Finite Group Databasebeta

← #7542935

Subgroups — #7542935

Order 512.

1727 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#32270C214yesmaximal, complemented
4#29206C212yesmaximal, complemented
5#26303C214yesmaximal, complemented
6#25347C214yesmaximal, complemented
7#25012C214yesmaximal, complemented
8#24104C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic, complemented
10#22663C214yesmaximal, complemented
11C4×#474C212yesmaximal, complemented
12C2×#559—21no—
13C2×#555C2212yes—
14C2×#481C2214yes—
15C2×#474C2212yes—
16C2×#436C2214yes—
17C2×#428C2214yes—
18#2868C2211yescharacteristic
19#2856—21no—
20#2724—22no—
21#2582—21no—
22#2563C2212yes—
23#2547C2214yes—
24#2537C2214yes—
25#2500—22no—
26#2491—24no—
27#2486C2214yes—
28#2447—24no—
29#2354C2214yes—
30D4×C4⋊C4C2214yes—
31C4×C4⋊Q8C2212yes—
32C4×#127C2211yescharacteristic
33#2285—21no—
34C42×D4C2211yescharacteristic
35#2077C2218yes—
36#2076C2218yes—
37#2058C2218yes—
38#2034C2218yes—
39#1990C2214yes—
40#1988C2212yes—
41#1954C2214yes—
42#1948C2214yes—
43#1948C2214yes—
44#1947C2212yes—
45#1918C2214yes—
46#1895C2214yes—
47#1870C2214yes—
48#1862C2214yes—
49#1854C2218yes—
50#1853C2218yes—
51#1844C2214yes—
52#1825C2214yes—
53#1809C2218yes—
54#1802C2212yes—
55C4×#107C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#102C2214yes—
58C22×M16C2314yes—
59#559—48no—
60#555—44no—
61#555—44no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#127D411yescharacteristic
66C2×#124—24no—
67C2×#124—24no—
68C2×#123—24no—
69C2×#121—22no—
70C2×#121—24no—
71C2×#121C2314yes—
72C2×C4×Q8—22no—
73C2×C4×D4—24no—
74C2×C4×D4C2312yes—
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2314yes—
77C2×C4×D4C2312yes—
78C2×#117—22no—
79C2×#117C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C42—21no—
82#481—48no—
83#474—28no—
84#436—216no—
85#428—48no—
86C2×#107C2314yes—
87C2×#107C2314yes—
88C2×#106C2318yes—
89C2×#106C2318yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#102C2314yes—
93C2×#102C2314yes—
94C2×C22⋊C8C2318yes—
95C2×C22⋊C8C2314yes—
96C2×C4×C8C2314yes—
97#398—24no—
98#397—24no—
99#397—28no—
100#394—28no—
101#393—24no—
102#391—24no—
103#391—22no—
104#390—22no—
105#390—24no—
106#390D414yes—
107#390D412yes—
108#389—24no—
109#389—22no—
110#389C2314yes—
111#388—28no—
112#385—28no—
113#384—21no—
114#384—21no—
115#384D411yescharacteristic
116#384C2312yes—
117#384C2314yes—
118#382—22no—
119#381—24no—
120#380—24no—
121#380C2318yes—
122#380C2318yes—
123#380C2314yes—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C43C2311yescharacteristic
132#340C2314yes—
133#340C2318yes—
134#340C2318yes—
135#340C2314yes—
136#340C2314yes—
137#340C2314yes—
138#336C2314yes—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×D4—24no—
142C22×D4C2×D414yes—
143C22×D4C2×D412yes—
144C22×D4C2411yescharacteristic
145C23×C4—21no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4C2414yes—
149C2×M16—216no—
150C2×M16—44no—
151C22×C8C2414yes—
152C22×C8C2414yes—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—84no—
158#127—24no—
159#124—416no—
160#124—416no—
161#123—416no—
162#121—28no—
163#121—44no—
164#121—48no—
165#121—88no—
166C4×Q8—48no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—28no—
170C4×D4—24no—
171C4×D4—28no—
172C4×D4—24no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—416no—
176#117—28no—
177#117—44no—
178#117—48no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2412yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2411yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4C2×D418yes—
204C2×C22⋊C4C2×D414yes—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—44no—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214C2×C42C2412yes—
215#107—28no—
216#107—28no—
217#106—48no—
218#106—48no—
219C4⋊C8—24no—
220C4⋊C8#4118yes—
221#102—28no—
222#102—28no—
223C22⋊C8—216no—
224C22⋊C8—44no—
225C4×C8—28no—
226#95—22no—
227#95—21no—
228#95—28no—
229#95—24no—
230#95—24no—
231#95—28no—
232#95—24no—
233C24—21no—
234C24C22×D414yes—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—84no—
240C2×D4—28no—
241C2×D4—28no—
242C2×D4—24no—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—48no—
250C2×D4—88no—
251C2×D4H(5,𝔽2)12yes—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4C22×D414yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D412yes—
275C22×C4C2511yescharacteristic
276M16—416no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8C2×#4118yes—
283C4⋊C4—28no—
284C4⋊C4—21no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4H(5,𝔽2)12yes—
303C22⋊C4—24no—
304C22⋊C4—216no—
305C22⋊C4—44no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—416no—
309C22⋊C4—48no—
310C22⋊C4—44no—
311C22⋊C4—88no—
312C42—24no—
313C42—21no—
314C42—21no—
315C42—41no—
316C42—48no—
317C42—44no—
318C42—42no—
319C42—48no—
320C42AGL(1,ℤ/8)14yes—
321C42C2×#4114yes—
322C422−1+414yes—
323C23—24no—
324C23—28no—
325C23—22no—
326C23—24no—
327C23—24no—
328C23—28no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—42no—
333C23C23×D411yescharacteristic
334Q8—88no—
335Q8—164no—
336D4—24no—
337D4—416no—
338D4—41no—
339D4—41no—
340D4—84no—
341D4—168no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—88no—
369C2×C4—168no—
370C2×C4#54714yes—
371C2×C4#54714yes—
372C2×C4C2×AGL(1,ℤ/8)12yes—
373C2×C4C2×AGL(1,ℤ/8)14yes—
374C2×C4#57714yes—
375C2×C4C22×#4112yes—
376C2×C4C2×H(5,𝔽2)11yescharacteristic
377C2×C4C2×H(5,𝔽2)11yescharacteristic
378C2×C4C2×2−1+412yes—
379C8—28no—
380C8—44no—
381C22—48no—
382C22—48no—
383C22—42no—
384C22—44no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22—44no—
389C22—82no—
390C22—82no—
391C22C2×#54712yes—
392C22#348411yescharacteristic
393C22C22×AGL(1,ℤ/8)11yescharacteristic
394C22C2×#57712yes—
395C22#358611yescharacteristic
396C4—24no—
397C4—22no—
398C4—42no—
399C4—44no—
400C4—41no—
401C4—82no—
402C4—84no—
403C4—84no—
404C4—84no—
405C4#335418yes—
406C4#348111yescharacteristic
407C4#352212yes—
408C4#355011yescharacteristic
409C2—84no—
410C2—82no—
411C2#5907011yescharacteristic
412C2C2×#335412yes—
413C2#6128112yes—
414C2#6158511yescharacteristic
415C2#6166211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
3. #32270C2semidirect
4. #29206C2semidirect
5. #26303C2semidirect
6. #25347C2semidirect
7. #25012C2semidirect
8. #24104C2semidirect
9. #23176C2semidirect
10. #22663C2semidirect
11. C4×#474C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#361872
#322702
#292062
#263032
#253472
#250122
#241042
#231762
#226632
C4×#4742