Finite Group Databasebeta

← #501420

Subgroups — #501420

Order 512.

543 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
1C2×#2235C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic
3#27927C211yesmaximal, characteristic
4#22612C211yesmaximal, characteristic
5#13507C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13475C211yesmaximal, characteristic, complemented
9#13178C211yesmaximal, characteristic, complemented
10#13177C211yesmaximal, characteristic, complemented
11#13133C211yesmaximal, characteristic, complemented
12#13132C211yesmaximal, characteristic, complemented
13#13062C211yesmaximal, characteristic
14#12913C211yesmaximal, characteristic
15#12868C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#485—21no—
18C2×#467C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2499—21no—
22#2345—21no—
23#2235—24no—
24#2159—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38#2097—21no—
39#2096—21no—
40#1969C2211yescharacteristic
41#1969C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1921—21no—
45#1921—21no—
46#1846C2211yescharacteristic
47#1845C2211yescharacteristic
48#1824—21no—
49#1808C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C22⋊Q8—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C2×M32—22no—
67C2×M32—22no—
68C2×M32—22no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#485—44no—
74#467—22no—
75#467—22no—
76#427—22no—
77#427—22no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#107—21no—
81C2×#107D414yes—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#397—41no—
98#397—41no—
99#391—21no—
100#391—21no—
101#389—21no—
102#389—21no—
103#387—41no—
104#387—41no—
105#387—41no—
106#387—41no—
107#384—21no—
108#382—21no—
109#380—21no—
110#380—41no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C22⋊C4—41no—
114C4×C22⋊C4D411yescharacteristic
115#367—22no—
116#367—22no—
117#367—22no—
118#367—22no—
119#366—22no—
120#366—22no—
121C4⋊C16—22no—
122C4⋊C16—22no—
123#358—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127C22⋊C16—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D412yes—
152C22×C8#4112yes—
153C22×C8C2411yescharacteristic
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156#117—42no—
157#117—42no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4D811yescharacteristic
177C2×C22⋊C4D811yescharacteristic
178C2×C42—41no—
179C2×C42—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183M32—28no—
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#4112yes—
195C2×C16#4112yes—
196#107—24no—
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—21no—
209#107—42no—
210#107C2×D412yes—
211#107C2×D412yes—
212#106—42no—
213#106—42no—
214#103—42no—
215#103—42no—
216#103—42no—
217C22⋊C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222#95—41no—
223#95—41no—
224#95—41no—
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
235C24C2×D811yescharacteristic
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—44no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—161no—
256C22×C4—161no—
257C22×C4#12112yes—
258C22×C4C2×D811yescharacteristic
259C22×C4C2×D811yescharacteristic
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4C22×D411yescharacteristic
265Q16—41no—
266Q16—41no—
267Q16—41no—
268Q16—41no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8#12112yes—
282C2×C8C22×D411yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C82−1+411yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—44no—
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—41no—
302C4⋊C4—41no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—161no—
307C4⋊C4—161no—
308C22⋊C4—22no—
309C22⋊C4—41no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—82no—
318C42—161no—
319C42—161no—
320C16—22no—
321C16—22no—
322C16—41no—
323C16—41no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#45912yes—
331C23C22×D811yescharacteristic
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
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—42no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
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—164no—
378C2×C4#45912yes—
379C2×C4#51012yes—
380C2×C4C2×#12111yescharacteristic
381C2×C4#54711yescharacteristic
382C2×C4#54711yescharacteristic
383C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
384C2×C4#57811yescharacteristic
385C2×C4#57811yescharacteristic
386C8—22no—
387C8—41no—
388C8—41no—
389C8#52511yescharacteristic
390C8#53611yescharacteristic
391C8#55211yescharacteristic
392C8#55211yescharacteristic
393C22—22no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22C2×#45911yescharacteristic
401C22#333411yescharacteristic
402C22#333411yescharacteristic
403C22C2×#51011yescharacteristic
404C22#341411yescharacteristic
405C22#341611yescharacteristic
406C22#341611yescharacteristic
407C4—22no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—82no—
413C4—82no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4#241311yescharacteristic
421C4#304411yescharacteristic
422C4#334311yescharacteristic
423C4#334311yescharacteristic
424C2—42no—
425C2#2457611yescharacteristic
426C2#3358111yescharacteristic
427C2#3358611yescharacteristic
428C2#3391111yescharacteristic
429C2#3391711yescharacteristic
430C2#3391811yescharacteristic
431C2#3392311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13507C2semidirect
6. #13506C2semidirect
7. #13477C2semidirect
8. #13475C2semidirect
9. #13178C2semidirect
10. #13177C2semidirect
11. #13133C2semidirect
12. #13132C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
C2×#21592
#279272
#226122
#135072
#135062
#134772
#134752
#131782
#131772
#131332
#131322
#130622
#129132
#128682