Finite Group Databasebeta

← #129542

Subgroups — #129542

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2965C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3#12599C211yesmaximal, characteristic, complemented
4C2×#1338C212yesmaximal, complemented
5C2×#1330C212yesmaximal, complemented
6C2×C4×#41C411yescharacteristic
7#2965—24no—
8#2965—24no—
9C2×#446C411yescharacteristic
10C2×#446C2211yescharacteristic
11C2×#436C411yescharacteristic
12C2×#436C2211yescharacteristic
13C2×#435—21no—
14C2×#435—21no—
15C2×#431—21no—
16C2×#431C411yescharacteristic
17C2×C4×M16—21no—
18C22×C4×C8C2211yescharacteristic
19C2×C4⋊C16C2212yes—
20C2×C22⋊C16—21no—
21C2×C22⋊C16—21no—
22C2×#346C2212yes—
23#1374—21no—
24#1374—21no—
25#1338—24no—
26#1330—24no—
27C22×#41C811yescharacteristic
28C22×M16—21no—
29C22×M16—21no—
30C23×C8D411yescharacteristic
31C23×C8D411yescharacteristic
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
46C22×C16—22no—
47C22×C16—22no—
48#446—22no—
49#446—22no—
50#446—22no—
51#446D414yes—
52#436—22no—
53#436—22no—
54#436—22no—
55#436D414yes—
56#435—28no—
57#435—28no—
58#431—28no—
59#431—22no—
60#431—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—
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—
82#346—22no—
83#346—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—42no—
87C2×#41—42no—
88C22×Q8C2×C811yescharacteristic
89C22×D4—21no—
90C22×D4C2×C811yescharacteristic
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4C22⋊C411yescharacteristic
95C23×C4C2×C811yescharacteristic
96C23×C4M1611yescharacteristic
97C2×M16—24no—
98C2×M16—24no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—41no—
113C22×C8C22⋊C411yescharacteristic
114C22×C8C22⋊C411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8C22⋊C412yes—
122C4×Q8M1612yes—
123C4×D4—24no—
124C4×D4—24no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—21no—
130C4×D4—21no—
131C4×D4C22⋊C412yes—
132C4×D4M1612yes—
133#117—24no—
134#117—24no—
135#117—22no—
136#117—22no—
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×C42—24no—
145C2×C42—24no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42C22⋊C411yescharacteristic
154C2×C42C2×C811yescharacteristic
155C2×C42M1611yescharacteristic
156C2×C42C22×C411yescharacteristic
157C2×C16—24no—
158C2×C16—42no—
159C2×C16—42no—
160C2×C16—42no—
161C2×C16—42no—
162C2×C16—42no—
163C4⋊C8—24no—
164C4⋊C8—24no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8C22⋊C412yes—
168C4⋊C8C22⋊C412yes—
169C22⋊C8—24no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172#97—24no—
173#97—24no—
174C4×C8—24no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8C2×D412yes—
180C4×C8C2×D412yes—
181C24—21no—
182C24C22⋊C811yescharacteristic
183#41—48no—
184#41—48no—
185C2×Q8—21no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8C22⋊C812yes—
189C2×D4—24no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4C22⋊C812yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4C22⋊C811yescharacteristic
226C22×C4C22⋊C811yescharacteristic
227C22×C4C22⋊C811yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C22×C811yescharacteristic
230C22×C4C2×M1611yescharacteristic
231M16—48no—
232M16—44no—
233M16—44no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—44no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8#10412yes—
251C2×C8#12012yes—
252C2×C8#12012yes—
253C2×C8#13512yes—
254C2×C8AGL(1,ℤ/8)11yescharacteristic
255C2×C8#13711yescharacteristic
256C4⋊C4—24no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4C22⋊C812yes—
262C4⋊C4C22⋊C812yes—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C42—24no—
268C42—24no—
269C42—21no—
270C42—22no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—21no—
276C42—21no—
277C42C2×C22⋊C411yescharacteristic
278C42C2×C22⋊C411yescharacteristic
279C42C2×M1611yescharacteristic
280C42C2×M1611yescharacteristic
281C16—44no—
282C16—82no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—22no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23#34912yes—
297C23C2×C22⋊C811yescharacteristic
298Q8—42no—
299Q8—42no—
300D4—42no—
301D4—42no—
302D4—48no—
303C2×C4—24no—
304C2×C4—24no—
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—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4#34912yes—
338C2×C4#35012yes—
339C2×C4#37911yescharacteristic
340C2×C4C2×C22⋊C811yescharacteristic
341C2×C4C2×C22⋊C811yescharacteristic
342C2×C4#40711yescharacteristic
343C2×C4#40711yescharacteristic
344C2×C4#40711yescharacteristic
345C2×C4#41611yescharacteristic
346C2×C4#41711yescharacteristic
347C2×C4C2×#10411yescharacteristic
348C2×C4#42111yescharacteristic
349C8—22no—
350C8—22no—
351C8—41no—
352C8—42no—
353C8—42no—
354C8#44912yes—
355C8#45212yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#147611yescharacteristic
371C22#147711yescharacteristic
372C22#177911yescharacteristic
373C22C2×#34911yescharacteristic
374C22C2×#34911yescharacteristic
375C22C2×#35011yescharacteristic
376C22#211711yescharacteristic
377C4—22no—
378C4—21no—
379C4—21no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4#148811yescharacteristic
388C4#148911yescharacteristic
389C4#178811yescharacteristic
390C4#178911yescharacteristic
391C2—42no—
392C2—82no—
393C2#831211yescharacteristic
394C2#1161711yescharacteristic
395C2#1161911yescharacteristic
396C2#1162611yescharacteristic
397C2#1162811yescharacteristic
398C2#1268111yescharacteristic
399C2#1268311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC22×#41C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #12599C2semidirect
4. C2×#1338C2semidirect
5. C2×#1330C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29652
#127832
#125992
C2×#13382
C2×#13302