Finite Group Databasebeta

← #127149

Subgroups — #127149

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21303C211yesmaximal, characteristic
2C2×C4×#101C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#7206C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#423C411yescharacteristic
8C22×#101C412yes—
9C2×C4×M16—21no—
10C2×C4×M16C411yescharacteristic
11#1985—21no—
12#1985—21no—
13#1984C412yes—
14#1984C2212yes—
15C4×#101—21no—
16C4×#101—21no—
17C4×#101C412yes—
18C4×#101C2212yes—
19C2×#343—21no—
20#1728—21no—
21#1728—21no—
22#1467—21no—
23#1467C412yes—
24#1463—21no—
25#1463C412yes—
26#1461—21no—
27#1461C2212yes—
28C23×Q8C2×C411yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×M16—22no—
32C2×C4×Q8—41no—
33C2×C4×Q8—41no—
34C2×C4×Q8—41no—
35C2×C4×Q8—41no—
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×Q8C2×C411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×Q8D411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C2×C4×Q8Q811yescharacteristic
43C2×C4×Q8C2311yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4Q811yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55#423—41no—
56#423—41no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2×C412yes—
59C2×#101—22no—
60C2×#101—28no—
61C2×#101C2×C414yes—
62#407—22no—
63C4×M16—22no—
64C4×M16—22no—
65C4×M16—41no—
66C4×M16—41no—
67C4×M16—41no—
68C4×M16—41no—
69C4×M16—41no—
70C2×#97—21no—
71C2×#97—41no—
72C2×C4×C8—41no—
73C2×C4×C8C2×C412yes—
74#343—22no—
75#343—22no—
76#336—22no—
77#336—22no—
78#336—22no—
79#336—22no—
80#324—22no—
81#324—22no—
82#324—22no—
83#324C2×C414yes—
84C22×Q8—24no—
85C22×Q8—41no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8C4211yescharacteristic
89C22×Q8C22⋊C411yescharacteristic
90C22×Q8C4⋊C411yescharacteristic
91C22×Q8C22×C411yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C2×M16—24no—
100C2×M16—28no—
101C2×M16—24no—
102C2×M16—42no—
103C2×M16—44no—
104C2×M16—42no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—44no—
108C2×M16—42no—
109C2×M16—41no—
110C2×M16—41no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—42no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×Q8—41no—
119C4×Q8—41no—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8—41no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4C4211yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C4⋊C411yescharacteristic
141C2×C4⋊C4C4⋊C411yescharacteristic
142C2×C4⋊C4C4⋊C411yescharacteristic
143C2×C4⋊C4C4⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C4⋊C4C2×Q811yescharacteristic
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42C4211yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C4⋊C8—42no—
173C4⋊C8—42no—
174C4⋊C8—42no—
175C4⋊C8—42no—
176#101—28no—
177#101—24no—
178#101M1618yescomplemented
179C22⋊C8—24no—
180C22⋊C8—42no—
181#97—42no—
182#97—42no—
183#97—42no—
184#97—42no—
185C4×C8—42no—
186C4×C8—42no—
187C4×C8—42no—
188C4×C8—42no—
189C24#9511yescharacteristic
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—48no—
202C2×Q8—48no—
203C2×Q8C2×M1614yes—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4#9511yescharacteristic
228C22×C4#9511yescharacteristic
229C22×C4#9511yescharacteristic
230C22×C4#9511yescharacteristic
231C22×C4#9511yescharacteristic
232C22×C4#9511yescharacteristic
233C22×C4#9511yescharacteristic
234C22×C4#9911yescharacteristic
235C22×C4#9911yescharacteristic
236C22×C4#9911yescharacteristic
237C22×C4#9911yescharacteristic
238C22×C4C2×C4211yescharacteristic
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245M16—28no—
246M16—44no—
247M16—44no—
248M16—48no—
249M16—44no—
250M16—82no—
251M16—84no—
252M16—82no—
253C2×C8—24no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—48no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—44no—
260C2×C8—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C23—22no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23#34211yescharacteristic
302C23#34211yescharacteristic
303C23C2×#9511yescharacteristic
304Q8—44no—
305Q8—44no—
306Q8—44no—
307Q8—44no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4#34211yescharacteristic
354C2×C4#34211yescharacteristic
355C2×C4#34412yes—
356C2×C4#35512yes—
357C2×C4#35612yes—
358C2×C4C4×M1612yes—
359C2×C4#40712yes—
360C2×C4C2×#9911yescharacteristic
361C2×C4C2×#9911yescharacteristic
362C2×C4#42312yes—
363C8—44no—
364C8—48no—
365C8—84no—
366C8—82no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#139412yes—
377C22#172811yescharacteristic
378C22C2×#34211yescharacteristic
379C22C2×#34411yescharacteristic
380C4—21no—
381C4—21no—
382C4—22no—
383C4—22no—
384C4—22no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4#139412yes—
395C4C2×#35511yescharacteristic
396C4C2×#35611yescharacteristic
397C2—22no—
398C2—21no—
399C2—21no—
400C2—41no—
401C2#792911yescharacteristic
402C2C2×#139411yescharacteristic
403C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
socleC22C2×#342
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7206C2semidirect
178. #101M16semidirect

Maximal subgroups

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

subgroupindex
#213032
C2×C4×#1012
#173652
#72062