Finite Group Databasebeta

← #120374

Subgroups — #120374

Order 512.

477 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21363C211yesmaximal, characteristic
2#17818C211yesmaximal, characteristic
3#8014C211yesmaximal, characteristic
4#7984C211yesmaximal, characteristic
5#7952C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7926C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18#1776—22no—
19#1573—22no—
20#1573C414yescomplemented
21#1568—22no—
22#1294—22no—
23#1289—21no—
24#1289—21no—
25#1285—22no—
26#1285—22no—
27#1282—22no—
28C2×#117—21no—
29C22×C4⋊C4—21no—
30C22×C42—21no—
31C22×C42C2311yescharacteristic
32#432—21no—
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432C2×C412yes—
43#432D412yes—
44#431—21no—
45#431—21no—
46#431C2×C412yes—
47#431Q812yes—
48C2×#107—22no—
49C2×#106—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8C2×C412yes—
53C2×C4⋊C8D412yes—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D412yes—
58C2×#97—21no—
59C2×#97—21no—
60C2×#97Q812yes—
61C2×C4×C8—21no—
62#384—21no—
63#383—21no—
64#382—21no—
65#382—21no—
66#381—21no—
67#380—21no—
68#380—21no—
69#380D411yescharacteristic
70#380D411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4D412yes—
77C4×C4⋊C4Q812yes—
78C4×C22⋊C4—21no—
79#376—21no—
80C2×#95—21no—
81#340—22no—
82#340—22no—
83#335—24no—
84#334—24no—
85C23×C4—41no—
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93#117—81no—
94#117—81no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—41no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—81no—
111C2×C42—81no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×Q811yescharacteristic
117C2×C42C2×Q811yescharacteristic
118C2×C42C2×Q811yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42#4111yescharacteristic
122C2×C42#4111yescharacteristic
123C2×C42#4111yescharacteristic
124#107—44no—
125#106—44no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8C22⋊C414yes—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8C4⋊C414yes—
142#97—22no—
143#97—22no—
144#97—22no—
145#97—22no—
146#97—41no—
147#97—41no—
148C4×C8—22no—
149C4×C8—22no—
150#95—22no—
151#95—22no—
152#95—21no—
153#95—21no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159C24#12111yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—82no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×C4⋊C411yescharacteristic
175C22×C4C4×D411yescharacteristic
176C22×C4C4×Q811yescharacteristic
177C22×C4#12011yescharacteristic
178C22×C4#12111yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12411yescharacteristic
189C22×C4#12511yescharacteristic
190C22×C4#12611yescharacteristic
191C22×C4C4⋊Q811yescharacteristic
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—24no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—44no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—82no—
216C42—82no—
217C42—82no—
218C42—82no—
219C42C2×C22⋊C411yescharacteristic
220C42#11711yescharacteristic
221C42C4×D411yescharacteristic
222C42C4×Q811yescharacteristic
223C42#12111yescharacteristic
224C42C22⋊Q811yescharacteristic
225C42C22⋊Q811yescharacteristic
226C42C22⋊Q811yescharacteristic
227C42C22⋊Q811yescharacteristic
228C42#12311yescharacteristic
229C42#12511yescharacteristic
230C42#12511yescharacteristic
231C42#12511yescharacteristic
232C42#12611yescharacteristic
233C42C4⋊Q811yescharacteristic
234C42C4⋊Q811yescharacteristic
235C23—41no—
236C23—41no—
237C23#38911yescharacteristic
238C23#39411yescharacteristic
239C23#39611yescharacteristic
240C23#47111yescharacteristic
241C23#47111yescharacteristic
242C23#47211yescharacteristic
243C23#47311yescharacteristic
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—82no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4#38011yescharacteristic
283C2×C4#38611yescharacteristic
284C2×C4#38711yescharacteristic
285C2×C4#38711yescharacteristic
286C2×C4#38911yescharacteristic
287C2×C4#39111yescharacteristic
288C2×C4#39511yescharacteristic
289C2×C4#39611yescharacteristic
290C2×C4#39811yescharacteristic
291C2×C4#39811yescharacteristic
292C2×C4#39811yescharacteristic
293C2×C4#40011yescharacteristic
294C8—44no—
295C8—44no—
296C8—82no—
297C8—82no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#144111yescharacteristic
309C22#195111yescharacteristic
310C22#195211yescharacteristic
311C22#205111yescharacteristic
312C22#205211yescharacteristic
313C22#206311yescharacteristic
314C22#206411yescharacteristic
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—84no—
327C4—161no—
328C4—162no—
329C4—161no—
330C4#190612yes—
331C2—21no—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2#1121511yescharacteristic
338C2#1123411yescharacteristic
339C2#1124911yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
20. #1573C4semidirect

Maximal subgroups

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

subgroupindex
#213632
#178182
#80142
#79842
#79522
C2×#12852
#79262