Finite Group Databasebeta

← #96739

Subgroups — #96739

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#391C211yesmaximal, characteristic
2#10693C212yesmaximal
3C2×#1272C212yesmaximal
4#7025C212yesmaximal
5C22×C4×Q8C411yescharacteristic
6C22×#97—22no—
7C2×#391C411yescharacteristic
8C2×#391C2211yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10C2×C43C2211yescharacteristic
11C2×#336—22no—
12C2×#336C2214yes—
13#1436—22no—
14C4⋊#95C411yescharacteristic
15C4⋊#95C2211yescharacteristic
16C4×#95—22no—
17#1272C418yescomplemented
18C23×Q8C2×C411yescharacteristic
19C23×C8—24no—
20C2×C4×Q8—22no—
21C2×C4×Q8—22no—
22C2×C4×Q8—42no—
23C2×C4×Q8—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C42—21no—
29C22×C42—21no—
30C22×C42C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D412yes—
33C22×C42D412yes—
34C22×C42C2311yescharacteristic
35C2×#97—28no—
36C2×#97—44no—
37C2×#97—44no—
38#391—22no—
39#391—22no—
40#391—22no—
41#391C2×C414yes—
42C4×C4⋊C4—42no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—21no—
47C2×#95—22no—
48C43—42no—
49#336—28no—
50#336—24no—
51#336C2×C418yes—
52C22×Q8—21no—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—41no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8C4212yes—
60C22×Q8M1612yes—
61C22×Q8M1612yes—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C412yes—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D412yes—
71C22×C8—28no—
72C22×C8—28no—
73C22×C8—28no—
74C22×C8—44no—
75C22×C8—44no—
76C22×C8—44no—
77C22×C8—44no—
78C4×Q8—48no—
79C4×Q8—48no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4C4212yes—
93C2×C4⋊C4M1612yes—
94C2×C4⋊C4M1612yes—
95C2×C42—22no—
96C2×C42—24no—
97C2×C42—22no—
98C2×C42—24no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—24no—
104C2×C42—24no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—24no—
108C2×C42—41no—
109C2×C42—42no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42—42no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—44no—
117C2×C42C22×C412yes—
118C2×C42#4112yes—
119#97—416no—
120#95—28no—
121#95—28no—
122#95—42no—
123#95—44no—
124#95—42no—
125#95—42no—
126C24C2×C22⋊C411yescharacteristic
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8#9714yescomplemented
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#9912yes—
187C22×C4#10012yes—
188C22×C4#10414yes—
189C22×C4#10414yes—
190C22×C4C2×C4211yescharacteristic
191C22×C4#11711yescharacteristic
192C22×C4C4×D414yes—
193C22×C4C2×M1611yescharacteristic
194C22×C4C2×M1611yescharacteristic
195C22×C4C2×M1611yescharacteristic
196C22×C4C2×M1611yescharacteristic
197C22×C4AGL(1,ℤ/8)12yes—
198C22×C4#13714yes—
199C22×C4#13712yes—
200C2×C8—216no—
201C2×C8—48no—
202C2×C8—48no—
203C2×C8—48no—
204C2×C8—48no—
205C2×C8—48no—
206C2×C8—48no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—48no—
214C42—28no—
215C42—28no—
216C42—24no—
217C42—24no—
218C42—42no—
219C42—42no—
220C42—48no—
221C42—42no—
222C42—42no—
223C42—48no—
224C42—84no—
225C23C4×C22⋊C411yescharacteristic
226C23#40711yescharacteristic
227C23#40711yescharacteristic
228C23C2×#9911yescharacteristic
229C23#41011yescharacteristic
230C23C2×#10011yescharacteristic
231C23#41311yescharacteristic
232C23#41712yes—
233C23#41912yes—
234C23C2×#10412yes—
235C23C2×#10412yes—
236Q8—44no—
237Q8—44no—
238Q8—44no—
239Q8—44no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#32914yes—
299C2×C4C2×#9711yescharacteristic
300C2×C4C4×M1611yescharacteristic
301C2×C4#43211yescharacteristic
302C2×C4#43211yescharacteristic
303C2×C4#44014yes—
304C2×C4#44114yes—
305C8—416no—
306C22#146211yescharacteristic
307C22#146411yescharacteristic
308C22#146511yescharacteristic
309C22#146611yescharacteristic
310C22#148912yes—
311C22#148912yes—
312C22#149112yes—
313C22#149112yes—
314C22#150412yes—
315C22#150612yes—
316C22#150812yes—
317C22#150912yes—
318C22C2×#32911yescharacteristic
319C22#152412yes—
320C22#152511yescharacteristic
321C22#155811yescharacteristic
322C22#155912yes—
323C22#156011yescharacteristic
324C22#174911yescharacteristic
325C22C4×#9911yescharacteristic
326C22C4×#10011yescharacteristic
327C22C4×#10412yes—
328C22#176312yes—
329C4—24no—
330C4—24no—
331C4—24no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—44no—
338C4—44no—
339C4—82no—
340C4—82no—
341C4—82no—
342C2#762911yescharacteristic
343C2#763012yes—
344C2#763111yescharacteristic
345C2#763412yes—
346C2#763511yescharacteristic
347C2#763911yescharacteristic
348C2#815211yescharacteristic
349C2#817312yes—
350C2#821912yes—
351C2C4×#32911yescharacteristic
352C2#823211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C2×C22⋊C4
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #1272C4semidirect
142. C2×Q8#97semidirect

Maximal subgroups

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

subgroupindex
C4×#3912
#106932
C2×#12722
#70252