Finite Group Databasebeta

← #507703

Subgroups — #507703

Order 512.

509 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33247C211yesmaximal, characteristic
2#21723C211yesmaximal, characteristic, complemented
3#13481C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic
5#13366C211yesmaximal, characteristic, complemented
6#13365C211yesmaximal, characteristic
7#13276C211yesmaximal, characteristic, complemented
8#13275C211yesmaximal, characteristic, complemented
9#13254C211yesmaximal, characteristic
10#13254C211yesmaximal, characteristic
11#13150C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic
13#13148C211yesmaximal, characteristic, complemented
14#13147C211yesmaximal, characteristic
15#12654C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×#489—21no—
20#2640—21no—
21#2330—21no—
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×C4×C16C2211yescharacteristic
34#1992—21no—
35#1973—21no—
36#1973—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1944—21no—
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1845C2211yescharacteristic
45#1776—21no—
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48#1726C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#125—21no—
64C2×#124—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#500—22no—
70#500—22no—
71#499—22no—
72#499—22no—
73#495—22no—
74#495—22no—
75#489—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#106—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#97D411yescharacteristic
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#397—41no—
96#397—41no—
97#393—41no—
98#393—41no—
99#391—21no—
100#391—21no—
101#390—21no—
102#390—21no—
103#389—21no—
104#389—21no—
105#389—41no—
106#389—41no—
107#387—41no—
108#387—41no—
109#384—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112#376D411yescharacteristic
113C4⋊C16—22no—
114C4⋊C16—22no—
115#358—22no—
116#358—22no—
117#358—41no—
118#358—41no—
119#357—22no—
120#357—22no—
121#357—41no—
122#357—41no—
123#346—41no—
124C4×C16—41no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129C22×Q8—21no—
130C22×Q8—41no—
131C22×D4—21no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×SD16—42no—
139C2×SD16—42no—
140C2×SD16—42no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2411yescharacteristic
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153#125—24no—
154#125—42no—
155#124—42no—
156#124—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
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—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—42no—
204#107—42no—
205#107#4112yes—
206#107#4112yes—
207#106—24no—
208#103—42no—
209#103—42no—
210#102—42no—
211#102—42no—
212#97—41no—
213#97—41no—
214#97SD1614yes—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8#4112yes—
220C4×C8#4112yes—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95D811yescharacteristic
230#95D811yescharacteristic
231#95D811yescharacteristic
232#95D811yescharacteristic
233C24—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×Q8—161no—
242C2×Q8—161no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C22×D411yescharacteristic
263Q16—41no—
264Q16—41no—
265Q16—41no—
266Q16—41no—
267SD16—84no—
268D8—41no—
269D8—41no—
270D8—41no—
271D8—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8#12412yes—
279C2×C8C2×SD1612yes—
280C2×C8C2×SD1612yes—
281C2×C8C2×#4111yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8H(5,𝔽2)11yescharacteristic
286C2×C82−1+411yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—161no—
303C4⋊C4—161no—
304C4⋊C4—161no—
305C4⋊C4—161no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—161no—
309C22⋊C4—161no—
310C22⋊C4—161no—
311C22⋊C4—161no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—161no—
317C42—161no—
318C42#12412yes—
319C42C2×SD1612yes—
320C16—41no—
321C16—41no—
322C16—41no—
323C16—41no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23C22×D811yescharacteristic
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336Q8—164no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
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#48612yes—
379C2×C4#48612yes—
380C2×C4#52511yescharacteristic
381C2×C4C2×#12411yescharacteristic
382C2×C4#53711yescharacteristic
383C2×C4C22×SD1611yescharacteristic
384C2×C4#57511yescharacteristic
385C2×C4#57511yescharacteristic
386C8—22no—
387C8—22no—
388C8—22no—
389C8#55611yescharacteristic
390C8#55611yescharacteristic
391C8#55911yescharacteristic
392C8#55911yescharacteristic
393C22—161no—
394C22—161no—
395C22—161no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22#310011yescharacteristic
401C22C2×#48611yescharacteristic
402C22#324511yescharacteristic
403C22#341511yescharacteristic
404C22#341611yescharacteristic
405C22#341611yescharacteristic
406C22#341711yescharacteristic
407C4—22no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—82no—
414C4—82no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4#259211yescharacteristic
421C4#310311yescharacteristic
422C4#314011yescharacteristic
423C4#324611yescharacteristic
424C2—321no—
425C2#2934611yescharacteristic
426C2#3368411yescharacteristic
427C2#3368511yescharacteristic
428C2#3370211yescharacteristic
429C2#3370311yescharacteristic
430C2#3382311yescharacteristic
431C2#3382411yescharacteristic
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
2. #21723C2semidirect
3. #13481C2semidirect
5. #13366C2semidirect
7. #13276C2semidirect
8. #13275C2semidirect
11. #13150C2semidirect
13. #13148C2semidirect
15. #12654C2semidirect

Maximal subgroups

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

subgroupindex
#332472
#217232
#134812
#134802
#133662
#133652
#132762
#132752
#132542
#131502
#131492
#131482
#131472
#126542