121#357—22no—
122#357—41no—
123#357—41no—
124C22⋊C16—41no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127C22⋊C16—41no—
128#346—22no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×D4—21no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4D811yescharacteristic
138C23×C4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8#4112yes—
148C22×C8—21no—
149C22×C8—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×C8C2411yescharacteristic
164#121—42no—
165#121—42no—
166#121—81no—
167#121—81no—
168#121—84no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—42no—
172C4×D4—42no—
173#117—42no—
174#117—81no—
175C2×C4⋊C4—21no—
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—42no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4D811yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—42no—
227#107—81no—
228#107#4112yes—
229#106—42no—
230#106—81no—
231C4⋊C8—22no—
232#102—42no—
233#102—42no—
234#102—42no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C4×C8#4112yes—
238C4×C8#4112yes—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246C24—41no—
247C24C2×D811yescharacteristic
248C24C2×D811yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—161no—
263C2×D4—161no—
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—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C22×D411yescharacteristic
294D8—22no—
295D8—22no—
296D8—41no—
297D8—41no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8#12312yes—
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
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—82no—
337C4⋊C4—82no—
338C4⋊C4—161no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—161no—
349C42—82no—
350C42—161no—
351C42#12312yes—
352C42#13712yes—
353C16—41no—
354C16—41no—
355C16—41no—
356C16—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23#50912yes—
378C23C22×D811yescharacteristic
379D4—44no—
380D4—42no—
381D4—42no—
382D4—81no—
383D4—81no—
384D4—164no—
385D4—164no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
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—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
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—81no—
425C2×C4—164no—
426C2×C4—164no—
427C2×C4#48012yes—
428C2×C4#48012yes—
429C2×C4C2×#12311yescharacteristic
430C2×C4#54611yescharacteristic
431C2×C4#54611yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C8—22no—
436C8—41no—
437C8—41no—
438C8#52911yescharacteristic
439C8#54011yescharacteristic
440C8#55311yescharacteristic
441C8#55511yescharacteristic
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22C2×#48011yescharacteristic
464C22#329411yescharacteristic
465C22#329411yescharacteristic
466C22C2×#50911yescharacteristic
467C22#341411yescharacteristic
468C22#341511yescharacteristic
469C22#341711yescharacteristic
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—82no—
475C4—161no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4#242011yescharacteristic
482C4#309911yescharacteristic
483C4#332911yescharacteristic
484C4#333111yescharacteristic
485C2—42no—
486C2—81no—
487C2—162no—
488C2#2496511yescharacteristic
489C2#3364811yescharacteristic
490C2#3365211yescharacteristic
491C2#3387511yescharacteristic
492C2#3388211yescharacteristic
493C2#3388511yescharacteristic
494C2#3388611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28227C2
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
2. #22656C2semidirect
3. #13493C2semidirect
4. #13492C2semidirect
5. #13380C2semidirect
6. #13380C2semidirect
7. #13190C2semidirect
8. #13176C2semidirect
9. #12992C2semidirect
10. #12991C2semidirect
11. #12890C2semidirect
12. #12868C2semidirect
13. #12867C2semidirect
14. #12813C2semidirect
15. #12637C2semidirect

Maximal subgroups

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

subgroupindex
#282272
#226562
#134932
#134922
#133802
#131902
#131762
#129922
#129912
#128902
#128682
#128672
#128132
#126372