Finite Group Databasebeta

← #188129

Subgroups — #188129

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2#21753C212yesmaximal
3#17397C212yesmaximal
4C2×#1463C212yesmaximal
5#8554C214yesmaximal, complemented
6#8551C214yesmaximal, complemented
7C2×#544—28no—
8C2×#544—28no—
9C2×#544—24no—
10C2×#544—24no—
11C22×C4⋊Q8C412yes—
12C22×C4⋊Q8C2212yes—
13C22×#125C412yes—
14C22×#125C2212yes—
15C22×C22⋊Q8C412yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C412yes—
19C23×C4⋊C4C2211yescharacteristic
20C2×#407C2214yes—
21#1901—28no—
22#1821C2214yes—
23#1784C2214yes—
24#1472—28no—
25#1463—24no—
26#1463C2218yes—
27#1463C2218yes—
28C2×#324—24no—
29C23×Q8C2×C412yes—
30C23×Q8C2×C411yescharacteristic
31C23×Q8C2311yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C22×M16—24no—
36#544—232no—
37#544—232no—
38#544—232no—
39#544—232no—
40C2×C4⋊Q8—28no—
41C2×C4⋊Q8—28no—
42C2×C4⋊Q8—28no—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8—28no—
45C2×C4⋊Q8—28no—
46C2×#125—28no—
47C2×#125—28no—
48C2×#125—28no—
49C2×#125—28no—
50C2×#125—28no—
51C2×#125—28no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8D418yes—
63C2×C22⋊Q8D418yes—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117—28no—
67C2×#117—28no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4C2×C412yes—
75C22×C4⋊C4C2×C412yes—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4D414yes—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C412yes—
85C22×C42C2×C412yes—
86C22×C42C2×C412yes—
87C2×#101—28no—
88#407—24no—
89#407—24no—
90#407—24no—
91#407C2318yes—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94#343—28no—
95#324—216no—
96#324—216no—
97C25C22×C411yescharacteristic
98C22×Q8—22no—
99C22×Q8—24no—
100C22×Q8—22no—
101C22×Q8—24no—
102C22×Q8—24no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—82no—
106C22×Q8—82no—
107C22×Q8—84no—
108C22×Q8C22⋊C414yes—
109C22×Q8C2×D414yes—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—28no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—82no—
116C23×C4—82no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22×C412yes—
122C23×C4C22×C412yes—
123C23×C4C22×C412yes—
124C23×C4C22×C412yes—
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D414yes—
129C23×C4C2×D414yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2411yescharacteristic
133C2×M16—28no—
134C2×M16—44no—
135C2×M16—88no—
136C22×C8—44no—
137C4⋊Q8—264no—
138C4⋊Q8—264no—
139#125—264no—
140#125—264no—
141C22⋊Q8—232no—
142C22⋊Q8—232no—
143C22⋊Q8—28no—
144C22⋊Q8—28no—
145C22⋊Q8—28no—
146C22⋊Q8—28no—
147C22⋊Q8—28no—
148C22⋊Q8—28no—
149C22⋊Q8—28no—
150C22⋊Q8—28no—
151#117—232no—
152#117—232no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4C22⋊C418yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22⋊C418yes—
187C2×C4⋊C4C2×D418yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D418yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4C22⋊C418yes—
197C2×C22⋊C4C22⋊C418yes—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204#101—88no—
205C22⋊C8—28no—
206C22⋊C8—28no—
207C22⋊C8—44no—
208C22⋊C8—44no—
209C24—24no—
210C24—42no—
211C24—42no—
212C24C2×C22⋊C412yes—
213C24C2×C22⋊C412yes—
214C24C2×C22⋊C411yescharacteristic
215C24C2×C22⋊C411yescharacteristic
216C24C23×C411yescharacteristic
217C2×Q8—216no—
218C2×Q8—28no—
219C2×Q8—28no—
220C2×Q8—48no—
221C2×Q8—84no—
222C2×Q8—88no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—84no—
227C2×Q8—88no—
228C2×Q8—88no—
229C22×C4—216no—
230C22×C4—216no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—48no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4#10118yescomplemented
266C22×C4C2×C22⋊C414yes—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C414yes—
270C22×C4C2×C22⋊C414yes—
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C414yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4#12014yes—
276C22×C4#12014yes—
277C22×C4#12014yes—
278C22×C4#12014yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D412yes—
282M16—416no—
283M16—88no—
284C2×C8—48no—
285C2×C8—88no—
286C4⋊C4—216no—
287C4⋊C4—216no—
288C4⋊C4—216no—
289C4⋊C4—216no—
290C4⋊C4—216no—
291C4⋊C4—216no—
292C4⋊C4—28no—
293C4⋊C4—216no—
294C4⋊C4—216no—
295C4⋊C4—28no—
296C4⋊C4—216no—
297C22⋊C4—28no—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—28no—
301C42—216no—
302C42—216no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—24no—
307C23—22no—
308C23—24no—
309C23—24no—
310C23—24no—
311C23—41no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—41no—
317C23—416no—
318C23#37912yes—
319C23#37912yes—
320C23C2×#10114yes—
321C23C22×C22⋊C411yescharacteristic
322C23C22×C22⋊C411yescharacteristic
323C23C22×C22⋊C411yescharacteristic
324Q8—88no—
325Q8—88no—
326Q8—816no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—216no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—216no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4#37914yes—
366C2×C4C2×#10118yes—
367C2×C4C2×#12014yes—
368C8—88no—
369C22—28no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—44no—
382C22—48no—
383C22—48no—
384C22—44no—
385C22#178414yes—
386C22C2×#37911yescharacteristic
387C22#288312yes—
388C22C22×#10112yes—
389C22#288912yes—
390C4—24no—
391C4—24no—
392C4—28no—
393C4—44no—
394C4—44no—
395C4—44no—
396C4—44no—
397C4—44no—
398C2—22no—
399C2—22no—
400C2—22no—
401C2—24no—
402C2—42no—
403C2#2058511yescharacteristic
404C2C2×#178411yescharacteristic
405C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8554C2semidirect
6. #8551C2semidirect
265. C22×C4#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
#217532
#173972
C2×#14632
#85542
#85512