Finite Group Databasebeta

← #122134

Subgroups — #122134

Order 512.

605 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35202C211yesmaximal, characteristic
2#8094C211yesmaximal, characteristic
3#8069C211yesmaximal, characteristic
4#7984C211yesmaximal, characteristic
5#7949C211yesmaximal, characteristic
6#7944C211yesmaximal, characteristic
7#7940C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11#2864C412yescomplemented
12#2859C412yescomplemented
13#2667—22no—
14#2606—22no—
15#2552—21no—
16#2552—21no—
17#2441—22no—
18#2441—22no—
19#2330—22no—
20#2285C2211yescharacteristic
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1303—22no—
25#1300—22no—
26#1289—22no—
27#1285—21no—
28#1285—21no—
29#1284—22no—
30#1284—22no—
31C2×#125—22no—
32C2×#125—22no—
33C2×#117—21no—
34C2×#117—21no—
35C22×C42C2311yescharacteristic
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432Q812yes—
43#432Q812yes—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#423—21no—
49#423—21no—
50C2×C4⋊C8Q812yes—
51#407—21no—
52#407—21no—
53C2×C22⋊C8—21no—
54C4×M16—21no—
55C4×M16—21no—
56C2×#97—21no—
57#399—24no—
58#398—24no—
59#398—24no—
60#396—24no—
61#389—24no—
62#389—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—24no—
67#389—24no—
68#387—24no—
69#387—24no—
70#387—24no—
71#384—22no—
72#384—22no—
73#384—21no—
74#383—21no—
75#383—21no—
76#382—21no—
77#382C2×C411yescharacteristic
78#382C2×C412yes—
79#382C2×C411yescharacteristic
80#381—21no—
81#381D412yes—
82#380—21no—
83#380—21no—
84#380—21no—
85#380D412yes—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4D411yescharacteristic
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4D412yes—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4D412yes—
97#376—21no—
98#376D412yes—
99#376Q812yes—
100C43C2×C412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—41no—
105C2×M16—41no—
106C22×C8—22no—
107#125—48no—
108#125—84no—
109#117—44no—
110#117—82no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—41no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—41no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42C22⋊C412yes—
135C2×C42C22⋊C412yes—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×Q811yescharacteristic
141C2×C42C2×Q811yescharacteristic
142C2×C42C2×Q811yescharacteristic
143C2×C42C2×Q811yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42#4111yescharacteristic
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154C22⋊C8—22no—
155C22⋊C8—22no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160#97—22no—
161#97—22no—
162#97—41no—
163#97—41no—
164C4×C8—41no—
165C4×C8—41no—
166#95—24no—
167#95—24no—
168#95—21no—
169#95—21no—
170#95—22no—
171#95—21no—
172#95—21no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—41no—
179#95—41no—
180#95C22⋊C412yes—
181#95C22⋊C412yes—
182#95C22⋊C414yes—
183C24#12011yescharacteristic
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4C2×C22⋊C411yescharacteristic
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4C2×C22⋊C411yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12011yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4C22⋊Q811yescharacteristic
205C22×C4C22⋊Q811yescharacteristic
206C22×C4#12411yescharacteristic
207C22×C4#12411yescharacteristic
208C22×C4#12411yescharacteristic
209C22×C4C4⋊Q811yescharacteristic
210C22×C4C4⋊Q811yescharacteristic
211C22×C4C4⋊Q811yescharacteristic
212M16—81no—
213M16—81no—
214M16—82no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—82no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C4⋊C4—42no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C22⋊C4—44no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—42no—
239C42—42no—
240C42—44no—
241C42—44no—
242C42—82no—
243C42—82no—
244C42—82no—
245C42C4×Q811yescharacteristic
246C42C4×Q811yescharacteristic
247C42C4×Q811yescharacteristic
248C42C4×Q811yescharacteristic
249C42C22⋊Q811yescharacteristic
250C42C22⋊Q811yescharacteristic
251C42C22⋊Q811yescharacteristic
252C42C22⋊Q811yescharacteristic
253C42C22⋊Q811yescharacteristic
254C42C22⋊Q811yescharacteristic
255C42#12411yescharacteristic
256C42#12411yescharacteristic
257C42#12411yescharacteristic
258C42C4⋊Q811yescharacteristic
259C42C4⋊Q811yescharacteristic
260C42C4⋊Q811yescharacteristic
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23#37911yescharacteristic
266C23#39311yescharacteristic
267C23#39511yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#39111yescharacteristic
307C2×C4#39111yescharacteristic
308C2×C4#39111yescharacteristic
309C2×C4#39111yescharacteristic
310C2×C4#39111yescharacteristic
311C2×C4#39111yescharacteristic
312C2×C4#39311yescharacteristic
313C2×C4#39311yescharacteristic
314C2×C4#39311yescharacteristic
315C2×C4#39511yescharacteristic
316C2×C4#39511yescharacteristic
317C2×C4#39511yescharacteristic
318C8—82no—
319C8—82no—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—81no—
330C22—81no—
331C22#144611yescharacteristic
332C22#179511yescharacteristic
333C22#179611yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—44no—
346C4—82no—
347C4—82no—
348C4—82no—
349C4—84no—
350C4—162no—
351C4#203112yes—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2#1138711yescharacteristic
357C2#1138811yescharacteristic
358C2#1138911yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
11. #2864C4semidirect
12. #2859C4semidirect

Maximal subgroups

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

subgroupindex
#352022
#80942
#80692
#79842
#79492
#79442
#79402