Finite Group Databasebeta

← #498608

Subgroups — #498608

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2242C211yesmaximal, characteristic
2#23186C211yesmaximal, characteristic
3#23160C211yesmaximal, characteristic, complemented
4#13277C211yesmaximal, characteristic, complemented
5#13254C211yesmaximal, characteristic
6#13150C211yesmaximal, characteristic, complemented
7#13149C211yesmaximal, characteristic
8#13133C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic
10#12936C211yesmaximal, characteristic, complemented
11#12935C211yesmaximal, characteristic
12C4×#366C211yesmaximal, characteristic, complemented
13#12662C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2370—21no—
21#2367—21no—
22#2242C418yescomplemented
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#346C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1989—21no—
38#1983—21no—
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1920—21no—
44#1919—21no—
45#1880—21no—
46#1877—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×SD16—21no—
62C22×D8C2311yescharacteristic
63C2×#125—21no—
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#499—22no—
72#499C2×C414yescomplemented
73#495—22no—
74#495C2×C414yes—
75C4×Q16—22no—
76C4×Q16—41no—
77C4×D8—22no—
78C4×D8—41no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×C4⋊C8D411yescharacteristic
87C2×#103—21no—
88C2×#103—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—21no—
95#390—21no—
96#389—21no—
97#389—21no—
98#389—41no—
99#388—41no—
100#388—41no—
101#387—41no—
102#387—41no—
103#385—41no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—41no—
109C4×C4⋊C4D411yescharacteristic
110C4×C22⋊C4—41no—
111#366—22no—
112#366—22no—
113C4⋊C16—22no—
114#358—22no—
115#358—41no—
116#358C2×C414yescomplemented
117#358C2×C414yescomplemented
118#357—22no—
119#357—41no—
120#357C2×C414yes—
121#357C2×C414yes—
122#346—41no—
123#346—41no—
124C4×C16C2×C414yescomplemented
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#336D411yescharacteristic
132#336D411yescharacteristic
133C22×Q8—21no—
134C22×D4—21no—
135C23×C4—41no—
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C22×C412yes—
142C2×SD16—24no—
143C2×SD16—81no—
144C2×SD16—81no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8C22×C412yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2411yescharacteristic
155#125—24no—
156#125—42no—
157#124—24no—
158#124—42no—
159C4×Q8—42no—
160C4×Q8—81no—
161C4×D4—42no—
162C4×D4—81no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C4⋊C4D811yescharacteristic
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—41no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C22×C412yes—
200C2×C16C22×C412yes—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107C22×C412yes—
210#107C22×C412yes—
211#107#4112yes—
212#107#4112yes—
213#106—24no—
214C4⋊C8—22no—
215#103—42no—
216#103—81no—
217#102—42no—
218#102—81no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8C22×C412yes—
224C4×C8#4112yes—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—161no—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C22×D411yescharacteristic
265Q16—22no—
266Q16—22no—
267Q16—41no—
268Q16—41no—
269SD16—84no—
270D8—22no—
271D8—22no—
272D8—41no—
273D8—41no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8C4×D412yes—
283C2×C8C23×C411yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—161no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—161no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—82no—
318C42—82no—
319C42—161no—
320C42—161no—
321C42—161no—
322C42C4×D412yes—
323C42#13512yes—
324C16—22no—
325C16—22no—
326C16—41no—
327C16—41no—
328C23—42no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23C22×D811yescharacteristic
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340D4—42no—
341D4—42no—
342D4—81no—
343D4—81no—
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—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
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—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—164no—
382C2×C4C4×D812yes—
383C2×C4C4×D812yes—
384C2×C4C2×C4×D411yescharacteristic
385C2×C4#54611yescharacteristic
386C2×C4#54811yescharacteristic
387C2×C4C2×#13511yescharacteristic
388C2×C4#57711yescharacteristic
389C2×C4#57711yescharacteristic
390C8—22no—
391C8—41no—
392C8—41no—
393C8C4×#4111yescharacteristic
394C8#52011yescharacteristic
395C8#55611yescharacteristic
396C8#55911yescharacteristic
397C22—82no—
398C22—82no—
399C22—82no—
400C22—161no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22C2×C4×D811yescharacteristic
405C22#329411yescharacteristic
406C22#335811yescharacteristic
407C22#341411yescharacteristic
408C22#341411yescharacteristic
409C22#341511yescharacteristic
410C22#341711yescharacteristic
411C4—22no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—82no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4#236711yescharacteristic
425C4#294911yescharacteristic
426C4#330411yescharacteristic
427C4#336511yescharacteristic
428C2—162no—
429C2#2321611yescharacteristic
430C2#3345111yescharacteristic
431C2#3345911yescharacteristic
432C2#3388911yescharacteristic
433C2#3389411yescharacteristic
434C2#3393711yescharacteristic
435C2#3394411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23160C2semidirect
4. #13277C2semidirect
6. #13150C2semidirect
8. #13133C2semidirect
10. #12936C2semidirect
12. C4×#366C2semidirect
13. #12662C2semidirect
14. #12638C2semidirect
22. #2242C4semidirect
72. #499C2×C4semidirect
116. #358C2×C4semidirect
117. #358C2×C4semidirect
124. C4×C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22422
#231862
#231602
#132772
#132542
#131502
#131492
#131332
#131302
#129362
#129352
C4×#3662
#126622
#126382
#126372