Finite Group Databasebeta

← #121456

Subgroups — #121456

Order 512.

566 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21988C211yesmaximal, characteristic
2#21986C211yesmaximal, characteristic
3#20801C211yesmaximal, characteristic
4C2×#1291C211yesmaximal, characteristic
5#7914C211yesmaximal, characteristic
6#7900C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#428—21no—
12C2×#428—21no—
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#2285C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1949—22no—
18#1948—21no—
19#1947—21no—
20#1854—22no—
21#1853—21no—
22#1852—21no—
23#1847—21no—
24#1847—21no—
25#1847—21no—
26#1847—21no—
27#1846—22no—
28#1836—21no—
29#1836—21no—
30#1291C414yescomplemented
31#1280—22no—
32#1279—22no—
33#1279—22no—
34C22×M16—21no—
35C2×#117—21no—
36C2×#117—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C42C2311yescharacteristic
40#432—21no—
41#432—21no—
42#432—21no—
43#432—21no—
44#432C2×C412yes—
45#432C2×C412yes—
46#432Q812yes—
47#432Q812yes—
48#431—21no—
49#431—21no—
50#431—21no—
51#431Q812yes—
52#428—44no—
53#428—44no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8D412yes—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8D412yes—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16C2×C412yes—
71C4×M16Q812yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8D412yes—
77#384D411yescharacteristic
78#384D411yescharacteristic
79#384D411yescharacteristic
80#384D411yescharacteristic
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C2×#95—21no—
96C2×#95—21no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—22no—
109C2×M16—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114#117—42no—
115#117—42no—
116#117—82no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42C22×C411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×Q811yescharacteristic
145C2×C42C2×Q811yescharacteristic
146C2×C42C2×Q811yescharacteristic
147C2×C42C2×Q811yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151#107—44no—
152#107—44no—
153#107—82no—
154#107—82no—
155#106—44no—
156#106—44no—
157#106—82no—
158#106—82no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—41no—
164C4⋊C8C4⋊C414yes—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—41no—
170C22⋊C8C4⋊C414yes—
171#97—22no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—22no—
176#97—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—41no—
180C4×C8C4⋊C414yes—
181#95—41no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189C24#12111yescharacteristic
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×Q811yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12511yescharacteristic
221C22×C4#12511yescharacteristic
222C22×C4#12511yescharacteristic
223C22×C4#12711yescharacteristic
224C22×C4C4⋊Q811yescharacteristic
225C22×C4C4⋊Q811yescharacteristic
226M16—28no—
227M16—28no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—81no—
239C2×C8—81no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C42—82no—
262C42—82no—
263C42C2×C4⋊C411yescharacteristic
264C42C2×C4⋊C411yescharacteristic
265C42C4×D411yescharacteristic
266C42C4×D411yescharacteristic
267C42C22⋊Q811yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42#12311yescharacteristic
274C42#12311yescharacteristic
275C42C4⋊Q811yescharacteristic
276C42C4⋊Q811yescharacteristic
277C42C4⋊Q811yescharacteristic
278C42C4⋊Q811yescharacteristic
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23#38911yescharacteristic
286C23#39211yescharacteristic
287C23#39811yescharacteristic
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—162no—
329C2×C4—162no—
330C2×C4#38411yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38911yescharacteristic
335C2×C4#38911yescharacteristic
336C2×C4#39511yescharacteristic
337C2×C4#39511yescharacteristic
338C2×C4#39511yescharacteristic
339C2×C4#39811yescharacteristic
340C2×C4#39811yescharacteristic
341C2×C4#39911yescharacteristic
342C2×C4#45312yes—
343C2×C4#45612yes—
344C8—28no—
345C8—44no—
346C8—44no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22#144411yescharacteristic
359C22#200211yescharacteristic
360C22#200511yescharacteristic
361C22#200811yescharacteristic
362C22#200911yescharacteristic
363C22#201711yescharacteristic
364C22#201711yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—161no—
376C4—161no—
377C4—161no—
378C4—161no—
379C4—161no—
380C4—161no—
381C4#185912yes—
382C4#206012yes—
383C2—41no—
384C2—41no—
385C2#1012011yescharacteristic
386C2#1012111yescharacteristic
387C2#1012611yescharacteristic
388C2#1012711yescharacteristic
389C2#1133311yescharacteristic
390C2#1133711yescharacteristic
391C2#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
30. #1291C4semidirect

Maximal subgroups

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

subgroupindex
#219882
#219862
#208012
C2×#12912
#79142
#79002
#78962