Finite Group Databasebeta

← #146721

Subgroups — #146721

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C22×#356C212yesmaximal
3C2×#1394C212yesmaximal
4#10763C212yesmaximal
5C22×C4⋊Q8C2211yescharacteristic
6C22×C4×Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×#101C2214yes—
9C4×C4⋊Q8—24no—
10C4×C4⋊Q8—24no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#389—22no—
14C2×#384C411yescharacteristic
15C2×C4×C4⋊C4—22no—
16C2×C43C411yescharacteristic
17C2×#356—24no—
18C2×#356—24no—
19C2×#356C418yescomplemented
20C2×#343—22no—
21#1728—22no—
22#1394—24no—
23C23×Q8C2×C411yescharacteristic
24C23×Q8C2311yescharacteristic
25C22×M16—24no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8C2×C414yes—
30C2×C4⋊Q8D414yes—
31C2×C4⋊Q8D412yes—
32C2×C4⋊Q8Q812yes—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—42no—
38C2×C4×Q8—42no—
39C2×C4×Q8—42no—
40C2×C4×Q8—42no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C42—21no—
48C22×C42—21no—
49C22×C42—21no—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D412yes—
54C2×#101—24no—
55C2×#101—24no—
56C2×#101—24no—
57C2×#101—24no—
58C2×#101—24no—
59C2×#101C2×C418yes—
60#391—22no—
61#391—22no—
62#391—22no—
63#391D414yes—
64#389—216no—
65#384—22no—
66#384—22no—
67C4×C4⋊C4—216no—
68C2×#95—22no—
69C2×#95—22no—
70C43—22no—
71C43—22no—
72#356—216no—
73#343—44no—
74#336—44no—
75C22×Q8—21no—
76C22×Q8—21no—
77C22×Q8—21no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—41no—
86C22×Q8C4212yes—
87C22×Q8C22⋊C412yes—
88C22×Q8C4⋊C412yes—
89C22×Q8C22×C412yes—
90C22×Q8C2×D412yes—
91C22×Q8C2×D412yes—
92C22×Q8C2×D411yescharacteristic
93C22×Q8C2×Q811yescharacteristic
94C22×Q8#4112yes—
95C22×Q8#4112yes—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C2×M16—28no—
106C2×M16—44no—
107C2×M16—44no—
108C2×M16—44no—
109C2×M16—44no—
110C2×M16—44no—
111C22×C8—44no—
112C4⋊Q8—28no—
113C4⋊Q8—24no—
114C4⋊Q8—24no—
115C4⋊Q8—28no—
116C4⋊Q8C22⋊C418yes—
117C4⋊Q8C4⋊C418yes—
118C4×Q8—48no—
119C4×Q8—48no—
120C4×Q8—48no—
121C4×Q8—48no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C42—24no—
143C2×C42—28no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—21no—
155C2×C42—24no—
156C2×C42—42no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—42no—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42C4212yes—
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C4⋊C412yes—
166#101—28no—
167#101—28no—
168#101—44no—
169#101—44no—
170#95—28no—
171#95—28no—
172C24C2×C22⋊C411yescharacteristic
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8#9514yes—
203C2×Q8C2×C22⋊C412yes—
204C2×Q8C2×C4⋊C412yes—
205C2×Q8C4×D414yes—
206C2×Q8#12112yes—
207C2×Q8C22⋊Q812yes—
208C2×Q8#12314yes—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4#9912yes—
249C22×C4#9912yes—
250C22×C4C2×C4211yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C22×C4#11711yescharacteristic
255C22×C4#11711yescharacteristic
256C22×C4C4×D414yes—
257C22×C4#12014yes—
258C22×C4#12112yes—
259C22×C4C22⋊Q812yes—
260M16—48no—
261M16—48no—
262M16—84no—
263M16—84no—
264C2×C8—48no—
265C2×C8—84no—
266C2×C8—84no—
267C4⋊C4—216no—
268C4⋊C4—216no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—416no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—48no—
276C42—216no—
277C42—24no—
278C42—24no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—48no—
283C42—42no—
284C42—48no—
285C42—42no—
286C42—48no—
287C42#9514yes—
288C23—22no—
289C23—22no—
290C23C4×C22⋊C411yescharacteristic
291C23#37911yescharacteristic
292C23#38011yescharacteristic
293C23C2×#9911yescharacteristic
294C23C2×#9911yescharacteristic
295C23#41011yescharacteristic
296C23#41011yescharacteristic
297Q8—42no—
298Q8—42no—
299Q8—44no—
300Q8—42no—
301Q8—42no—
302Q8—44no—
303Q8—44no—
304Q8—44no—
305Q8—44no—
306Q8—44no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#35614yescomplemented
361C2×C4C2×#9511yescharacteristic
362C2×C4C4×C22⋊C411yescharacteristic
363C2×C4#38011yescharacteristic
364C2×C4#38111yescharacteristic
365C2×C4#38514yes—
366C2×C4#38614yes—
367C8—88no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22C22⋊#9511yescharacteristic
377C22C4×#9911yescharacteristic
378C22C4×#9911yescharacteristic
379C22#178111yescharacteristic
380C22#178211yescharacteristic
381C22#179911yescharacteristic
382C22#179911yescharacteristic
383C22C2×#35612yes—
384C22#213212yes—
385C4—28no—
386C4—24no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—24no—
392C4—24no—
393C4—44no—
394C4—44no—
395C4—42no—
396C4—82no—
397C4—82no—
398C2—21no—
399C2—21no—
400C2—21no—
401C2—21no—
402C2#1076111yescharacteristic
403C2C4×#35612yes—
404C2#1270011yescharacteristic
405C2#1270111yescharacteristic
406C2#1280212yes—
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. C2×#356C4semidirect
360. C2×C4#356semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C22×#3562
C2×#13942
#107632