Finite Group Databasebeta

← #508883

Subgroups — #508883

Order 512.

517 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33319C211yesmaximal, characteristic, complemented
2#33247C211yesmaximal, characteristic
3#13483C211yesmaximal, characteristic, complemented
4#13482C211yesmaximal, characteristic
5#13481C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic
7#13366C211yesmaximal, characteristic, complemented
8#13365C211yesmaximal, characteristic
9#13364C211yesmaximal, characteristic, complemented
10#13363C211yesmaximal, characteristic
11#13283C211yesmaximal, characteristic, complemented
12#13282C211yesmaximal, characteristic, complemented
13#13205C211yesmaximal, characteristic
14#13204C211yesmaximal, characteristic
15#12592C211yesmaximal, characteristic, complemented
16C2×#501—21no—
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#499C2211yescharacteristic
20C2×#495C2211yescharacteristic
21C2×#489—21no—
22#2660—21no—
23#2640—21no—
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34#1973—21no—
35#1973—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1969C2211yescharacteristic
39#1969C2211yescharacteristic
40#1847—21no—
41#1847—21no—
42#1846C2211yescharacteristic
43#1846C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1726C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×#125—21no—
65C2×#124—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#501—44no—
71#500—22no—
72#500—22no—
73#500—41no—
74#500—41no—
75#499—41no—
76#499—41no—
77#495—22no—
78#495—22no—
79#489—44no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#97D411yescharacteristic
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#398—41no—
104#398—41no—
105#397—41no—
106#397—41no—
107#395—41no—
108#395—41no—
109#393—41no—
110#393—41no—
111#391—21no—
112#391—21no—
113#390—21no—
114#390—21no—
115#384—21no—
116#384—21no—
117#382—21no—
118#382—21no—
119#376D411yescharacteristic
120#358—41no—
121#358—41no—
122#358—41no—
123#358—41no—
124#357—41no—
125#357—41no—
126#357—41no—
127#357—41no—
128#346—22no—
129C4×C16—22no—
130C22×Q8—21no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×D4—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×SD16—42no—
141C2×SD16—42no—
142C2×SD16—42no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2411yescharacteristic
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157#125—42no—
158#125—42no—
159#124—42no—
160#124—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—42no—
209#107—42no—
210#107—42no—
211#107—42no—
212#106—42no—
213#106—42no—
214#106—42no—
215#103—42no—
216#103—42no—
217#102—42no—
218#102—42no—
219#97—22no—
220#97—22no—
221#97SD1614yes—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8#4112yes—
227C4×C8#4112yes—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95D811yescharacteristic
237#95D811yescharacteristic
238#95D811yescharacteristic
239#95D811yescharacteristic
240C24—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—161no—
250C2×Q8—161no—
251C2×Q8—161no—
252C2×Q8—161no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C22×D411yescharacteristic
273Q16—41no—
274Q16—41no—
275Q16—41no—
276Q16—41no—
277SD16—84no—
278D8—41no—
279D8—41no—
280D8—41no—
281D8—41no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8#12412yes—
287C2×C8C2×SD1612yes—
288C2×C8C2×SD1612yes—
289C2×C8AGL(1,ℤ/8)12yes—
290C2×C8#13712yes—
291C2×C8C2×#4111yescharacteristic
292C2×C8C2×#4111yescharacteristic
293C2×C8H(5,𝔽2)11yescharacteristic
294C2×C8H(5,𝔽2)11yescharacteristic
295C2×C82−1+411yescharacteristic
296C2×C82−1+411yescharacteristic
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C22⋊C4—161no—
320C22⋊C4—161no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42#12412yes—
328C42C2×SD1612yes—
329C16—41no—
330C16—41no—
331C16—41no—
332C16—41no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23C22×D811yescharacteristic
341Q8—81no—
342Q8—81no—
343Q8—81no—
344Q8—81no—
345Q8—164no—
346Q8—164no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—81no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4#48612yes—
389C2×C4#48612yes—
390C2×C4C2×#12411yescharacteristic
391C2×C4#53511yescharacteristic
392C2×C4#53611yescharacteristic
393C2×C4C22×SD1611yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C8—22no—
397C8—22no—
398C8#48914yes—
399C8#54211yescharacteristic
400C8#54211yescharacteristic
401C8#54211yescharacteristic
402C8#54211yescharacteristic
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22—161no—
410C22C2×#48611yescharacteristic
411C22#320511yescharacteristic
412C22#322611yescharacteristic
413C22#341511yescharacteristic
414C22#341611yescharacteristic
415C22#341611yescharacteristic
416C22#341711yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4#264011yescharacteristic
431C4C2×#48911yescharacteristic
432C4#320611yescharacteristic
433C4#322711yescharacteristic
434C2—321no—
435C2#3020311yescharacteristic
436C2#3370211yescharacteristic
437C2#3370311yescharacteristic
438C2#3379011yescharacteristic
439C2#3379111yescharacteristic
440C2#3380911yescharacteristic
441C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33319C2semidirect
3. #13483C2semidirect
5. #13481C2semidirect
7. #13366C2semidirect
9. #13364C2semidirect
11. #13283C2semidirect
12. #13282C2semidirect
15. #12592C2semidirect

Maximal subgroups

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

subgroupindex
#333192
#332472
#134832
#134822
#134812
#134802
#133662
#133652
#133642
#133632
#132832
#132822
#132052
#132042
#125922