Finite Group Databasebeta

← #106521

Subgroups — #106521

Order 512.

878 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic
2C2×#1467C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#8030C212yesmaximal
5C2×#1285C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7#2983—24no—
8#2983—22no—
9#2983C414yescomplemented
10C2×#446—24no—
11C2×#432—21no—
12C2×#432C2212yes—
13C2×#432C2212yes—
14C2×#431—21no—
15C2×#391C412yes—
16C2×#382C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1467—22no—
19#1467C414yescomplemented
20C2×#324—21no—
21C2×#324—21no—
22#1441—21no—
23#1441—21no—
24C4⋊#95C412yes—
25#1434—21no—
26#1434—21no—
27#1295—24no—
28#1285—24no—
29C23×Q8C2×C411yescharacteristic
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8C2×C412yes—
35C2×C4×Q8C2×C412yes—
36C2×C4×Q8D412yes—
37C2×C4×Q8Q812yes—
38C22×C4⋊C4—21no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C42C2×C412yes—
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42C2311yescharacteristic
47#446—216no—
48#446—216no—
49#432—24no—
50#432—24no—
51#432—22no—
52#432—22no—
53#432—22no—
54#432—22no—
55#432—22no—
56#432C2×C414yes—
57#432C2×C414yes—
58#432D414yes—
59#431—24no—
60#431—24no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×#97—22no—
68C2×#97—22no—
69C2×#97D414yes—
70C2×C4×C8—22no—
71#391—44no—
72#382—22no—
73#382—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83#324—24no—
84#324—24no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—41no—
90C22×Q8—41no—
91C22×Q8C4212yes—
92C22×Q8C4⋊C412yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—24no—
103C22×C8—42no—
104C4×Q8—24no—
105C4×Q8—24no—
106C4×Q8—24no—
107C4×Q8—24no—
108C4×Q8—24no—
109C4×Q8—24no—
110C4×Q8—24no—
111C4×Q8—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4C4212yes—
124C2×C4⋊C4C22⋊C412yes—
125C2×C4⋊C4C22⋊C412yes—
126C2×C4⋊C4C4⋊C412yes—
127C2×C4⋊C4C4⋊C412yes—
128C2×C4⋊C4C4⋊C412yes—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—24no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42C22⋊C412yes—
141C2×C42C22⋊C412yes—
142C2×C42C4⋊C412yes—
143C2×C42C4⋊C412yes—
144C2×C42C22×C411yescharacteristic
145C2×C42C22×C411yescharacteristic
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D411yescharacteristic
149C2×C42C2×Q811yescharacteristic
150C2×C42#4112yes—
151C2×C42#4112yes—
152C4⋊C8—28no—
153C4⋊C8—28no—
154C4⋊C8—24no—
155C4⋊C8—24no—
156C4⋊C8—24no—
157C4⋊C8—42no—
158C22⋊C8—24no—
159C22⋊C8—24no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162#97—28no—
163#97—24no—
164#97—24no—
165#97—42no—
166#97C22⋊C418yes—
167C4×C8—28no—
168#95—44no—
169#95—44no—
170#95—44no—
171#95—44no—
172#95—41no—
173#95—41no—
174#95—42no—
175#95—41no—
176#95—41no—
177C24C2×C22⋊C411yescharacteristic
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—44no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4#9512yes—
225C22×C4#9512yes—
226C22×C4#9512yes—
227C22×C4#9512yes—
228C22×C4#10012yes—
229C22×C4#10112yes—
230C22×C4C2×C4211yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C411yescharacteristic
236C22×C4C4×D412yes—
237C22×C4C4×D412yes—
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12112yes—
241C22×C4#12112yes—
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C2×C8—28no—
245C2×C8—48no—
246C2×C8—44no—
247C2×C8—84no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—48no—
255C4⋊C4—84no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—84no—
265C42C2×C22⋊C412yes—
266C42#11712yes—
267C42C4×D412yes—
268C42C4×D412yes—
269C42#12012yes—
270C42C22⋊Q812yes—
271C42C22⋊Q812yes—
272C42#12312yes—
273C23—22no—
274C23—22no—
275C23C2×#9511yescharacteristic
276C23#38611yescharacteristic
277C23#38611yescharacteristic
278C23C2×#10011yescharacteristic
279C23C2×#10111yescharacteristic
280C23#41311yescharacteristic
281C23#41311yescharacteristic
282Q8—48no—
283Q8—48no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4#34314yes—
338C2×C4C4×C22⋊C412yes—
339C2×C4#37911yescharacteristic
340C2×C4#38012yes—
341C2×C4#38111yescharacteristic
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38512yes—
345C2×C4#38612yes—
346C8—48no—
347C8—84no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22C22⋊#9511yescharacteristic
357C22C2×#34311yescharacteristic
358C22#174211yescharacteristic
359C22#188412yes—
360C22#190011yescharacteristic
361C22#190011yescharacteristic
362C22#190111yescharacteristic
363C22#190111yescharacteristic
364C22#191212yes—
365C4—24no—
366C4—22no—
367C4—22no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—84no—
378C4—162no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2#944812yes—
384C2#945012yes—
385C2#1073911yescharacteristic
386C2#1074111yescharacteristic
387C2#1075111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2983C4semidirect
19. #1467C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
C2×#14672
#139422
#80302
C2×#12852