Finite Group Databasebeta

← #498615

Subgroups — #498615

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2243C211yesmaximal, characteristic
2#23186C211yesmaximal, characteristic
3#23158C211yesmaximal, characteristic, complemented
4#13275C211yesmaximal, characteristic, complemented
5#13254C211yesmaximal, characteristic
6#13149C211yesmaximal, characteristic
7#13148C211yesmaximal, characteristic, complemented
8#13133C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic
10#12938C211yesmaximal, characteristic, complemented
11#12937C211yesmaximal, characteristic
12#12674C211yesmaximal, characteristic, complemented
13#12663C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2369—21no—
21#2367—21no—
22#2243C418yescomplemented
23C2×#367C2211yescharacteristic
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#1987—21no—
38#1983—21no—
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1921—21no—
44#1921—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×C4⋊Q8—21no—
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500C2×C414yescomplemented
73#495—22no—
74#495C2×C414yes—
75C4×Q16—22no—
76C4×Q16—41no—
77C4×D8—22no—
78C4×D8—41no—
79C2×#107—21no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
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#391—41no—
96#391—41no—
97#390—21no—
98#389—21no—
99#389—21no—
100#389—41no—
101#388—41no—
102#388—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#367—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#346C2×C414yescomplemented
124C4×C16—41no—
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×Q8—41no—
135C22×D4—21no—
136C23×C4—41no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16C22×C412yes—
143C2×SD16—24no—
144C2×SD16—81no—
145C2×SD16—81no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8C22×C412yes—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2411yescharacteristic
156C4⋊Q8—24no—
157C4⋊Q8—42no—
158#124—24no—
159#124—42no—
160C4×Q8—42no—
161C4×Q8—81no—
162C4×D4—42no—
163C4×D4—81no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
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—24no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107C22×C412yes—
211#107C22×C412yes—
212#107#4112yes—
213#107#4112yes—
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—44no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—161no—
244C2×Q8—161no—
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—161no—
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C22×D411yescharacteristic
268Q16—22no—
269Q16—22no—
270Q16—41no—
271Q16—41no—
272SD16—84no—
273D8—22no—
274D8—22no—
275D8—41no—
276D8—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8C4×D412yes—
286C2×C8C23×C411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C2×C82−1+411yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—44no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—82no—
308C4⋊C4—161no—
309C4⋊C4—161no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—161no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—82no—
320C42—82no—
321C42—161no—
322C42—161no—
323C42—161no—
324C42C4×D412yes—
325C42#13512yes—
326C16—22no—
327C16—22no—
328C16—41no—
329C16—41no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23C22×D811yescharacteristic
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342Q8—164no—
343D4—42no—
344D4—42no—
345D4—81no—
346D4—81no—
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—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
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—164no—
384C2×C4C4×D812yes—
385C2×C4C4×D812yes—
386C2×C4#51012yes—
387C2×C4C2×C4×D411yescharacteristic
388C2×C4#54611yescharacteristic
389C2×C4#54811yescharacteristic
390C2×C4C2×#13511yescharacteristic
391C2×C4#57711yescharacteristic
392C2×C4#57711yescharacteristic
393C8—22no—
394C8—41no—
395C8—41no—
396C8C4×#4111yescharacteristic
397C8#52011yescharacteristic
398C8#55611yescharacteristic
399C8#55911yescharacteristic
400C22—82no—
401C22—82no—
402C22—82no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22C2×C4×D811yescharacteristic
408C22#329411yescharacteristic
409C22#335811yescharacteristic
410C22C2×#51011yescharacteristic
411C22#341411yescharacteristic
412C22#341511yescharacteristic
413C22#341611yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4#236711yescharacteristic
428C4#294911yescharacteristic
429C4#330411yescharacteristic
430C4#336511yescharacteristic
431C2—162no—
432C2#2321611yescharacteristic
433C2C4×#51011yescharacteristic
434C2#3345411yescharacteristic
435C2#3388111yescharacteristic
436C2#3388811yescharacteristic
437C2#3393611yescharacteristic
438C2#3393711yescharacteristic
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. #23158C2semidirect
4. #13275C2semidirect
7. #13148C2semidirect
8. #13133C2semidirect
10. #12938C2semidirect
12. #12674C2semidirect
13. #12663C2semidirect
14. #12638C2semidirect
22. #2243C4semidirect
72. #500C2×C4semidirect
116. #358C2×C4semidirect
117. #358C2×C4semidirect
123. #346C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
#231862
#231582
#132752
#132542
#131492
#131482
#131332
#131302
#129382
#129372
#126742
#126632
#126382
C4×#3572