Finite Group Databasebeta

← #121464

Subgroups — #121464

Order 512.

594 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4C2×#1302C211yesmaximal, characteristic
5#8040C211yesmaximal, characteristic
6#8034C211yesmaximal, characteristic
7#7914C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×#107—21no—
12C22×#106—21no—
13C22×#97C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1949—22no—
18#1947—22no—
19#1853—21no—
20#1852—22no—
21#1852—21no—
22#1847—22no—
23#1847—22no—
24#1847—21no—
25#1845—21no—
26#1836—21no—
27#1834—21no—
28#1302C414yescomplemented
29#1296—22no—
30#1295—22no—
31#1280—22no—
32C23×C8—21no—
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C42C2311yescharacteristic
40#432—21no—
41#432—21no—
42#432—21no—
43#432—21no—
44#432—21no—
45#432C2×C412yes—
46#432Q812yes—
47#432Q812yes—
48#431—21no—
49#431—21no—
50#431C2×C412yes—
51#431Q812yes—
52C2×#107—24no—
53C2×#107—24no—
54C2×#107—22no—
55C2×#107—44no—
56C2×#106—24no—
57C2×#106—22no—
58C2×#106—44no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8D412yes—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8D412yes—
65C2×#97—21no—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97C2×C412yes—
71C2×#97D412yes—
72C2×#97Q812yes—
73C2×C4×C8—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384D411yescharacteristic
78#384D411yescharacteristic
79#384D412yes—
80#382—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C2×#95—21no—
90C2×#95—21no—
91#340—24no—
92#340—22no—
93#340—22no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—41no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C22×C8—24no—
101C22×C8—24no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108#117—82no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—82no—
130C2×C4⋊C4—82no—
131C2×C22⋊C4—41no—
132C2×C42—41no—
133C2×C42C22×C411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×Q811yescharacteristic
139C2×C42C2×Q811yescharacteristic
140C2×C42C2×Q811yescharacteristic
141C2×C42C2×Q811yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145#107—48no—
146#107—84no—
147#107—82no—
148#106—48no—
149#106—82no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—41no—
155C4⋊C8C4⋊C414yes—
156C22⋊C8—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—41no—
161C22⋊C8C4⋊C414yes—
162#97—22no—
163#97—22no—
164#97—22no—
165#97—22no—
166#97—22no—
167#97—22no—
168#97—41no—
169#97C4⋊C414yes—
170C4×C8—22no—
171C4×C8—22no—
172#95—41no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—41no—
177#95—41no—
178C24#12111yescharacteristic
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4C4×D411yescharacteristic
201C22×C4C4×Q811yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4C22⋊Q811yescharacteristic
208C22×C4C22⋊Q811yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4#12311yescharacteristic
211C22×C4#12511yescharacteristic
212C22×C4#12511yescharacteristic
213C22×C4#12511yescharacteristic
214C22×C4#12711yescharacteristic
215C22×C4C4⋊Q811yescharacteristic
216C22×C4C4⋊Q811yescharacteristic
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—28no—
220C2×C8—28no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—81no—
230C2×C8—81no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—82no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C22⋊C4—82no—
249C42—82no—
250C42C2×C4⋊C411yescharacteristic
251C42C2×C4⋊C411yescharacteristic
252C42C4×D411yescharacteristic
253C42C4×D411yescharacteristic
254C42C22⋊Q811yescharacteristic
255C42C22⋊Q811yescharacteristic
256C42C22⋊Q811yescharacteristic
257C42C22⋊Q811yescharacteristic
258C42C22⋊Q811yescharacteristic
259C42C22⋊Q811yescharacteristic
260C42#12311yescharacteristic
261C42#12311yescharacteristic
262C42C4⋊Q811yescharacteristic
263C42C4⋊Q811yescharacteristic
264C42C4⋊Q811yescharacteristic
265C42C4⋊Q811yescharacteristic
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#38911yescharacteristic
271C23#39211yescharacteristic
272C23#39811yescharacteristic
273C23#47212yes—
274C23#47312yes—
275C2×C4—22no—
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—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—164no—
313C2×C4—162no—
314C2×C4—162no—
315C2×C4#38411yescharacteristic
316C2×C4#38511yescharacteristic
317C2×C4#38511yescharacteristic
318C2×C4#38511yescharacteristic
319C2×C4#38911yescharacteristic
320C2×C4#38911yescharacteristic
321C2×C4#39511yescharacteristic
322C2×C4#39511yescharacteristic
323C2×C4#39511yescharacteristic
324C2×C4#39811yescharacteristic
325C2×C4#39811yescharacteristic
326C2×C4#39911yescharacteristic
327C2×C4#45312yes—
328C2×C4#45612yes—
329C8—28no—
330C8—44no—
331C8—44no—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22#144411yescharacteristic
342C22#195212yes—
343C22#200211yescharacteristic
344C22#200511yescharacteristic
345C22#200811yescharacteristic
346C22#200911yescharacteristic
347C22#201811yescharacteristic
348C22#202011yescharacteristic
349C22#208912yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—161no—
361C4—161no—
362C4—162no—
363C4—161no—
364C4—161no—
365C4#185912yes—
366C4#206012yes—
367C2—22no—
368C2—22no—
369C2#1012811yescharacteristic
370C2#1013311yescharacteristic
371C2#1013911yescharacteristic
372C2#1014211yescharacteristic
373C2#1133111yescharacteristic
374C2#1133311yescharacteristic
375C2#1134011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1302C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#219822
C2×#18472
C2×#13022
#80402
#80342
#79142