Finite Group Databasebeta

← #4416935

Subgroups — #4416935

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic
2C2×#2859C211yesmaximal, characteristic
3#17067C212yesmaximal
4#17061C212yesmaximal
5#16992C212yesmaximal
6#16990C212yesmaximal
7#16981C214yesmaximal
8#15211C212yesmaximal
9#15207C212yesmaximal
10#14929C214yesmaximal
11#14818C214yesmaximal
12#14677C212yesmaximal
13#14677C212yesmaximal
14C4×#382C211yesmaximal, characteristic
15C22×#125C2212yes—
16#2864—28no—
17#2859—28no—
18C2×#398C2214yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2218yes—
28C2×#387C2212yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#382C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C43C2211yescharacteristic
40#1445C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2218yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2218yes—
54#1441C2218yes—
55#1441C2218yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1436C2218yes—
60C4⋊#95C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63C4×#95C2212yes—
64C4×#95C2212yes—
65C2×#125—28no—
66C2×#125—48no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2318yes—
84C22×C4⋊C4C2318yes—
85C22×C4⋊C4C2314yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2314yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#398—48no—
95#389—28no—
96#389—28no—
97#389—28no—
98#389—28no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—416no—
104#387—28no—
105#387—28no—
106#387—28no—
107#387—28no—
108#384—24no—
109#384—24no—
110#382—28no—
111#382—24no—
112#382—24no—
113#382Q8116yes—
114C4×C4⋊C4—44no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2318yes—
128C2×#95C2314yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C43D418yes—
137C23×C4C2412yes—
138C23×C4C2412yes—
139C23×C4C2414yes—
140C23×C4C2414yes—
141C23×C4C2412yes—
142C23×C4C2412yes—
143C23×C4C2411yescharacteristic
144C23×C4C2418yes—
145C23×C4C2411yescharacteristic
146C23×C4C2411yescharacteristic
147C23×C4C2411yescharacteristic
148C23×C4C2411yescharacteristic
149C23×C4C2411yescharacteristic
150C23×C4C2411yescharacteristic
151#125—432no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—216no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—48no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—48no—
194C2×C4⋊C4C2×Q8116yes—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—44no—
204C2×C42—44no—
205C2×C42C2×D414yes—
206C2×C42C2×D414yes—
207C2×C42C2×D414yes—
208C2×C42C2×D414yes—
209C2×C42C2×D414yes—
210C2×C42C2×D414yes—
211C2×C42C2×D414yes—
212C2×C42C2×Q818yes—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—28no—
218#95—28no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—28no—
224#95—28no—
225#95—44no—
226#95—48no—
227#95—48no—
228#95—48no—
229#95—48no—
230#95—48no—
231#95—48no—
232#95C2×Q8116yes—
233#95C2×Q8116yes—
234C24C2511yescharacteristic
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—28no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×Q8116yes—
294C22×C4C22×Q814yes—
295C22×C4C22×Q814yes—
296C22×C4C22×Q814yes—
297C22×C4H(5,𝔽2)14yes—
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+414yes—
300C22×C42−1+412yes—
301C4⋊C4—416no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—416no—
313C4⋊C4—432no—
314C4⋊C4—48no—
315C4⋊C4—416no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—816no—
320C42—28no—
321C42—28no—
322C42—28no—
323C42—44no—
324C42—44no—
325C42—44no—
326C42—88no—
327C42C4⋊Q8116yes—
328C23C23×D411yescharacteristic
329C23C23×Q812yes—
330C23C2×H(5,𝔽2)12yes—
331C23C2×H(5,𝔽2)11yescharacteristic
332C23C2×2−1+412yes—
333C23C2×2−1+411yescharacteristic
334C23#58512yes—
335C23#58512yes—
336C23#58512yes—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—216no—
362C2×C4—216no—
363C2×C4—216no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—48no—
396C2×C4—44no—
397C2×C4C2×C4⋊Q814yes—
398C2×C4C2×C4⋊Q814yes—
399C2×C4C2×C4⋊Q814yes—
400C2×C4#53514yes—
401C2×C4#53614yes—
402C2×C4#53714yes—
403C22C22×C4⋊Q811yescharacteristic
404C22C2×#53511yescharacteristic
405C22C2×#53611yescharacteristic
406C22C2×#53711yescharacteristic
407C22#344911yescharacteristic
408C22#344911yescharacteristic
409C22#344911yescharacteristic
410C22#350112yes—
411C22#350212yes—
412C22#350314yes—
413C22#350414yes—
414C22#352914yes—
415C22#355314yes—
416C22#355812yes—
417C22#355912yes—
418C22#355912yes—
419C22#356312yes—
420C4—24no—
421C4—24no—
422C4—24no—
423C4—24no—
424C4—24no—
425C4—24no—
426C4—42no—
427C4—416no—
428C4—84no—
429C4—82no—
430C4—82no—
431C4—84no—
432C4—82no—
433C4—82no—
434C2#3517311yescharacteristic
435C2#3528111yescharacteristic
436C2#3533511yescharacteristic
437C2#3668711yescharacteristic
438C2#3754911yescharacteristic
439C2#3755411yescharacteristic
440C2#3760711yescharacteristic
441C2#4600814yes—
442C2#4600914yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#28592
#170672
#170612
#169922
#169902
#169812
#152112
#152072
#149292
#148182
#146772
C4×#3822