Finite Group Databasebeta

← #127480

Subgroups — #127480

Order 512.

531 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21820C211yesmaximal, characteristic
2#17365C211yesmaximal, characteristic
3C2×#1467C211yesmaximal, characteristic
4#7248C214yesmaximal, complemented
5C22×C4×Q8C2211yescharacteristic
6#2920—21no—
7C2×#432—21no—
8C2×#432C411yescharacteristic
9C2×#431C411yescharacteristic
10#1985—21no—
11#1985—21no—
12#1984C412yes—
13#1984C2212yes—
14#1817—21no—
15#1817—21no—
16#1467—21no—
17#1467—21no—
18#1467—21no—
19#1467C412yes—
20#1467C2212yes—
21#1467C2212yes—
22#1463—21no—
23#1463C412yes—
24#1461—21no—
25#1461C412yes—
26C2×#324—21no—
27C2×#324C412yes—
28C23×Q8Q811yescharacteristic
29C2×C4×Q8—41no—
30C2×C4×Q8—41no—
31C2×C4×Q8—41no—
32C2×C4×Q8—41no—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×Q8Q811yescharacteristic
40C2×C4×Q8C2311yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47#432—22no—
48#432—22no—
49#432—22no—
50#432—41no—
51#432—41no—
52#432—41no—
53#432—41no—
54#432—41no—
55#431—22no—
56#431—41no—
57#431—41no—
58#431—41no—
59#423—41no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8C2×C412yes—
63#407—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—41no—
68C2×#97—21no—
69C2×#97—41no—
70C2×C4×C8C2×C412yes—
71#336—22no—
72#336—22no—
73#336—22no—
74#336—22no—
75#324—22no—
76#324—22no—
77#324—22no—
78#324—22no—
79#324—22no—
80#324—22no—
81#324C2×C414yes—
82#324C2×C414yes—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8C4⋊C411yescharacteristic
88C22×Q8C4⋊C411yescharacteristic
89C22×Q8C4⋊C411yescharacteristic
90C22×Q8C2×Q811yescharacteristic
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C2×M16—42no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—42no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
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—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4C4211yescharacteristic
125C2×C4⋊C4C4211yescharacteristic
126C2×C4⋊C4C22⋊C411yescharacteristic
127C2×C4⋊C4C22⋊C411yescharacteristic
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C22×C411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42C4211yescharacteristic
149C2×C42C22⋊C411yescharacteristic
150C2×C42C22⋊C411yescharacteristic
151C2×C42C22⋊C411yescharacteristic
152C2×C42C22⋊C411yescharacteristic
153C2×C42C22⋊C411yescharacteristic
154C2×C42C4⋊C411yescharacteristic
155C2×C42C4⋊C411yescharacteristic
156C2×C42C4⋊C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166C22⋊C8—24no—
167C22⋊C8—24no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174#97—42no—
175#97—42no—
176#97—42no—
177#97—42no—
178C4×C8—42no—
179C4×C8—42no—
180C24#9511yescharacteristic
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—48no—
194C2×Q8—48no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4#9511yescharacteristic
218C22×C4#9511yescharacteristic
219C22×C4#9511yescharacteristic
220C22×C4#9511yescharacteristic
221C22×C4#9511yescharacteristic
222C22×C4#9511yescharacteristic
223C22×C4#9511yescharacteristic
224C22×C4#9911yescharacteristic
225C22×C4#9911yescharacteristic
226C22×C4#9911yescharacteristic
227C22×C4#9911yescharacteristic
228C22×C4C2×C4211yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4C2×C4⋊C411yescharacteristic
235M16—84no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—84no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
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—
279C23—22no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23#34211yescharacteristic
284C23#34211yescharacteristic
285C23C2×#9511yescharacteristic
286Q8—44no—
287Q8—44no—
288Q8—44no—
289Q8—44no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
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—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4#33812yes—
337C2×C4#34211yescharacteristic
338C2×C4#34211yescharacteristic
339C2×C4C2×#9911yescharacteristic
340C2×C4C2×#9911yescharacteristic
341C2×C4#43012yes—
342C8—82no—
343C8—84no—
344C8—84no—
345C8—82no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22#139112yes—
356C22C2×#33811yescharacteristic
357C22#173811yescharacteristic
358C22C2×#34211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—22no—
363C4—22no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4#213312yes—
375C2—22no—
376C2—21no—
377C2—21no—
378C2—41no—
379C2#800911yescharacteristic
380C2C2×#139111yescharacteristic
381C2#1258611yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×Q8C23
prime radicalC24#95
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7248C2semidirect

Maximal subgroups

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

subgroupindex
#218202
#173652
C2×#14672
#72482