Finite Group Databasebeta

← #6433833

Subgroups — #6433833

Order 512.

1770 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic
3#30303C214yesmaximal, complemented
4#30301C214yesmaximal, complemented
5#30299C218yesmaximal, complemented
6#23658C212yesmaximal, complemented
7#23657C212yesmaximal, complemented
8#23554C212yesmaximal, complemented
9#23553C212yesmaximal, complemented
10#20661C214yesmaximal
11#20335C211yesmaximal, characteristic
12C23×M16C2212yes—
13C2×#533C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#428C2214yes—
17C2×#428C2214yes—
18#2629—24no—
19#2629C2214yes—
20#2629C2214yes—
21#2629C2212yes—
22#2629C2212yes—
23#2628C2214yes—
24#2627—22no—
25#2626—22no—
26#2396—24no—
27C2×#398—22no—
28C2×#396—22no—
29#2296C2212yes—
30C2×#389C2212yes—
31#2285C2211yescharacteristic
32C2×#384C2211yescharacteristic
33#2280C2212yes—
34C2×#380C2212yes—
35#2085C2218yescomplemented
36#2084C2218yescomplemented
37#2084C2214yescomplemented
38#2083C2218yescomplemented
39#2083C2214yescomplemented
40#2059C2218yescomplemented
41#2059C2218yescomplemented
42#2059C2218yescomplemented
43#2058C2214yescomplemented
44#2057C2214yescomplemented
45#1809C2218yes—
46#1809C2218yes—
47#1808C2218yes—
48#1807C2218yes—
49#1736C2214yes—
50#1736C2214yes—
51#1729C2214yes—
52#1729C2214yes—
53#1728C2212yes—
54#1728C2212yes—
55C24×C4C2311yescharacteristic
56C22×M16—44no—
57C22×M16—44no—
58C22×M16Q8116yescomplemented
59C22×M16C2314yes—
60C22×M16C2314yes—
61C23×C8C2314yes—
62#533—44no—
63C2×C4⋊Q8C2312yes—
64C2×#125—22no—
65C2×#125—44no—
66C2×#125C2314yes—
67C2×#125C2312yes—
68C2×C22⋊Q8—22no—
69C2×C4×Q8—22no—
70C2×#117—44no—
71C2×#117D414yes—
72C2×#117C2312yes—
73C2×#117C2312yes—
74C2×#117C2312yes—
75C2×#117C2312yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4D412yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#428—48no—
87#428—48no—
88C2×#107C2318yes—
89C2×#107C2318yes—
90C2×#106C2318yes—
91C2×#106C2318yes—
92#400—24no—
93#399—24no—
94#398—24no—
95#398—24no—
96#398—28no—
97#398—28no—
98#396—24no—
99#396—24no—
100#396—28no—
101#396—28no—
102#395—24no—
103#393—24no—
104#389—22no—
105#389—22no—
106#389C2314yes—
107#389C2314yes—
108#389C2314yes—
109#389C2314yes—
110#385—24no—
111#384—22no—
112#384C2311yescharacteristic
113#384C2314yes—
114#384C2311yescharacteristic
115#382C2312yes—
116#381—22no—
117#380—22no—
118#380—24no—
119#380C2314yes—
120#380C2318yes—
121#380C2314yes—
122#379—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C22⋊C4—24no—
127C2×#95—22no—
128C2×#95—22no—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#340C2318yes—
133#336C2314yes—
134#336C2318yes—
135#336C2314yes—
136C25C2×D411yescharacteristic
137C22×Q8—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q818yes—
148C23×C4C2412yes—
149C23×C4C2411yescharacteristic
150C2×M16—28no—
151C2×M16—28no—
152C2×M16—44no—
153C2×M16—44no—
154C2×M16—432no—
155C2×M16C2×Q8116yes—
156C2×M16C2×Q8116yes—
157C22×C8—44no—
158C22×C8—44no—
159C22×C8C2×Q8116yes—
160C22×C8C2418yes—
161C4⋊Q8—44no—
162#125—44no—
163#125—44no—
164#125—42no—
165#125—42no—
166#125—48no—
167C22⋊Q8—48no—
168C4×Q8—48no—
169#117—28no—
170#117—24no—
171#117—24no—
172#117—48no—
173#117—42no—
174#117—42no—
175#117—42no—
176#117—42no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4—84no—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2412yes—
202C2×C4⋊C4C2412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—84no—
210C2×C22⋊C4C2×D414yes—
211C2×C22⋊C4C2×D414yes—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2412yes—
219C2×C42C2411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—48no—
222#107—48no—
223#106—48no—
224#106—48no—
225#95—28no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233C24—22no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24C4⋊Q814yes—
238C24C22×D412yes—
239C24C22×D411yescharacteristic
240C2×Q8—44no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—216no—
248C22×C4—216no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4C4⋊Q814yes—
271C22×C4C4⋊Q818yes—
272C22×C4C22×D412yes—
273C22×C4C22×D412yes—
274C22×C4C22×D414yes—
275C22×C4C22×D412yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q814yes—
278C22×C4C22×Q814yes—
279C22×C4C22×Q814yes—
280C22×C4H(5,𝔽2)14yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283C22×C4C2511yescharacteristic
284M16—232no—
285M16—416no—
286C2×C8—28no—
287C2×C8—28no—
288C2×C8—416no—
289C2×C8C22×Q8116yes—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—416no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—82no—
307C4⋊C4—84no—
308C4⋊C4—82no—
309C4⋊C4—84no—
310C4⋊C4—88no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4—88no—
320C42—24no—
321C42—24no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—44no—
327C42—44no—
328C23—22no—
329C23—22no—
330C23—24no—
331C23—24no—
332C23—22no—
333C23—22no—
334C23—28no—
335C23—48no—
336C23—84no—
337C23—82no—
338C23—82no—
339C23C2×C4⋊Q814yes—
340C23C2×C4⋊Q812yes—
341C23#53412yes—
342C23#53712yes—
343C23C23×D411yescharacteristic
344Q8—88no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—216no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4—88no—
379C2×C4C2×C4⋊Q814yes—
380C2×C4C2×C4⋊Q812yes—
381C2×C4#53412yes—
382C2×C4#53712yes—
383C2×C4#54318yes—
384C2×C4#54418yes—
385C2×C4C23×Q812yes—
386C2×C4C2×H(5,𝔽2)11yescharacteristic
387C2×C4C2×H(5,𝔽2)12yes—
388C2×C4C2×2−1+411yescharacteristic
389C8—216no—
390C22—24no—
391C22—24no—
392C22—22no—
393C22—22no—
394C22—24no—
395C22—28no—
396C22—44no—
397C22—42no—
398C22—42no—
399C22—88no—
400C22#265914yes—
401C22C22×C4⋊Q811yescharacteristic
402C22C2×#53411yescharacteristic
403C22C2×#53711yescharacteristic
404C22#358512yes—
405C22#358612yes—
406C4—24no—
407C4—44no—
408C4—42no—
409C4—42no—
410C4—84no—
411C4—84no—
412C4—82no—
413C4—84no—
414C4—82no—
415C4#265914yes—
416C4C2×#54312yes—
417C4C2×#54412yes—
418C2—24no—
419C2—44no—
420C2C2×#265911yescharacteristic
421C2#6066412yes—
422C2#6075512yes—
423C2#6092011yescharacteristic
424C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30303C2semidirect
4. #30301C2semidirect
5. #30299C2semidirect
6. #23658C2semidirect
7. #23657C2semidirect
8. #23554C2semidirect
9. #23553C2semidirect
35. #2085C22semidirect
36. #2084C22semidirect
37. #2084C22semidirect
38. #2083C22semidirect
39. #2083C22semidirect
40. #2059C22semidirect
41. #2059C22semidirect
42. #2059C22semidirect
43. #2058C22semidirect
44. #2057C22semidirect
58. C22×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#26292
#303032
#303012
#302992
#236582
#236572
#235542
#235532
#206612
#203352