Finite Group Databasebeta

← #121457

Subgroups — #121457

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21988C211yesmaximal, characteristic
2#21986C211yesmaximal, characteristic
3#20765C211yesmaximal, characteristic
4C2×#1291C211yesmaximal, characteristic
5#7914C211yesmaximal, characteristic
6#7900C211yesmaximal, characteristic
7#7896C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1949—22no—
16#1948—21no—
17#1947—21no—
18#1854—22no—
19#1853—21no—
20#1852—21no—
21#1847—21no—
22#1847—21no—
23#1836—21no—
24#1836—21no—
25#1836—21no—
26#1836—21no—
27#1835—22no—
28#1736—21no—
29#1736—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—
52C2×#107—22no—
53C2×#107—22no—
54C2×#106—22no—
55C2×#106—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8D412yes—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8D412yes—
62C4×M16—21no—
63C4×M16—21no—
64C4×M16C2×C412yes—
65C4×M16Q812yes—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97—21no—
69C2×C4×C8—21no—
70C2×C4×C8D412yes—
71#384D411yescharacteristic
72#384D411yescharacteristic
73#384D411yescharacteristic
74#384D411yescharacteristic
75#382—21no—
76#382—21no—
77#382—21no—
78#382—21no—
79#382—21no—
80#382—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C2×#95—21no—
90C2×#95—21no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—42no—
107C2×M16—42no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—41no—
112#117—44no—
113#117—82no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
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—42no—
131C2×C4⋊C4—42no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C42—41no—
135C2×C42—41no—
136C2×C42C22×C411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×Q811yescharacteristic
142C2×C42C2×Q811yescharacteristic
143C2×C42C2×Q811yescharacteristic
144C2×C42C2×Q811yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42#4111yescharacteristic
148#107—82no—
149#107—82no—
150#106—82no—
151#106—82no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—41no—
157C4⋊C8C4⋊C414yes—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8C4⋊C414yes—
164#97—22no—
165#97—22no—
166#97—22no—
167#97—22no—
168#97—22no—
169#97—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—41no—
173C4×C8C4⋊C414yes—
174#95—41no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182C24#12111yescharacteristic
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4C2×C4⋊C411yescharacteristic
202C22×C4C4×D411yescharacteristic
203C22×C4C4×Q811yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4C22⋊Q811yescharacteristic
210C22×C4C22⋊Q811yescharacteristic
211C22×C4C22⋊Q811yescharacteristic
212C22×C4#12311yescharacteristic
213C22×C4#12511yescharacteristic
214C22×C4#12511yescharacteristic
215C22×C4#12511yescharacteristic
216C22×C4#12711yescharacteristic
217C22×C4C4⋊Q811yescharacteristic
218C22×C4C4⋊Q811yescharacteristic
219M16—44no—
220M16—44no—
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—42no—
230C2×C8—42no—
231C2×C8—81no—
232C2×C8—81no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C42—82no—
253C42—82no—
254C42C2×C4⋊C411yescharacteristic
255C42C2×C4⋊C411yescharacteristic
256C42C4×D411yescharacteristic
257C42C4×D411yescharacteristic
258C42C22⋊Q811yescharacteristic
259C42C22⋊Q811yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42C22⋊Q811yescharacteristic
263C42C22⋊Q811yescharacteristic
264C42#12311yescharacteristic
265C42#12311yescharacteristic
266C42C4⋊Q811yescharacteristic
267C42C4⋊Q811yescharacteristic
268C42C4⋊Q811yescharacteristic
269C42C4⋊Q811yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23#38911yescharacteristic
277C23#39211yescharacteristic
278C23#39811yescharacteristic
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—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
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—162no—
320C2×C4—162no—
321C2×C4#38411yescharacteristic
322C2×C4#38511yescharacteristic
323C2×C4#38511yescharacteristic
324C2×C4#38511yescharacteristic
325C2×C4#38911yescharacteristic
326C2×C4#38911yescharacteristic
327C2×C4#39511yescharacteristic
328C2×C4#39511yescharacteristic
329C2×C4#39511yescharacteristic
330C2×C4#39811yescharacteristic
331C2×C4#39811yescharacteristic
332C2×C4#39911yescharacteristic
333C2×C4#45412yes—
334C2×C4#45512yes—
335C8—44no—
336C8—44no—
337C8—44no—
338C8—82no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22#144411yescharacteristic
350C22#200311yescharacteristic
351C22#200411yescharacteristic
352C22#200811yescharacteristic
353C22#200911yescharacteristic
354C22#201711yescharacteristic
355C22#201711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—161no—
367C4—161no—
368C4—161no—
369C4—161no—
370C4—161no—
371C4—161no—
372C4#186012yes—
373C4#206112yes—
374C2—41no—
375C2—41no—
376C2#1012211yescharacteristic
377C2#1012311yescharacteristic
378C2#1012411yescharacteristic
379C2#1012511yescharacteristic
380C2#1133311yescharacteristic
381C2#1133711yescharacteristic
382C2#1133911yescharacteristic
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
#207652
C2×#12912
#79142
#79002
#78962