Finite Group Databasebeta

← #106739

Subgroups — #106739

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2#20688C211yesmaximal, characteristic
3#20685C211yesmaximal, characteristic
4#8024C212yesmaximal
5#7943C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C22×C22⋊C8C2211yescharacteristic
9#2343—24no—
10#2343—24no—
11#2315—21no—
12#2313C412yescomplemented
13#2313C412yescomplemented
14#2283—21no—
15#2283—21no—
16#2280C2212yes—
17C2×#381C2212yes—
18#2273—21no—
19#1820—22no—
20#1819—21no—
21#1819—21no—
22#1818C414yescomplemented
23#1817—21no—
24#1817—21no—
25#1740C414yescomplemented
26#1729—22no—
27#1294—24no—
28#1284—24no—
29C24×C4C2311yescharacteristic
30C22×M16—21no—
31C23×C8—21no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117C2×C414yes—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117D412yes—
40C2×#117Q812yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C22⋊C4—22no—
44C22×C22⋊C4D411yescharacteristic
45C22×C22⋊C4D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#407—22no—
49#407—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56#387—28no—
57#387—28no—
58#385—28no—
59#383—22no—
60#383—22no—
61#382—22no—
62#382—22no—
63#381—22no—
64#381—22no—
65#381—22no—
66#381C2×C414yes—
67#381D414yes—
68#380—22no—
69#380—22no—
70#380—22no—
71#380—22no—
72#379—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C22⋊C4—28no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80#376—22no—
81#376—22no—
82C2×#95—22no—
83C2×#95D414yes—
84#342—24no—
85#342—24no—
86#336—24no—
87#336—24no—
88C25C2×D411yescharacteristic
89C23×C4—24no—
90C23×C4—44no—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×Q811yescharacteristic
103C23×C4#4112yes—
104C23×C4#4112yes—
105C2×M16—44no—
106C2×M16—42no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—44no—
110C22×C8—42no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—24no—
115#117—24no—
116#117—24no—
117#117—24no—
118#117—24no—
119#117—48no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4C4212yes—
128C2×C4⋊C4C4212yes—
129C2×C4⋊C4C4⋊C412yes—
130C2×C4⋊C4C4⋊C412yes—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—48no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C4⋊C412yes—
144C2×C22⋊C4C4⋊C412yes—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42C22⋊C412yes—
157C2×C42C22⋊C412yes—
158C2×C42C4⋊C412yes—
159C2×C42C4⋊C412yes—
160C22⋊C8—24no—
161C22⋊C8—24no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—42no—
174#95—42no—
175#95C22⋊C418yes—
176C24—24no—
177C24—42no—
178C24—81no—
179C24C2×C22⋊C411yescharacteristic
180C24C2×C22⋊C411yescharacteristic
181C24C2×C4⋊C411yescharacteristic
182C24#12012yes—
183C24#12312yes—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—28no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—84no—
210C22×C4—84no—
211C22×C4#9512yes—
212C22×C4#9512yes—
213C22×C4#9512yes—
214C22×C4#9512yes—
215C22×C4C2×C4211yescharacteristic
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C4⋊C411yescharacteristic
219C22×C4C2×C4⋊C411yescharacteristic
220C22×C4#11712yes—
221C22×C4C4×D412yes—
222C22×C4C4×D412yes—
223C22×C4C4×D412yes—
224C22×C4C4×D412yes—
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12112yes—
228C22×C4#12112yes—
229C22×C4C22⋊Q812yes—
230C22×C4C22⋊Q812yes—
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233M16—84no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—84no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—48no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—84no—
264C42—84no—
265C42—84no—
266C23—24no—
267C23—22no—
268C23—24no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—88no—
281C23C2×#9511yescharacteristic
282C23#37911yescharacteristic
283C23#38111yescharacteristic
284C23#38512yes—
285C23#38612yes—
286C23#41014yes—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—48no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—84no—
327C2×C4—168no—
328C2×C4C4×C22⋊C412yes—
329C2×C4#38012yes—
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38611yescharacteristic
333C2×C4#38611yescharacteristic
334C8—82no—
335C8—82no—
336C22—21no—
337C22—22no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—22no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—22no—
348C22—44no—
349C22—48no—
350C22—44no—
351C22—84no—
352C22—82no—
353C22C22⋊#9511yescharacteristic
354C22#174012yes—
355C22#179412yes—
356C22#189914yes—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—21no—
362C4—21no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—88no—
368C4—164no—
369C2—22no—
370C2—24no—
371C2—22no—
372C2—41no—
373C2—42no—
374C2#1073611yescharacteristic
375C2#1074311yescharacteristic
376C2#1075211yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2313C4semidirect
13. #2313C4semidirect
22. #1818C4semidirect
25. #1740C4semidirect

Maximal subgroups

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

subgroupindex
#341772
#206882
#206852
#80242
#79432