Finite Group Databasebeta

← #120331

Subgroups — #120331

Order 512.

460 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21962C211yesmaximal, characteristic
2C2×#1573C211yesmaximal, characteristic
3#8044C211yesmaximal, characteristic
4#7984C211yesmaximal, characteristic
5#7981C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1949—22no—
16#1948—21no—
17#1947—21no—
18#1808—21no—
19#1807—21no—
20#1573—22no—
21#1573—21no—
22#1573—21no—
23C2×#335C412yescomplemented
24C2×#334C412yescomplemented
25#1296—22no—
26#1289—22no—
27#1289—21no—
28#1289—21no—
29#1285—22no—
30#1282—22no—
31C23×C8—21no—
32C2×#117—21no—
33C2×#117—21no—
34C22×C42C2311yescharacteristic
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432Q812yes—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47C2×#107—22no—
48C2×#106—22no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8C2×C411yescharacteristic
54C2×C4⋊C8C2×C411yescharacteristic
55C2×C4⋊C8C2×C412yes—
56C2×C4⋊C8D412yes—
57C2×C4⋊C8D411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59C2×C4⋊C8Q812yes—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8D412yes—
63C2×#97—21no—
64C2×#97—21no—
65C2×C4×C8C2×C412yes—
66#384—21no—
67#384—21no—
68#383—21no—
69#382—21no—
70#382—21no—
71#382Q812yes—
72#381—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380D411yescharacteristic
78#380D411yescharacteristic
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4D412yes—
84C4×C22⋊C4—21no—
85#376—21no—
86#340—22no—
87#340—22no—
88#335—22no—
89#335—22no—
90#334—22no—
91#334—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4#4111yescharacteristic
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—41no—
102C22×C8—82no—
103C22×C8C4⋊C414yes—
104#117—81no—
105#117—81no—
106#117—81no—
107#117—81no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C42—22no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×Q811yescharacteristic
128C2×C42C2×Q811yescharacteristic
129C2×C42C2×Q811yescharacteristic
130C2×C42#4111yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134C2×C42#4111yescharacteristic
135#107—82no—
136#106—82no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—22no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—41no—
149C4⋊C8C22⋊C412yes—
150C4⋊C8C22⋊C412yes—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157#97—22no—
158#97—41no—
159#97—41no—
160#97—41no—
161C4×C8—22no—
162C4×C8—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172C24#12111yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C2×C4⋊C411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×Q811yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12511yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4C4⋊Q811yescharacteristic
202C2×C8—24no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—44no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—81no—
215C2×C8—81no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C42—82no—
237C42—82no—
238C42—82no—
239C42C2×C22⋊C411yescharacteristic
240C42#11711yescharacteristic
241C42C4×D411yescharacteristic
242C42C4×Q811yescharacteristic
243C42#12111yescharacteristic
244C42C22⋊Q811yescharacteristic
245C42C22⋊Q811yescharacteristic
246C42C22⋊Q811yescharacteristic
247C42C22⋊Q811yescharacteristic
248C42#12311yescharacteristic
249C42#12511yescharacteristic
250C42#12511yescharacteristic
251C42#12511yescharacteristic
252C42#12611yescharacteristic
253C42C4⋊Q811yescharacteristic
254C42C4⋊Q811yescharacteristic
255C23—41no—
256C23—41no—
257C23#38911yescharacteristic
258C23#39411yescharacteristic
259C23#39611yescharacteristic
260C23#47111yescharacteristic
261C23#47111yescharacteristic
262C23#47211yescharacteristic
263C23#47311yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4#38011yescharacteristic
304C2×C4#38611yescharacteristic
305C2×C4#38711yescharacteristic
306C2×C4#38711yescharacteristic
307C2×C4#38911yescharacteristic
308C2×C4#39111yescharacteristic
309C2×C4#39511yescharacteristic
310C2×C4#39611yescharacteristic
311C2×C4#39811yescharacteristic
312C2×C4#39811yescharacteristic
313C2×C4#39811yescharacteristic
314C2×C4#40011yescharacteristic
315C8—44no—
316C8—42no—
317C8—42no—
318C8—82no—
319C8—81no—
320C8—81no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22#144111yescharacteristic
332C22#195111yescharacteristic
333C22#195211yescharacteristic
334C22#205111yescharacteristic
335C22#205211yescharacteristic
336C22#206311yescharacteristic
337C22#206411yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—161no—
349C4—162no—
350C4—162no—
351C4—161no—
352C4#181212yes—
353C4#190411yescharacteristic
354C4#190511yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2#1121311yescharacteristic
362C2#1122911yescharacteristic
363C2#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
23. C2×#335C4semidirect
24. C2×#334C4semidirect

Maximal subgroups

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

subgroupindex
#219622
C2×#15732
#80442
#79842
#79812
#79522
#79282