Finite Group Databasebeta

← #106070

Subgroups — #106070

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1533C212yesmaximal
3#8040C212yesmaximal, complemented
4#7910C212yesmaximal, complemented
5C22×#124C2211yescharacteristic
6C22×C4×Q8C411yescharacteristic
7C22×C4×D4C411yescharacteristic
8C2×#432C2214yes—
9#2326—24no—
10#2326—24no—
11C2×#391C411yescharacteristic
12C2×#390C411yescharacteristic
13C2×#388—22no—
14C2×#385—22no—
15C2×#384C2211yescharacteristic
16C2×#376C2211yescharacteristic
17#1533—24no—
18#1533C418yescomplemented
19C2×#329—24no—
20#1296—24no—
21#1280—22no—
22#1280—22no—
23C23×Q8C2×C411yescharacteristic
24C23×D4C2×C411yescharacteristic
25C24×C4—21no—
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124C2×C414yes—
30C2×#124D412yes—
31C2×#124D414yes—
32C2×#124Q812yes—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—42no—
36C2×C4×Q8—42no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—42no—
40C2×C4×D4—42no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
48C22×C42—21no—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52#432—24no—
53#432—24no—
54#432—24no—
55#432C2×C418yescomplemented
56C2×C4⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×#97—24no—
59#391—22no—
60#391—22no—
61#390—22no—
62#390—22no—
63#388—216no—
64#385—216no—
65#384—22no—
66#384D414yes—
67#376—22no—
68#376—22no—
69C2×#95—22no—
70C2×#95D412yes—
71C2×#95D412yes—
72#329—216no—
73C25—21no—
74C22×Q8—21no—
75C22×Q8—21no—
76C22×Q8—21no—
77C22×Q8—41no—
78C22×Q8—41no—
79C22×Q8—41no—
80C22×Q8—41no—
81C22×Q8C4212yes—
82C22×Q8C22⋊C412yes—
83C22×Q8C4⋊C412yes—
84C22×D4—21no—
85C22×D4—21no—
86C22×D4—21no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4C4212yes—
92C22×D4C22⋊C412yes—
93C22×D4C4⋊C412yes—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—24no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—84no—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C411yescharacteristic
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D412yes—
106C22×C8—44no—
107#124—28no—
108#124—24no—
109#124—24no—
110#124—28no—
111#124C22⋊C418yes—
112#124C4⋊C418yes—
113C4×Q8—48no—
114C4×Q8—48no—
115C4×D4—48no—
116C4×D4—48no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4C22⋊C412yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—28no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—84no—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—24no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—41no—
147C2×C42—41no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42C22×C412yes—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×Q811yescharacteristic
155C2×C42#4112yes—
156C2×C42#4112yes—
157C4⋊C8—28no—
158C4⋊C8—44no—
159C22⋊C8—28no—
160C22⋊C8—44no—
161#97—28no—
162#97—44no—
163#95—28no—
164#95—24no—
165#95—24no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—44no—
170C24—44no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24—41no—
175C24C2×C22⋊C411yescharacteristic
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8#9514yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4#9514yes—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—48no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—88no—
242C22×C4—88no—
243C22×C4—88no—
244C22×C4#10012yes—
245C22×C4#10012yes—
246C22×C4C2×C4211yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4#11711yescharacteristic
251C22×C4#11711yescharacteristic
252C22×C4C4×D414yes—
253C22×C4#12014yes—
254C22×C4#12112yes—
255C22×C4C22⋊Q812yes—
256C2×C8—48no—
257C2×C8—84no—
258C2×C8—84no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—48no—
264C4⋊C4—88no—
265C22⋊C4—216no—
266C22⋊C4—216no—
267C22⋊C4—48no—
268C22⋊C4—88no—
269C42—24no—
270C42—24no—
271C42—42no—
272C42—42no—
273C42—48no—
274C42C2×C22⋊C412yes—
275C42C2×C4⋊C412yes—
276C42C4×D414yes—
277C42#12112yes—
278C42C22⋊Q812yes—
279C42#12314yes—
280C23—22no—
281C23—22no—
282C23—24no—
283C23—48no—
284C23—48no—
285C23—44no—
286C23—44no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—82no—
294C23—82no—
295C23—84no—
296C23—84no—
297C23—82no—
298C23—82no—
299C23C4×C22⋊C411yescharacteristic
300C23#37911yescharacteristic
301C23#38011yescharacteristic
302C23C2×#10011yescharacteristic
303C23C2×#10011yescharacteristic
304C23#41311yescharacteristic
305C23#41311yescharacteristic
306Q8—44no—
307Q8—44no—
308Q8—44no—
309Q8—44no—
310D4—44no—
311D4—44no—
312D4—44no—
313D4—44no—
314C2×C4—24no—
315C2×C4—24no—
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—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—816no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4C2×#9511yescharacteristic
361C2×C4C4×C22⋊C411yescharacteristic
362C2×C4#38011yescharacteristic
363C2×C4#38111yescharacteristic
364C2×C4#38514yes—
365C2×C4#38614yes—
366C8—88no—
367C22—22no—
368C22—22no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—44no—
376C22—48no—
377C22—84no—
378C22—84no—
379C22—84no—
380C22—84no—
381C22—82no—
382C22—84no—
383C22—82no—
384C22C22⋊#9511yescharacteristic
385C22C4×#10011yescharacteristic
386C22C4×#10011yescharacteristic
387C22#178311yescharacteristic
388C22#178511yescharacteristic
389C22#179512yes—
390C22#179612yes—
391C22#180011yescharacteristic
392C22#180011yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—24no—
398C4—24no—
399C4—44no—
400C4—42no—
401C4—42no—
402C4—82no—
403C4—82no—
404C4—82no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—84no—
410C2#852211yescharacteristic
411C2#852312yes—
412C2#852411yescharacteristic
413C2#1076211yescharacteristic
414C2#1077512yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×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
3. #8040C2semidirect
4. #7910C2semidirect
18. #1533C4semidirect
55. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#15332
#80402
#79102