Finite Group Databasebeta

← #6132623

Subgroups — #6132623

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2C2×#2862C211yesmaximal, characteristic
3#17080C212yesmaximal
4#17079C212yesmaximal
5#16993C212yesmaximal
6#16964C212yesmaximal
7#16891C212yesmaximal
8#16705C214yesmaximal
9#16696C214yesmaximal
10#16695C212yesmaximal
11#16565C214yesmaximal
12#16560C212yesmaximal
13#16507C212yesmaximal
14#14689C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17#2863—28no—
18#2862—28no—
19C2×#400C2214yes—
20C2×#398C2214yes—
21C2×#395C2214yes—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#383C2212yes—
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×C43C2211yescharacteristic
40#1446C2214yes—
41#1446C2214yes—
42#1446C2214yes—
43#1446C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2218yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1443C2218yes—
51#1443C2218yes—
52#1442C2214yes—
53#1442C2218yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2218yes—
57#1441C2218yes—
58#1441C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61#1434C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65C23×Q8C2312yes—
66C23×Q8C2312yes—
67C23×Q8C2311yescharacteristic
68C2×C4⋊Q8—24no—
69C2×C4⋊Q8—24no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8—44no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2318yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#400—48no—
92#398—48no—
93#395—48no—
94#391—28no—
95#391—28no—
96#391—28no—
97#391—28no—
98#391—24no—
99#391—24no—
100#391—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#387—28no—
105#387—28no—
106#387—28no—
107#383—28no—
108#382—28no—
109#382—28no—
110#382—44no—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2312yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C43D418yes—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—28no—
144C22×Q8—24no—
145C22×Q8—44no—
146C22×Q8—44no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—44no—
150C22×Q8—44no—
151C23×C4C2412yes—
152C23×C4C2414yes—
153C23×C4C2412yes—
154C23×C4C2412yes—
155C23×C4C2414yes—
156C23×C4C2418yes—
157C23×C4C2412yes—
158C23×C4C2411yescharacteristic
159C23×C4C2411yescharacteristic
160C23×C4C2411yescharacteristic
161C23×C4C2411yescharacteristic
162C23×C4C2411yescharacteristic
163C23×C4C2411yescharacteristic
164C23×C4C2411yescharacteristic
165C4⋊Q8—416no—
166C4⋊Q8—416no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—48no—
193C2×C4⋊C4—48no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42C2×D414yes—
203C2×C42C2×D414yes—
204C2×C42C2×D414yes—
205C2×C42C2×D414yes—
206C2×C42C2×D414yes—
207C2×C42C2×D414yes—
208C2×C42C2×D414yes—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—44no—
227#95—48no—
228#95—48no—
229#95—48no—
230#95—44no—
231#95—48no—
232#95—48no—
233#95—48no—
234#95—48no—
235C24C2511yescharacteristic
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—44no—
243C2×Q8—48no—
244C2×Q8—48no—
245C2×Q8—88no—
246C2×Q8—816no—
247C2×Q8—88no—
248C2×Q8—88no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C22×D412yes—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4H(5,𝔽2)14yes—
309C22×C4H(5,𝔽2)12yes—
310C22×C42−1+414yes—
311C22×C42−1+414yes—
312C22×C42−1+414yes—
313C22×C42−1+412yes—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—88no—
325C4⋊C4—816no—
326C42—28no—
327C42—28no—
328C42—28no—
329C42—28no—
330C42—28no—
331C42—44no—
332C42—44no—
333C23C23×D411yescharacteristic
334C23C2×H(5,𝔽2)12yes—
335C23C2×H(5,𝔽2)11yescharacteristic
336C23C2×2−1+412yes—
337C23C2×2−1+412yes—
338C23C2×2−1+412yes—
339C23C2×2−1+411yescharacteristic
340C23#58512yes—
341C23#58512yes—
342Q8—816no—
343Q8—816no—
344Q8—88no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—416no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4#53514yes—
406C2×C4#53614yes—
407C2×C4#53614yes—
408C2×C4#53614yes—
409C2×C4#53714yes—
410C22C2×#53511yescharacteristic
411C22C2×#53611yescharacteristic
412C22C2×#53611yescharacteristic
413C22C2×#53611yescharacteristic
414C22C2×#53711yescharacteristic
415C22#344911yescharacteristic
416C22#344911yescharacteristic
417C22#354414yes—
418C22#355012yes—
419C22#355514yes—
420C22#355612yes—
421C22#355812yes—
422C22#356112yes—
423C22#356114yes—
424C22#356112yes—
425C22#356312yes—
426C22#356414yes—
427C4—24no—
428C4—24no—
429C4—24no—
430C4—24no—
431C4—24no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—88no—
436C4—84no—
437C4—82no—
438C4—84no—
439C4—82no—
440C4—82no—
441C2#3753711yescharacteristic
442C2#3754611yescharacteristic
443C2#3754811yescharacteristic
444C2#3755511yescharacteristic
445C2#3763211yescharacteristic
446C2#3763511yescharacteristic
447C2#3764111yescharacteristic
448C2#5335814yes—
449C2#5336614yes—
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×#28632
C2×#28622
#170802
#170792
#169932
#169642
#168912
#167052
#166962
#166952
#165652
#165602
#165072
#146892