Finite Group Databasebeta

← #129544

Subgroups — #129544

Order 512.

691 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2971C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic, complemented
3C4×#349C211yesmaximal, characteristic, complemented
4#11607C212yesmaximal, complemented
5#11587C212yesmaximal, complemented
6C2×C4×#41C411yescharacteristic
7#2971—24no—
8#2971—24no—
9C2×C8×Q8C411yescharacteristic
10C2×C8×Q8C2211yescharacteristic
11C2×#436—21no—
12C2×#436—21no—
13C2×#436C411yescharacteristic
14C2×#436C2211yescharacteristic
15C2×#432—21no—
16C2×#432C411yescharacteristic
17C2×C4×M16—21no—
18C2×C4×M16C2211yescharacteristic
19#2150C2212yes—
20C2×#349—21no—
21C2×#349—21no—
22C4×M32C2212yes—
23#1377—21no—
24#1377—21no—
25Q8⋊C16—24no—
26#1330—24no—
27C22×#41C811yescharacteristic
28C22×M16—21no—
29C22×M16—21no—
30C22×M16D411yescharacteristic
31C22×M16D411yescharacteristic
32C4×#41—22no—
33C4×#41—22no—
34C4×#41—22no—
35C4×#41—22no—
36C2×C4×Q8C811yescharacteristic
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×C4×D4C811yescharacteristic
41C2×C4×D4C2×C411yescharacteristic
42C2×#117—21no—
43C2×#117C811yescharacteristic
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C2×M32—22no—
47C2×M32—22no—
48C8×Q8—22no—
49C8×Q8—22no—
50C8×Q8—22no—
51C8×Q8D414yes—
52#436—28no—
53#436—28no—
54#436—22no—
55#436—22no—
56#436—22no—
57#436D414yes—
58#432—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—
68C4×M16—28no—
69C4×M16—22no—
70C4×M16—22no—
71C2×#97—21no—
72C2×#97D411yescharacteristic
73C2×#97D411yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8C2×C411yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C4⋊C16—22no—
79C4⋊C16—22no—
80#349—24no—
81#349—42no—
82#346—22no—
83C4×C16—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—44no—
87C22×Q8C2×C811yescharacteristic
88C22×D4—21no—
89C22×D4C2×C811yescharacteristic
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C2×C811yescharacteristic
95C23×C4M1611yescharacteristic
96C2×M16—24no—
97C2×M16—22no—
98C2×M16—24no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—41no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—41no—
109C2×M16—41no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8C22⋊C411yescharacteristic
113C22×C8C22⋊C411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8C22⋊C412yes—
121C4×Q8M1612yes—
122C4×D4—24no—
123C4×D4—24no—
124C4×D4—24no—
125C4×D4—24no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4C22⋊C412yes—
131C4×D4M1612yes—
132#117—24no—
133#117—24no—
134#117—22no—
135#117—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×C811yescharacteristic
139C2×C4⋊C4C2×C811yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C42—24no—
144C2×C42—24no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42C22⋊C411yescharacteristic
153C2×C42C2×C811yescharacteristic
154C2×C42M1611yescharacteristic
155C2×C42C22×C411yescharacteristic
156M32—24no—
157M32—42no—
158M32—42no—
159M32—42no—
160C2×C16—42no—
161C2×C16—42no—
162C4⋊C8—24no—
163C4⋊C8—24no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8C22⋊C412yes—
167C4⋊C8C22⋊C412yes—
168C22⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—24no—
171#97—24no—
172#97—22no—
173#97—22no—
174C4×C8—24no—
175C4×C8—24no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8C2×D412yes—
179C4×C8C2×D412yes—
180C24—21no—
181C24C22⋊C811yescharacteristic
182#41—48no—
183#41—44no—
184#41—44no—
185C2×Q8—24no—
186C2×Q8—21no—
187C2×Q8C22⋊C812yes—
188C2×D4—24no—
189C2×D4—21no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4C22⋊C812yes—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—22no—
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—21no—
209C22×C4—21no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—82no—
217C22×C4—81no—
218C22×C4C22⋊C811yescharacteristic
219C22×C4C22⋊C811yescharacteristic
220C22×C4C22⋊C811yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C22×C811yescharacteristic
223C22×C4C2×M1611yescharacteristic
224M16—44no—
225M16—48no—
226M16—42no—
227M16—44no—
228M16—42no—
229M16—44no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8#10412yes—
244C2×C8#12012yes—
245C2×C8#12012yes—
246C2×C8#13512yes—
247C2×C8AGL(1,ℤ/8)11yescharacteristic
248C2×C8#13711yescharacteristic
249C4⋊C4—24no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4C22⋊C812yes—
255C4⋊C4C22⋊C812yes—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C42—24no—
261C42—24no—
262C42—22no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—22no—
268C42—21no—
269C42—21no—
270C42C2×C22⋊C411yescharacteristic
271C42C2×C22⋊C411yescharacteristic
272C42C2×M1611yescharacteristic
273C42C2×M1611yescharacteristic
274C16—44no—
275C16—82no—
276C23—21no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C2×C22⋊C811yescharacteristic
287Q8—24no—
288Q8—22no—
289Q8#34914yescomplemented
290D4—42no—
291D4—48no—
292D4—42no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—48no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—84no—
323C2×C4#37911yescharacteristic
324C2×C4C2×C22⋊C811yescharacteristic
325C2×C4C2×C22⋊C811yescharacteristic
326C2×C4#40711yescharacteristic
327C2×C4#40711yescharacteristic
328C2×C4#40711yescharacteristic
329C2×C4#41611yescharacteristic
330C2×C4#41711yescharacteristic
331C2×C4C2×#10411yescharacteristic
332C2×C4#42111yescharacteristic
333C8—22no—
334C8—22no—
335C8—41no—
336C8—42no—
337C8—42no—
338C8#44912yes—
339C8#45212yes—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—42no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22—82no—
352C22#147611yescharacteristic
353C22#147711yescharacteristic
354C22#177911yescharacteristic
355C4—24no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—42no—
361C4—41no—
362C4—42no—
363C4—41no—
364C4#148811yescharacteristic
365C4#148911yescharacteristic
366C4#178811yescharacteristic
367C4#178911yescharacteristic
368C4C2×#34912yes—
369C2—21no—
370C2—21no—
371C2—81no—
372C2—82no—
373C2#831211yescharacteristic
374C2#1268111yescharacteristic
375C2#1268611yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radicalC22×#41C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×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. #12785C2semidirect
3. C4×#349C2semidirect
4. #11607C2semidirect
5. #11587C2semidirect
289. Q8#349semidirect

Maximal subgroups

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

subgroupindex
C2×#29712
#127852
C4×#3492
#116072
#115872