Finite Group Databasebeta

← #496869

Subgroups — #496869

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33645C211yesmaximal, characteristic
2#21544C211yesmaximal, characteristic
3C2×#1945C211yesmaximal, characteristic, complemented
4#13169C212yesmaximal, complemented
5#13151C212yesmaximal, complemented
6#13113C212yesmaximal, complemented
7#12990C211yesmaximal, characteristic
8#12939C212yesmaximal, complemented
9#12922C211yesmaximal, characteristic, complemented
10#12869C211yesmaximal, characteristic, complemented
11C4×C22⋊C16C211yesmaximal, characteristic, complemented
12#3050C2211yescharacteristic
13C2×#443C2211yescharacteristic
14#2929—21no—
15C22×#108C2211yescharacteristic
16C2×#427C2211yescharacteristic
17C2×#424—21no—
18C22×C4×C8C2211yescharacteristic
19#2237C412yescomplemented
20#2236C411yescharacteristic, complemented
21#2235C411yescharacteristic, complemented
22#2234C412yescomplemented
23#2233C411yescharacteristic, complemented
24#2232C411yescharacteristic, complemented
25C2×#368C2212yes—
26#2158C2212yes—
27C2×#359C2212yes—
28#2138C2212yes—
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2212yes—
32#1974C2211yescharacteristic
33#1945—21no—
34#1945—21no—
35#1945C2211yescharacteristic
36#1945C2212yes—
37#1945C2212yes—
38#1937—21no—
39#1896—21no—
40#1810C2211yescharacteristic
41#1772C2211yescharacteristic
42#1764—21no—
43C2×#339—21no—
44#1382C2212yes—
45#1382C2212yes—
46#1382C2212yes—
47#1381C2212yes—
48#1374C2212yes—
49C2×#135C2311yescharacteristic
50C2×#131—21no—
51C22×M16—21no—
52C23×C8D411yescharacteristic
53C23×C8C2311yescharacteristic
54#525—21no—
55C2×#117—21no—
56C22×C42D411yescharacteristic
57C22×C16C2312yes—
58C22×C16C2312yes—
59#467C2×C411yescharacteristic
60#467C2×C411yescharacteristic
61#466C2×C411yescharacteristic
62#466C2×C411yescharacteristic
63#465—21no—
64#465—21no—
65#443—42no—
66#435—41no—
67#435—41no—
68C8×D4—41no—
69C8×D4—41no—
70C2×#108—21no—
71C2×#108—42no—
72C2×#108C2311yescharacteristic
73C2×#108C2311yescharacteristic
74C2×#108C2312yes—
75#427—21no—
76#427—21no—
77#427C2×C412yes—
78#427C2312yes—
79#427C2311yescharacteristic
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#106—21no—
83#424—22no—
84#424—22no—
85#424—21no—
86#424—41no—
87#424—41no—
88C2×C4⋊C8—41no—
89C2×C4⋊C8—41no—
90C2×#104—21no—
91C2×#104—21no—
92#416—21no—
93#408—21no—
94#407—21no—
95#405—21no—
96C2×C4×C8—42no—
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#368—42no—
102#367C2×C412yescomplemented
103#367C2×C412yescomplemented
104#366C2×C412yescomplemented
105#366C2×C412yescomplemented
106#359—42no—
107#358C2×C412yescomplemented
108#358C2×C412yescomplemented
109#357C2×C412yescomplemented
110#357C2×C412yescomplemented
111C22⋊C16—21no—
112C22⋊C16—21no—
113C22⋊C16C2×C411yescharacteristic
114C22⋊C16C2×C411yescharacteristic
115C22⋊C16C2×C412yes—
116C4×C16—42no—
117#341—21no—
118#339—21no—
119#339—21no—
120#339—22no—
121#339—22no—
122C2×#41—21no—
123C23×C4—21no—
124C23×C4C2×D411yescharacteristic
125#135—22no—
126#135—22no—
127C2×Q16C22×C411yescharacteristic
128C2×SD16—21no—
129C2×D8C22×C411yescharacteristic
130#131—42no—
131#131—42no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8D811yescharacteristic
146C22×C8D811yescharacteristic
147C22×C8D811yescharacteristic
148C22×C8D811yescharacteristic
149C22×C8C22×C411yescharacteristic
150C22×C8C22×C411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8#4111yescharacteristic
155C22×C8#4111yescharacteristic
156C22×C8C2411yescharacteristic
157#123—41no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160#121—21no—
161#121—21no—
162#120—41no—
163C4×D4—41no—
164C4×D4—41no—
165#117—22no—
166#117—22no—
167#117—21no—
168#117—41no—
169#117—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C22⋊C4—41no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—41no—
176C2×C42—41no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16C22×C412yes—
185C2×C16C22×C412yes—
186#108—22no—
187#108—22no—
188#108—22no—
189#108—22no—
190#108—42no—
191#107—21no—
192#107—21no—
193#107C22×C412yes—
194#107C22×C412yes—
195#107C22×C411yescharacteristic
196#107C22×C411yescharacteristic
197#106—22no—
198#106—22no—
199#106—21no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206#104—42no—
207#104—42no—
208#103—21no—
209#103—21no—
210#102—21no—
211#102—21no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#97—41no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222C24#13511yescharacteristic
223#41—41no—
224#41—41no—
225#41—41no—
226#41—41no—
227C2×Q8—21no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4#13511yescharacteristic
249C22×C4C22×D411yescharacteristic
250Q16—22no—
251SD16—42no—
252D8—22no—
253M16—41no—
254M16—42no—
255M16—41no—
256M16—41no—
257M16—42no—
258M16—41no—
259M16—82no—
260M16—82no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—82no—
279C2×C8—82no—
280C2×C8C4×D411yescharacteristic
281C2×C8C4×D411yescharacteristic
282C2×C8C2×D811yescharacteristic
283C2×C8C2×D811yescharacteristic
284C2×C8C2×D811yescharacteristic
285C2×C8C2×D811yescharacteristic
286C2×C8C23×C411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8H(5,𝔽2)12yes—
289C2×C82−1+412yes—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—42no—
309C42—41no—
310C42—42no—
311C42—42no—
312C42#13712yes—
313C16—22no—
314C16—22no—
315C16—42no—
316C23—21no—
317C23—22no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23C4×D811yescharacteristic
322C23C4×D811yescharacteristic
323C23C2×#13511yescharacteristic
324Q8—41no—
325Q8—41no—
326D4—41no—
327D4—41no—
328D4—82no—
329D4—81no—
330D4—82no—
331D4—81no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—48no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4C4×D811yescharacteristic
359C2×C4C4×D811yescharacteristic
360C2×C4C2×C4×D411yescharacteristic
361C2×C4#53411yescharacteristic
362C2×C4#53611yescharacteristic
363C2×C4C22×D811yescharacteristic
364C2×C4C2×#13711yescharacteristic
365C2×C4#57511yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—81no—
373C8—82no—
374C8—81no—
375C8#52012yes—
376C8#53912yes—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—42no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#217812yes—
390C22C2×C4×D811yescharacteristic
391C22#319011yescharacteristic
392C22#323211yescharacteristic
393C4—24no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—82no—
399C4—81no—
400C4—82no—
401C4—81no—
402C4#232211yescharacteristic
403C4C4×#13711yescharacteristic
404C4#318611yescharacteristic
405C4#322611yescharacteristic
406C2—21no—
407C2—22no—
408C2—21no—
409C2—41no—
410C2—161no—
411C2#2207511yescharacteristic
412C2C2×#217811yescharacteristic
413C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#3050C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1945C2semidirect
4. #13169C2semidirect
5. #13151C2semidirect
6. #13113C2semidirect
8. #12939C2semidirect
9. #12922C2semidirect
10. #12869C2semidirect
11. C4×C22⋊C16C2semidirect
19. #2237C4semidirect
20. #2236C4semidirect
21. #2235C4semidirect
22. #2234C4semidirect
23. #2233C4semidirect
24. #2232C4semidirect
102. #367C2×C4semidirect
103. #367C2×C4semidirect
104. #366C2×C4semidirect
105. #366C2×C4semidirect
107. #358C2×C4semidirect
108. #358C2×C4semidirect
109. #357C2×C4semidirect
110. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
#336452
#215442
C2×#19452
#131692
#131512
#131132
#129902
#129392
#129222
#128692
C4×C22⋊C162