Finite Group Databasebeta

← #106755

Subgroups — #106755

Order 512.

850 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21303C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#8074C212yesmaximal
5#7960C212yesmaximal
6C22×C4×Q8C2211yescharacteristic
7#2920C2212yes—
8C2×#423C2211yescharacteristic
9C2×C4×M16C2211yescharacteristic
10#2356—24no—
11#2355—24no—
12#2313—21no—
13#2313C412yes—
14#2310C412yes—
15#2285—21no—
16#2283—21no—
17#2283C2212yes—
18#2274—21no—
19#1985C414yescomplemented
20#1984—22no—
21#1728—21no—
22#1728—21no—
23#1467—22no—
24#1463—21no—
25#1463—21no—
26#1461C414yescomplemented
27#1300—24no—
28#1286—24no—
29C23×Q8C2×C411yescharacteristic
30C22×M16—21no—
31C22×M16—21no—
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8C2×C412yes—
37C2×C4×Q8C2×C412yes—
38C2×C4×Q8D412yes—
39C2×C4×Q8Q812yes—
40C2×#117—22no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42C2311yescharacteristic
47#432—22no—
48#432—22no—
49#431—22no—
50#431C2×C414yes—
51#423—22no—
52#423—22no—
53#423—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yes—
56#407—22no—
57C4×M16—22no—
58C4×M16—22no—
59C4×M16—22no—
60C2×#97C2×C412yes—
61C2×C4×C8—21no—
62#391—28no—
63#387—28no—
64#387—28no—
65#384—22no—
66#383—22no—
67#383—22no—
68#382—22no—
69#382—22no—
70#382D414yes—
71#381—22no—
72#381C2×C414yes—
73#380—22no—
74#380—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—28no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82#376—22no—
83#376D414yes—
84#336—24no—
85#336—24no—
86#324—24no—
87#324—24no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—44no—
91C22×Q8C4212yes—
92C22×Q8C4⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—42no—
102C2×M16—41no—
103C2×M16—41no—
104C2×M16—44no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—42no—
108C22×C8—22no—
109C22×C8—22no—
110C4×Q8—24no—
111C4×Q8—24no—
112C4×Q8—24no—
113C4×Q8—24no—
114C4×Q8—24no—
115C4×Q8—24no—
116C4×Q8—24no—
117C4×Q8—24no—
118#117—48no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4C4212yes—
129C2×C4⋊C4C22⋊C412yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C4⋊C412yes—
132C2×C4⋊C4C4⋊C412yes—
133C2×C4⋊C4C4⋊C412yes—
134C2×C22⋊C4—24no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—24no—
146C2×C42C22⋊C412yes—
147C2×C42C22⋊C412yes—
148C2×C42C4⋊C412yes—
149C2×C42C4⋊C412yes—
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×Q811yescharacteristic
156C2×C42#4112yes—
157C2×C42#4112yes—
158C4⋊C8—24no—
159C4⋊C8—24no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C22⋊C8—24no—
163C22⋊C8—42no—
164#97—24no—
165#97—42no—
166C4×C8—24no—
167C4×C8—42no—
168#95—24no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95C22⋊C418yes—
176C24C2×C22⋊C411yescharacteristic
177C2×Q8—24no—
178C2×Q8—24no—
179C2×Q8—24no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—44no—
184C2×Q8—44no—
185C2×Q8—48no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—28no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4#9512yes—
204C22×C4#9512yes—
205C22×C4#9512yes—
206C22×C4#9512yes—
207C22×C4C2×C4211yescharacteristic
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4C2×C4⋊C411yescharacteristic
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4#11712yes—
215C22×C4#12011yescharacteristic
216C22×C4#12012yes—
217C22×C4#12011yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12312yes—
221M16—44no—
222M16—44no—
223M16—82no—
224M16—84no—
225M16—82no—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8—42no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—48no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—84no—
242C22⋊C4—48no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—48no—
252C42—84no—
253C42—84no—
254C42C4×D412yes—
255C42C4×D412yes—
256C42C4×D412yes—
257C42C4×D412yes—
258C42#12112yes—
259C42#12112yes—
260C42C22⋊Q812yes—
261C42C22⋊Q812yes—
262C23—22no—
263C23—22no—
264C23—41no—
265C23C2×#9511yescharacteristic
266C23#37911yescharacteristic
267C23#38111yescharacteristic
268Q8—48no—
269Q8—48no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4#34414yes—
307C2×C4C4×C22⋊C412yes—
308C2×C4#38012yes—
309C2×C4#38511yescharacteristic
310C2×C4#38512yes—
311C2×C4#38511yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38612yes—
314C2×C4#38611yescharacteristic
315C2×C4#45412yes—
316C2×C4#45612yes—
317C8—44no—
318C8—84no—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—44no—
328C22C22⋊#9511yescharacteristic
329C22C2×#34412yes—
330C22#179312yes—
331C4—24no—
332C4—22no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—21no—
337C4—21no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—88no—
342C4#185712yes—
343C4#188611yescharacteristic
344C4#188811yescharacteristic
345C2—22no—
346C2—22no—
347C2—41no—
348C2#1073811yescharacteristic
349C2#1074311yescharacteristic
350C2#1075011yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1985C4semidirect
26. #1461C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#213032
#173652
#80742
#79602