Finite Group Databasebeta

← #213531

Subgroups — #213531

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#387C211yesmaximal, characteristic
2#20715C212yesmaximal
3#9086C218yesmaximal, complemented
4#8698C212yesmaximal
5#8299C212yesmaximal
6C23×C4⋊C4C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#432C2214yes—
10C2×#387—22no—
11C2×#387—22no—
12C2×#387—22no—
13C2×#387—22no—
14C2×#387C414yes—
15C2×#387C2214yes—
16C22×#95C411yescharacteristic
17C22×#95C411yescharacteristic
18C22×#95C411yescharacteristic
19C22×#95C2211yescharacteristic
20#1817C2214yes—
21#1797—22no—
22#1797—22no—
23#1779C2214yes—
24#1749—22no—
25#1749—22no—
26C2×#324C2218yes—
27#1280C2218yes—
28C24×C4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—24no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—44no—
43C22×C4⋊C4C2×C414yes—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2312yes—
46C22×C42—44no—
47C22×C42C2×C412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42C2312yes—
53#432—48no—
54C2×C4⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8C2318yes—
58C2×#97—24no—
59#387—24no—
60#387—24no—
61#387—24no—
62#387—24no—
63#387—24no—
64#387—24no—
65#387—24no—
66#387—24no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95C2×C414yes—
79C2×#95C2×C414yes—
80C2×#95C2×C414yes—
81C2×#95C2×C412yes—
82C2×#95C2312yes—
83#336—24no—
84#336—24no—
85#324—48no—
86C25C22×C411yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22⋊C411yescharacteristic
123C23×C4M1614yes—
124C23×C4M1614yes—
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2411yescharacteristic
135C22×C8—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—216no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—48no—
149C2×C42—28no—
150C2×C42—44no—
151C2×C42—44no—
152C2×C42—44no—
153C2×C42—44no—
154C2×C42—44no—
155C2×C42C22⋊C412yes—
156C2×C42C22⋊C412yes—
157C2×C42C22⋊C412yes—
158C2×C42C22⋊C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C4⋊C8—48no—
164C22⋊C8—216no—
165C22⋊C8—48no—
166C22⋊C8—48no—
167#97—48no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95—24no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—22no—
180C24—22no—
181C24—22no—
182C24C2×C22⋊C411yescharacteristic
183C24C2×C22⋊C411yescharacteristic
184C24C2×M1612yes—
185C24C2×M1612yes—
186C24C23×C411yescharacteristic
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—48no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—416no—
240C22×C4—416no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—48no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—88no—
271C22×C4—88no—
272C22×C4#10114yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×M1614yes—
279C22×C4C2×M1614yes—
280C22×C4C22×D412yes—
281C22×C4H(5,𝔽2)14yes—
282C2×C8—48no—
283C2×C8—48no—
284C2×C8—48no—
285C4⋊C4—216no—
286C4⋊C4—216no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C42—24no—
290C42—24no—
291C42—24no—
292C42—24no—
293C42—42no—
294C42—42no—
295C42—42no—
296C42—42no—
297C23—28no—
298C23—28no—
299C23—28no—
300C23—28no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23#40712yes—
320C23#40712yes—
321C23#41012yes—
322C23#41012yes—
323C23C2×#10112yes—
324C23#41312yes—
325C23C22×C22⋊C411yescharacteristic
326C23#51812yes—
327C23C22×M1611yescharacteristic
328C23C22×M1611yescharacteristic
329C23#56812yes—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4C2×#10114yes—
386C2×C4#53414yes—
387C8—88no—
388C22—216no—
389C22—216no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22#146414yes—
411C22#146714yes—
412C22#230811yescharacteristic
413C22C2×#40711yescharacteristic
414C22#287911yescharacteristic
415C22C2×#41011yescharacteristic
416C22C2×#41011yescharacteristic
417C22C22×#10111yescharacteristic
418C22C2×#41311yescharacteristic
419C22#297312yes—
420C22#297512yes—
421C4—24no—
422C4—24no—
423C4—44no—
424C4—42no—
425C4—42no—
426C4—44no—
427C4—42no—
428C4—42no—
429C4—82no—
430C4—82no—
431C2—22no—
432C2—22no—
433C2—24no—
434C2—24no—
435C2C2×#146411yescharacteristic
436C2C2×#146711yescharacteristic
437C2#1739411yescharacteristic
438C2#1739811yescharacteristic
439C2#2174211yescharacteristic
440C2#2174711yescharacteristic
441C2#2175311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9086C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3872
#207152
#90862
#86982
#82992