Finite Group Databasebeta

← #127147

Subgroups — #127147

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2C2×C4×#101C211yesmaximal, characteristic
3#17365C211yesmaximal, characteristic
4#7206C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic
8C2×#431C411yescharacteristic
9C22×#101C412yes—
10C2×C4×M16—21no—
11#1985—21no—
12#1985—21no—
13#1984C412yes—
14#1984C2212yes—
15#1817—21no—
16#1817—21no—
17C4×#101—21no—
18C4×#101—21no—
19C4×#101C412yes—
20C4×#101C2212yes—
21C2×#343—21no—
22#1467—21no—
23#1467C412yes—
24#1463—21no—
25#1463C412yes—
26#1461—21no—
27#1461C2212yes—
28C23×Q8C2×C411yescharacteristic
29C22×M16—22no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×Q8Q811yescharacteristic
41C2×C4×Q8C2311yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4Q811yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—22no—
49#432—41no—
50#432—41no—
51#432—41no—
52#431—22no—
53#431—41no—
54#431—41no—
55#431—41no—
56#423—41no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2×C412yes—
59C2×#101—22no—
60C2×#101—28no—
61C2×#101C2×C414yes—
62#407—22no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C4×M16—22no—
66C4×M16—22no—
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—28no—
100C2×M16—24no—
101C2×M16—44no—
102C2×M16—42no—
103C2×M16—42no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—42no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×Q8—41no—
112C4×Q8—41no—
113C4×Q8—41no—
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—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4C4211yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×Q811yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
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×C42C4211yescharacteristic
154C2×C42C22⋊C411yescharacteristic
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C4⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C4⋊C8—42no—
166C4⋊C8—42no—
167C4⋊C8—42no—
168C4⋊C8—42no—
169#101—28no—
170#101—24no—
171#101M1618yescomplemented
172C22⋊C8—24no—
173C22⋊C8—42no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178#97—42no—
179#97—42no—
180#97—42no—
181#97—42no—
182C4×C8—42no—
183C4×C8—42no—
184C4×C8—42no—
185C4×C8—42no—
186C24#9511yescharacteristic
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—48no—
199C2×Q8—48no—
200C2×Q8C2×M1614yes—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4#9511yescharacteristic
224C22×C4#9511yescharacteristic
225C22×C4#9511yescharacteristic
226C22×C4#9511yescharacteristic
227C22×C4#9511yescharacteristic
228C22×C4#9511yescharacteristic
229C22×C4#9511yescharacteristic
230C22×C4#9911yescharacteristic
231C22×C4#9911yescharacteristic
232C22×C4#9911yescharacteristic
233C22×C4#9911yescharacteristic
234C22×C4C2×C4211yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C4⋊C411yescharacteristic
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4C2×C4⋊C411yescharacteristic
241M16—28no—
242M16—48no—
243M16—44no—
244M16—84no—
245C2×C8—44no—
246C2×C8—48no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
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—
288C23—22no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23#34211yescharacteristic
293C23#34211yescharacteristic
294C23C2×#9511yescharacteristic
295Q8—44no—
296Q8—44no—
297Q8—44no—
298Q8—44no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
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—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4#34211yescharacteristic
344C2×C4#34211yescharacteristic
345C2×C4#34412yes—
346C2×C4C4×M1612yes—
347C2×C4#40712yes—
348C2×C4C2×#9911yescharacteristic
349C2×C4C2×#9911yescharacteristic
350C2×C4#42312yes—
351C8—48no—
352C8—82no—
353C8—84no—
354C8—82no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22#139512yes—
365C22#172811yescharacteristic
366C22C2×#34211yescharacteristic
367C22C2×#34411yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4#213312yes—
384C2—22no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2#792911yescharacteristic
389C2#1258611yescharacteristic
390C2C2×#139511yescharacteristic
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
171. #101M16semidirect

Maximal subgroups

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

subgroupindex
#218202
C2×C4×#1012
#173652
#72062