Finite Group Databasebeta

← #97546

Subgroups — #97546

Order 512.

793 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic
2#11207C212yesmaximal
3#11050C211yesmaximal, characteristic
4C2×#1272C211yesmaximal, characteristic
5C2×#1271C212yesmaximal
6C22×#103—21no—
7#2809—22no—
8#2809—24no—
9#2809C414yescomplemented
10C2×#400—21no—
11C2×#398—22no—
12C2×#391—21no—
13C2×#391C2211yescharacteristic
14C2×#389C2212yes—
15C2×#387—22no—
16C2×#384C2211yescharacteristic
17C2×#340—21no—
18C2×#340—22no—
19C2×#340—22no—
20C2×#336C2211yescharacteristic
21C2×#336C2212yes—
22#1272—22no—
23#1271—24no—
24C23×Q8D411yescharacteristic
25C23×C8—21no—
26C23×C8—22no—
27C22×C4⋊C4—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4D412yes—
33C22×C4⋊C4D411yescharacteristic
34C22×C4⋊C4D412yes—
35C22×C42—21no—
36C22×C42C2311yescharacteristic
37C2×#103—42no—
38C2×#103—42no—
39C2×#103—42no—
40#400—24no—
41#400—24no—
42#398—28no—
43#398—28no—
44#391—24no—
45#391—24no—
46#391—22no—
47#391C2×C412yes—
48#391D412yes—
49#389—22no—
50#389—22no—
51#389C2×C414yes—
52#389D414yes—
53#387—28no—
54#387—28no—
55#384—42no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61#340—44no—
62#340—44no—
63#340—42no—
64#336—22no—
65#336—22no—
66#336—22no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—21no—
70C22×Q8—41no—
71C22×Q8—41no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8C22⋊C412yes—
75C22×Q8SD1612yes—
76C22×Q8Q1612yes—
77C23×C4—21no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—41no—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C22×C8—24no—
85C22×C8—22no—
86C22×C8—42no—
87C22×C8—41no—
88C22×C8—42no—
89C22×C8—41no—
90C22×C8—42no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—42no—
94C22×C8—41no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—41no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—81no—
130C2×C4⋊C4—81no—
131C2×C4⋊C4—81no—
132C2×C4⋊C4—81no—
133C2×C4⋊C4C22⋊C414yes—
134C2×C4⋊C4C22⋊C414yes—
135C2×C4⋊C4C22⋊C412yes—
136C2×C42—24no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42—81no—
140C2×C42—81no—
141C2×C42C22×C411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D412yes—
144#103—48no—
145#95—24no—
146#95—28no—
147#95—28no—
148#95—24no—
149#95—24no—
150#95—24no—
151#95—24no—
152C24#12011yescharacteristic
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—81no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×Q8—81no—
168C2×Q8#10314yes—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4—82no—
206C22×C4—81no—
207C22×C4—82no—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4#12011yescharacteristic
211C22×C4#12012yes—
212C22×C4C2×SD1611yescharacteristic
213C22×C4C2×Q1611yescharacteristic
214C22×C4#13512yes—
215C22×C4#13512yes—
216C22×C4AGL(1,ℤ/8)12yes—
217C22×C4#13712yes—
218C22×C4#13711yescharacteristic
219C22×C4#13711yescharacteristic
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—42no—
226C2×C8—82no—
227C2×C8—81no—
228C2×C8—81no—
229C2×C8—82no—
230C2×C8—82no—
231C2×C8—81no—
232C2×C8—81no—
233C2×C8—82no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—81no—
249C4⋊C4—82no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C42—41no—
257C42—42no—
258C42—41no—
259C42—84no—
260C23#37911yescharacteristic
261C23#44811yescharacteristic
262C23#44912yes—
263C23#45111yescharacteristic
264C23#45212yes—
265C23#45311yescharacteristic
266C23#45412yes—
267C23#45511yescharacteristic
268C23#45612yes—
269C23B(4,𝔽2)11yescharacteristic
270C23#45811yescharacteristic
271Q8—44no—
272Q8—82no—
273Q8—82no—
274Q8—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—84no—
327C2×C4—161no—
328C2×C4—162no—
329C2×C4—161no—
330C2×C4C2×#10311yescharacteristic
331C2×C4#41612yes—
332C2×C4#41712yes—
333C2×C4#41911yescharacteristic
334C2×C4#44811yescharacteristic
335C2×C4#44912yes—
336C2×C4#45111yescharacteristic
337C2×C4#45212yes—
338C8—82no—
339C8—84no—
340C22#159611yescharacteristic
341C22#159712yes—
342C22#160211yescharacteristic
343C22#160312yes—
344C22#160811yescharacteristic
345C22#160912yes—
346C22#161411yescharacteristic
347C22#161512yes—
348C22#162411yescharacteristic
349C22#162612yes—
350C22#163411yescharacteristic
351C22#163612yes—
352C22#164612yes—
353C22#164812yes—
354C22#165312yes—
355C22#165711yescharacteristic
356C22#166011yescharacteristic
357C22#166412yes—
358C22#178711yescharacteristic
359C22#178812yes—
360C22#179011yescharacteristic
361C22#179211yescharacteristic
362C22#179312yes—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—22no—
368C4—22no—
369C4—22no—
370C4—22no—
371C4—44no—
372C4—84no—
373C4—82no—
374C4—84no—
375C4—162no—
376C2#774311yescharacteristic
377C2#776011yescharacteristic
378C2#777112yes—
379C2#780211yescharacteristic
380C2#781312yes—
381C2#783111yescharacteristic
382C2#840511yescharacteristic
383C2#840712yes—
384C2#844812yes—
385C2#845211yescharacteristic
386C2#850711yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2809C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
#112072
#110502
C2×#12722
C2×#12712