Finite Group Databasebeta

← #120160

Subgroups — #120160

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20429C211yesmaximal, characteristic
2#14164C211yesmaximal, characteristic
3C2×#1288C212yesmaximal
4#7910C211yesmaximal, characteristic
5#7904C212yesmaximal
6C2×#432C2212yes—
7C2×#423—21no—
8C2×C4×M16C2212yes—
9C2×#383C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11C2×#376C2211yescharacteristic
12#1768—24no—
13#1728—21no—
14#1444C411yescharacteristic
15#1444C411yescharacteristic
16#1442C411yescharacteristic
17#1442C411yescharacteristic
18#1436—21no—
19#1436—21no—
20#1436—21no—
21#1436—21no—
22#1288—24no—
23#1280—21no—
24#1280—21no—
25#1279—24no—
26C22×M16—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C42—21no—
32C22×C42D411yescharacteristic
33C22×C42D411yescharacteristic
34C22×C42C2311yescharacteristic
35#432—22no—
36#432—22no—
37#432—22no—
38#432Q814yes—
39#423—24no—
40#423—24no—
41C2×C4⋊C8—24no—
42C2×C4⋊C8—22no—
43C2×C22⋊C8—22no—
44C4×M16—22no—
45C4×M16—22no—
46C4×M16—22no—
47C4×M16—22no—
48C2×#97—22no—
49C2×#97—22no—
50C2×C4×C8D414yes—
51#383—21no—
52#383—21no—
53#383D412yes—
54#383Q812yes—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57#376—22no—
58#376—22no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95C2×C411yescharacteristic
69C2×#95C2×C411yescharacteristic
70C2×#95C2×C411yescharacteristic
71C2×#95C2×C411yescharacteristic
72C2×#95D411yescharacteristic
73C2×#95D411yescharacteristic
74#336—24no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×M16—24no—
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—42no—
91C2×M16—42no—
92C22×C8—24no—
93C22×C8—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—81no—
103C2×C4⋊C4—81no—
104C2×C42—22no—
105C2×C42—22no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—41no—
111C2×C42—41no—
112C2×C42—42no—
113C2×C42—81no—
114C2×C42—81no—
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2×Q812yes—
118C2×C42C2×Q811yescharacteristic
119C2×C42#4112yes—
120C2×C42#4111yescharacteristic
121C2×C42#4112yes—
122C2×C42#4111yescharacteristic
123C4⋊C8—216no—
124C4⋊C8—24no—
125C4⋊C8—42no—
126C22⋊C8—24no—
127C22⋊C8—42no—
128#97—24no—
129#97—42no—
130#97—42no—
131#97—42no—
132C4×C8—24no—
133C4×C8—24no—
134#95—21no—
135#95—21no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95—22no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—41no—
149#95—41no—
150#95—81no—
151#95—81no—
152#95C4⋊C412yes—
153#95C4⋊C412yes—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
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—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4#10412yes—
210C22×C4#10412yes—
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4#11711yescharacteristic
213C22×C4C4×D412yes—
214C22×C4C4×Q812yes—
215C22×C4#12011yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12112yes—
218C22×C4#12111yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q812yes—
221C22×C4C22⋊Q811yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4C4⋊Q811yescharacteristic
225M16—44no—
226M16—44no—
227M16—44no—
228M16—44no—
229C2×C8—28no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—82no—
234C2×C8—82no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C42—24no—
242C42—24no—
243C42—42no—
244C42—42no—
245C42—84no—
246C42#12112yes—
247C42C22⋊Q812yes—
248C42C22⋊Q812yes—
249C42#12312yes—
250C42#12512yes—
251C42#12512yes—
252C42#12612yes—
253C42C4⋊Q812yes—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23#38011yescharacteristic
259C23#38611yescharacteristic
260C23#39111yescharacteristic
261C23C2×#10411yescharacteristic
262C23C2×#10411yescharacteristic
263C23#42111yescharacteristic
264C23#42111yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
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—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4#38712yes—
336C2×C4#38912yes—
337C2×C4#39411yescharacteristic
338C2×C4#39511yescharacteristic
339C2×C4#39612yes—
340C2×C4#39811yescharacteristic
341C2×C4#39812yes—
342C2×C4#40011yescharacteristic
343C8—48no—
344C8—84no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22#144111yescharacteristic
358C22#180611yescharacteristic
359C22#180611yescharacteristic
360C22#189111yescharacteristic
361C22#189111yescharacteristic
362C22#189611yescharacteristic
363C22#189711yescharacteristic
364C22#190911yescharacteristic
365C22#191011yescharacteristic
366C22#198611yescharacteristic
367C22#198611yescharacteristic
368C4—22no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—82no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—82no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2#949111yescharacteristic
389C2#949211yescharacteristic
390C2#949311yescharacteristic
391C2#949411yescharacteristic
392C2#1122311yescharacteristic
393C2#1122511yescharacteristic
394C2#1122611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#204292
#141642
C2×#12882
#79102
#79042