Finite Group Databasebeta

← #129538

Subgroups — #129538

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2966C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3C4×C22⋊C16C211yesmaximal, characteristic, complemented
4C2×Q8⋊C16C212yesmaximal, complemented
5C2×D4⋊C16C212yesmaximal, complemented
6C2×#529C411yescharacteristic
7#2966—24no—
8#2966—24no—
9C2×#446C411yescharacteristic
10C2×C8×Q8C2211yescharacteristic
11C2×#436C411yescharacteristic
12C2×#435—21no—
13C2×C8×D4—21no—
14C2×C8×D4C2211yescharacteristic
15C2×#433—21no—
16C2×#432C411yescharacteristic
17C2×#405—21no—
18C22×C4×C8C2211yescharacteristic
19C2×C4⋊C16C2212yes—
20C2×C22⋊C16—21no—
21C2×C22⋊C16—21no—
22C2×C4×C16C2212yes—
23#1374—21no—
24#1374—21no—
25Q8⋊C16—24no—
26D4⋊C16—24no—
27C22×M16—21no—
28C23×C8—21no—
29C23×C8D411yescharacteristic
30C23×C8D411yescharacteristic
31#529—22no—
32#529—22no—
33#529—22no—
34#529—22no—
35C2×#126—21no—
36C2×#125C811yescharacteristic
37C2×#124C811yescharacteristic
38C2×#123—21no—
39C2×C22⋊Q8C811yescharacteristic
40C2×#121C811yescharacteristic
41C2×C4×Q8C2×C411yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4C2×C411yescharacteristic
44C2×#117—21no—
45C22×C42C2×C411yescharacteristic
46C22×C16—22no—
47C22×C16—22no—
48#446—22no—
49#446—22no—
50C8×Q8—22no—
51C8×Q8D414yes—
52#436—22no—
53#436—22no—
54#435—28no—
55C8×D4—28no—
56C8×D4—22no—
57C8×D4D414yes—
58#433—28no—
59#432—22no—
60#432—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2×C411yescharacteristic
64C2×C4⋊C8C2×C411yescharacteristic
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68#405—28no—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8C2×C411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C4⋊C16—22no—
79C4⋊C16—22no—
80C22⋊C16—24no—
81C22⋊C16—42no—
82C4×C16—22no—
83C4×C16—22no—
84C22×Q8C2×C811yescharacteristic
85C22×D4—21no—
86C22×D4C2×C811yescharacteristic
87C23×C4—21no—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C2×C811yescharacteristic
90C23×C4M1611yescharacteristic
91C2×M16—24no—
92C2×M16—42no—
93C2×M16—42no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—22no—
97C22×C8—24no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—41no—
104C22×C8—42no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8C22⋊C411yescharacteristic
108C22×C8C22⋊C411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111#126—24no—
112#126—24no—
113#125—22no—
114#125—22no—
115#124—22no—
116#124—22no—
117#123—24no—
118#123—24no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121#121—22no—
122#121—22no—
123C4×Q8—21no—
124C4×Q8—21no—
125C4×Q8C22⋊C412yes—
126C4×Q8M1612yes—
127C4×D4—24no—
128C4×D4—24no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4C22⋊C412yes—
132C4×D4M1612yes—
133#117—24no—
134#117—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C2×C811yescharacteristic
140C2×C4⋊C4C2×C811yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42C22⋊C411yescharacteristic
152C2×C42C2×C811yescharacteristic
153C2×C42M1611yescharacteristic
154C2×C42C22×C411yescharacteristic
155C2×C16—24no—
156C2×C16—42no—
157C2×C16—42no—
158C2×C16—42no—
159C2×C16—42no—
160C2×C16—42no—
161C4⋊C8—24no—
162C4⋊C8—24no—
163C4⋊C8—21no—
164C4⋊C8—21no—
165C4⋊C8C22⋊C412yes—
166C4⋊C8C22⋊C412yes—
167C22⋊C8—24no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170#97—24no—
171#97—24no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—24no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8C2×D412yes—
178C4×C8C2×D412yes—
179C24—21no—
180C24C22⋊C811yescharacteristic
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8C22⋊C812yes—
185C2×D4—24no—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4C22⋊C812yes—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4C22⋊C811yescharacteristic
216C22×C4C22⋊C811yescharacteristic
217C22×C4C22⋊C811yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C22×C811yescharacteristic
220C22×C4C2×M1611yescharacteristic
221M16—48no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—44no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—44no—
240C2×C8#10412yes—
241C2×C8#12012yes—
242C2×C8#12012yes—
243C2×C8#13512yes—
244C2×C8AGL(1,ℤ/8)11yescharacteristic
245C2×C8#13711yescharacteristic
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4C22⋊C812yes—
254C4⋊C4C22⋊C812yes—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C42—21no—
262C42—22no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—22no—
269C42C2×C22⋊C411yescharacteristic
270C42C2×C22⋊C411yescharacteristic
271C42C2×M1611yescharacteristic
272C42C2×M1611yescharacteristic
273C16—44no—
274C16—82no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#35012yes—
289C23C2×C22⋊C811yescharacteristic
290Q8—24no—
291Q8—42no—
292D4—24no—
293D4—48no—
294D4—42no—
295C2×C4—21no—
296C2×C4—21no—
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—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4#37911yescharacteristic
328C2×C4C2×C22⋊C811yescharacteristic
329C2×C4C2×C22⋊C811yescharacteristic
330C2×C4#40711yescharacteristic
331C2×C4#40711yescharacteristic
332C2×C4#40711yescharacteristic
333C2×C4#41611yescharacteristic
334C2×C4#41711yescharacteristic
335C2×C4C2×#10411yescharacteristic
336C2×C4#42111yescharacteristic
337C2×C4#50412yes—
338C2×C4#50412yes—
339C8—22no—
340C8—22no—
341C8—41no—
342C8—42no—
343C8—42no—
344C8#44912yes—
345C8#45212yes—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#147611yescharacteristic
361C22#147711yescharacteristic
362C22#177911yescharacteristic
363C22#211311yescharacteristic
364C22#211311yescharacteristic
365C22C2×#35011yescharacteristic
366C22#211711yescharacteristic
367C4—21no—
368C4—21no—
369C4—41no—
370C4—42no—
371C4—41no—
372C4—42no—
373C4—41no—
374C4—42no—
375C4#148811yescharacteristic
376C4#148911yescharacteristic
377C4#178811yescharacteristic
378C4#178911yescharacteristic
379C4#211312yes—
380C4#211312yes—
381C2—42no—
382C2—82no—
383C2#831211yescharacteristic
384C2#1161411yescharacteristic
385C2#1161611yescharacteristic
386C2#1162211yescharacteristic
387C2#1162411yescharacteristic
388C2#1268011yescharacteristic
389C2#1268311yescharacteristic
subgroupG/H
centerC22×C8C2×D4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. C4×C22⋊C16C2semidirect
4. C2×Q8⋊C16C2semidirect
5. C2×D4⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29662
#127832
C4×C22⋊C162
C2×Q8⋊C162
C2×D4⋊C162