Finite Group Databasebeta

← #357121

Subgroups — #357121

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C212yesmaximal
2C2×#1770C212yesmaximal, complemented
3C2×#1749C212yesmaximal
4C4×#385C211yesmaximal, characteristic
5#10922C211yesmaximal, characteristic
6#10922C211yesmaximal, characteristic
7#10877C212yesmaximal, complemented
8#10875C212yesmaximal, complemented
9#10696C212yesmaximal, complemented
10C23×C42C2211yescharacteristic
11C22×C4⋊C8C2212yes—
12C22×C22⋊C8C2212yes—
13C22×C22⋊C8C2212yes—
14C22×#97C2212yes—
15C22×#97C2212yes—
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×C4×C8C2212yes—
19C2×#385C412yes—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1923C4116yescomplemented
24#1770—28no—
25#1749—28no—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1436C414yes—
35C22⋊#95C412yes—
36C4×#95C2212yes—
37C24×C4C2×C412yes—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4C2×C414yes—
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2×C411yescharacteristic
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C42—24no—
48C22×C42—24no—
49C22×C42—41no—
50C22×C42—41no—
51C22×C42C2×C412yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2312yes—
55C22×C42C2312yes—
56C2×C4⋊C8—24no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8C2×C418yescomplemented
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8C2×C418yes—
64C2×C22⋊C8C2×C418yes—
65C2×#97—24no—
66C2×#97—24no—
67C2×#97—24no—
68C2×#97—24no—
69C2×#97—24no—
70C2×#97—24no—
71C2×#97—42no—
72C2×#97—42no—
73C2×#97—41no—
74C2×#97—41no—
75C2×C4×C8—24no—
76C2×C4×C8—24no—
77C2×C4×C8C2×C418yescomplemented
78#385—28no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C22⋊C4—41no—
82C4×C22⋊C4—41no—
83C4×C22⋊C4C2×C418yes—
84C2×#95C2×C414yes—
85C2×#95C2×C414yes—
86C2×#95C2×C414yes—
87C2×#95C2×C412yes—
88C2×#95C2×C412yes—
89#336—24no—
90#336—24no—
91#336—24no—
92#336—41no—
93#336—41no—
94#336C2×C418yescomplemented
95#336C2×C418yescomplemented
96#336D418yes—
97C25C22×C411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—24no—
106C23×C4—28no—
107C23×C4—44no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4C22×C412yes—
111C23×C4C22×C412yes—
112C23×C4C22×C412yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C414yes—
115C23×C4C22×C414yes—
116C23×C4C22×C412yes—
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8C22×C414yes—
139C22×C8C22×C414yes—
140C22×C8C22×C414yes—
141C22×C8C22×C414yes—
142C22×C8C2×D414yes—
143C22×C8C2×D414yes—
144C22×C8#4114yes—
145C22×C8#4114yes—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—44no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4C4214yes—
154C2×C22⋊C4C4214yes—
155C2×C22⋊C4M1618yes—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—28no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—44no—
174C2×C42—48no—
175C2×C42—44no—
176C2×C42—42no—
177C2×C42C22×C414yes—
178C2×C42C22×C414yes—
179C2×C42C2×D414yes—
180C2×C42#4114yes—
181C4⋊C8—216no—
182C22⋊C8—216no—
183C22⋊C8—216no—
184#97—48no—
185#97—48no—
186#97—44no—
187#97—44no—
188C4×C8—216no—
189#95—28no—
190#95—28no—
191#95—44no—
192#95—42no—
193#95C4218yes—
194C24—22no—
195C24—22no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—24no—
200C24C2×C4212yes—
201C24C2×M1614yes—
202C24C23×C411yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—21no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—48no—
237C22×C4—416no—
238C22×C4—416no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4C2×C4212yes—
252C22×C4C2×C4214yes—
253C22×C4C2×C4214yes—
254C22×C4C4×D418yes—
255C22×C4C2×M1618yes—
256C22×C4C23×C412yes—
257C22×C4C22×D412yes—
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4C2×#4112yes—
261C22×C4H(5,𝔽2)11yescharacteristic
262C22×C4H(5,𝔽2)11yescharacteristic
263C22×C42−1+412yes—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8C4×D418yes—
291C2×C8C4×D418yes—
292C4⋊C4—416no—
293C22⋊C4—44no—
294C22⋊C4—48no—
295C22⋊C4#97116yescomplemented
296C42—24no—
297C42—24no—
298C42—24no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—48no—
305C42—41no—
306C42—42no—
307C42—41no—
308C42—41no—
309C42—42no—
310C42—41no—
311C23—22no—
312C23—24no—
313C23—28no—
314C23—28no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—22no—
319C23—41no—
320C23—42no—
321C23—41no—
322C23—44no—
323C23—44no—
324C23—42no—
325C23—42no—
326C23C2×#9714yes—
327C23C22×C4211yescharacteristic
328C23C2×C4×D412yes—
329C23C4×#4112yes—
330C23#51811yescharacteristic
331C23#51911yescharacteristic
332C23C22×M1612yes—
333C23#56812yes—
334C23#56814yes—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—432no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4C2×#9718yes—
372C2×C4C2×C4×D412yes—
373C2×C4C2×C4×D412yes—
374C2×C4C4×#4112yes—
375C2×C4C4×#4112yes—
376C2×C4#52012yes—
377C2×C4#52012yes—
378C2×C4D4211yescharacteristic
379C2×C4#54612yes—
380C2×C4#54712yes—
381C2×C4#54812yes—
382C2×C4#55111yescharacteristic
383C2×C4#55111yescharacteristic
384C2×C4#55212yes—
385C2×C4#55312yes—
386C2×C4#55612yes—
387C2×C4#55911yescharacteristic
388C8—28no—
389C8—28no—
390C8—44no—
391C8—44no—
392C22—28no—
393C22—24no—
394C22—44no—
395C22—44no—
396C22—48no—
397C22—42no—
398C22—44no—
399C22—42no—
400C22C42×D412yes—
401C22#227411yescharacteristic
402C22#235811yescharacteristic
403C22#236212yes—
404C22#237411yescharacteristic
405C22C22×#9711yescharacteristic
406C22#287311yescharacteristic
407C22#287512yes—
408C22#293014yes—
409C22D4×M1614yes—
410C22#296714yes—
411C22#297014yes—
412C22#297512yes—
413C22#297812yes—
414C22#298312yes—
415C22#298712yes—
416C4—28no—
417C4—44no—
418C4—48no—
419C4—42no—
420C4—82no—
421C4—82no—
422C4#235912yes—
423C4#236112yes—
424C4#236712yes—
425C4#237312yes—
426C2—44no—
427C2#1382311yescharacteristic
428C2#2048712yes—
429C2D4×#9712yes—
430C2#2053611yescharacteristic
431C2#2054211yescharacteristic
432C2#2285212yes—
433C2#2305314yes—
434C2#2330412yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1770C2semidirect
7. #10877C2semidirect
8. #10875C2semidirect
9. #10696C2semidirect
23. #1923C4semidirect
58. C2×C4⋊C8C2×C4semidirect
77. C2×C4×C8C2×C4semidirect
94. #336C2×C4semidirect
95. #336C2×C4semidirect
295. C22⋊C4#97semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#17702
C2×#17492
C4×#3852
#109222
#108772
#108752
#106962