Finite Group Databasebeta

← #215620

Subgroups — #215620

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#9883C214yesmaximal
5#9882C214yesmaximal
6#8701C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C22×C4×Q8C412yes—
9C4×C4⋊Q8—21no—
10C4×C4⋊Q8—21no—
11C4×C4⋊Q8—21no—
12C4×C4⋊Q8—21no—
13C4×C4⋊Q8—21no—
14C4×C4⋊Q8—21no—
15C4×C4⋊Q8—21no—
16C4×C4⋊Q8—21no—
17C2×#391C2212yes—
18C2×#389C412yes—
19C2×#389C2212yes—
20C2×#384C411yescharacteristic
21C2×C4×C4⋊C4C412yes—
22C2×C4×C4⋊C4C2212yes—
23C2×C43C2211yescharacteristic
24#1839—21no—
25#1839—21no—
26#1831—21no—
27#1831—21no—
28#1817C2212yes—
29#1817C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1767—21no—
33#1767—21no—
34#1749C2212yes—
35#1749C2212yes—
36#1745—21no—
37#1745—21no—
38#1280C22116yes—
39C23×Q8C2×C412yes—
40C2×C4⋊Q8—28no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—48no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C414yes—
60C22×C42C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×C22⋊C8C2314yes—
68C2×#97—22no—
69C2×#97—22no—
70#391—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C2×#95C2×C414yes—
95C2×#95C2314yes—
96C43D411yescharacteristic
97C43D411yescharacteristic
98C43D411yescharacteristic
99C43D411yescharacteristic
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43D411yescharacteristic
103C43D411yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—28no—
115C22×Q8—44no—
116C22×Q8—44no—
117C23×C4C22×C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C414yes—
124C23×C4C22×C414yes—
125C23×C4C2411yescharacteristic
126C22×C8—24no—
127C22×C8—24no—
128C4⋊Q8—28no—
129C4⋊Q8—28no—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4×Q8—48no—
133C4×Q8—48no—
134C4×Q8—48no—
135C4×Q8—48no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4M1618yes—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—24no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—48no—
165C2×C42C22⋊C411yescharacteristic
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C4⋊C8—44no—
194C4⋊C8—44no—
195C22⋊C8—28no—
196C22⋊C8—28no—
197#97—44no—
198#97—44no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203C24C23×C411yescharacteristic
204C2×Q8—28no—
205C2×Q8—28no—
206C2×Q8—44no—
207C2×Q8—432no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×M1614yes—
251C22×C4C2×M1618yes—
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C42−1+414yes—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—44no—
261C4⋊C4—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—28no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—48no—
300C42—48no—
301C42#10014yes—
302C42#10114yes—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23C22×C22⋊C411yescharacteristic
308C23#51912yes—
309C23C22×M1612yes—
310C23#56812yes—
311Q8—416no—
312Q8—416no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—88no—
359C2×C4#40712yes—
360C2×C4#40712yes—
361C2×C4#41012yes—
362C2×C4#41012yes—
363C2×C4C2×#10012yes—
364C2×C4C2×#10012yes—
365C2×C4C2×#10012yes—
366C2×C4C2×#10112yes—
367C2×C4C2×#10112yes—
368C2×C4C2×#10112yes—
369C2×C4#53612yes—
370C2×C4#53612yes—
371C8—84no—
372C8—84no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22#231011yescharacteristic
380C22C2×#40711yescharacteristic
381C22#287911yescharacteristic
382C22C2×#41011yescharacteristic
383C22C2×#41011yescharacteristic
384C22C22×#10011yescharacteristic
385C22C22×#10111yescharacteristic
386C22#298314yes—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—48no—
400C4—41no—
401C4—41no—
402C4—84no—
403C2—24no—
404C2#1736411yescharacteristic
405C2#1736511yescharacteristic
406C2#1740311yescharacteristic
407C2#1740411yescharacteristic
408C2#2182211yescharacteristic
409C2#2182611yescharacteristic
410C2#2182911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
#207782
#98832
#98822
#87012