Finite Group Databasebeta

← #120589

Subgroups — #120589

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21371C211yesmaximal, characteristic
2C4×#428C211yesmaximal, characteristic
3#8098C211yesmaximal, characteristic
4#8071C211yesmaximal, characteristic
5#7948C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7937C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#428—21no—
13C2×C4×M16C2211yescharacteristic
14C4×#117C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1950C412yescomplemented
17#1950C412yescomplemented
18#1855—21no—
19#1855—21no—
20#1778—22no—
21#1776—22no—
22C4×#107—21no—
23C4×#106—21no—
24#1736—21no—
25#1303—22no—
26#1300—22no—
27#1285—22no—
28#1285—21no—
29#1285—21no—
30#1283—22no—
31C22×M16D411yescharacteristic
32C22×M16D411yescharacteristic
33C2×#117—21no—
34C22×C4⋊C4—21no—
35C22×C42—21no—
36C22×C42C2311yescharacteristic
37#432—21no—
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432—21no—
43#432—21no—
44#432—21no—
45#432—21no—
46#432D412yes—
47#432Q812yes—
48#432Q812yes—
49#431—21no—
50#431—21no—
51#431—21no—
52#431—21no—
53C2×#108—22no—
54C2×#108—22no—
55#428—28no—
56C2×#107—22no—
57C2×#106—22no—
58#423—21no—
59#423C2×C411yescharacteristic
60#423C2×C411yescharacteristic
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63#407—21no—
64#407—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16C2×C412yes—
72C4×M16D412yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97Q812yes—
76C2×C4×C8—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4D411yescharacteristic
82C4×C4⋊C4D411yescharacteristic
83C4×C4⋊C4D412yes—
84C4×C22⋊C4—21no—
85C4×C22⋊C4—21no—
86#376—21no—
87C2×#95—21no—
88C43—21no—
89#341—22no—
90#341—22no—
91#340—22no—
92#340—22no—
93C23×C4—41no—
94C23×C4C2×D411yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C2×M16—21no—
98C2×M16—24no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—42no—
103C2×M16—41no—
104C2×M16C22⋊C412yes—
105C2×M16C22⋊C412yes—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8—22no—
110#117—24no—
111#117—42no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C42—22no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42—82no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×Q811yescharacteristic
132C2×C42C2×Q811yescharacteristic
133C2×C42C2×Q811yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4111yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138#108—44no—
139#108—44no—
140#107—28no—
141#106—28no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—22no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8C4⋊C414yes—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159C22⋊C8—41no—
160#97—22no—
161#97—22no—
162#97—22no—
163#97—22no—
164#97—22no—
165#97—22no—
166#97—22no—
167#97—41no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—41no—
171C4×C8—41no—
172#95—21no—
173#95—21no—
174#95—22no—
175#95—41no—
176#95—41no—
177C24#12411yescharacteristic
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—44no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—81no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4C2×C22⋊C411yescharacteristic
190C22×C4C4×Q811yescharacteristic
191C22×C4C4×Q811yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4C4⋊Q811yescharacteristic
207M16—24no—
208M16—22no—
209M16—22no—
210M16—22no—
211M16—42no—
212M16—42no—
213M16—42no—
214M16—81no—
215M16—81no—
216M16#10414yes—
217C2×C8—24no—
218C2×C8—22no—
219C2×C8—24no—
220C2×C8—22no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—82no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—44no—
237C42—44no—
238C42—81no—
239C42—81no—
240C42—82no—
241C42—82no—
242C42—82no—
243C42C2×C4⋊C411yescharacteristic
244C42#11711yescharacteristic
245C42C4×D411yescharacteristic
246C42C4×D411yescharacteristic
247C42#12111yescharacteristic
248C42#12111yescharacteristic
249C42#12111yescharacteristic
250C42C22⋊Q811yescharacteristic
251C42C22⋊Q811yescharacteristic
252C42C22⋊Q811yescharacteristic
253C42C22⋊Q811yescharacteristic
254C42C22⋊Q811yescharacteristic
255C42#12511yescharacteristic
256C42#12511yescharacteristic
257C42C4⋊Q811yescharacteristic
258C42C4⋊Q811yescharacteristic
259C23—21no—
260C23—21no—
261C23—22no—
262C23—41no—
263C23#39111yescharacteristic
264C23#39411yescharacteristic
265C23#40011yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—88no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4#38011yescharacteristic
298C2×C4#38611yescharacteristic
299C2×C4#38711yescharacteristic
300C2×C4#38711yescharacteristic
301C2×C4#38911yescharacteristic
302C2×C4#38911yescharacteristic
303C2×C4#39511yescharacteristic
304C2×C4#39611yescharacteristic
305C2×C4#39611yescharacteristic
306C2×C4#39811yescharacteristic
307C2×C4#39811yescharacteristic
308C2×C4#39811yescharacteristic
309C2×C4C2×#10412yes—
310C2×C4#42112yes—
311C8—24no—
312C8—44no—
313C8—42no—
314C8—82no—
315C8—82no—
316C8—81no—
317C8—81no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#144111yescharacteristic
328C22#198612yes—
329C22#204811yescharacteristic
330C22#204911yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—82no—
342C4—82no—
343C4—84no—
344C4#180612yes—
345C4#189611yescharacteristic
346C4#189711yescharacteristic
347C2—21no—
348C2—21no—
349C2—22no—
350C2—41no—
351C2#1122611yescharacteristic
352C2#1124711yescharacteristic
353C2#1125311yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
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
16. #1950C4semidirect
17. #1950C4semidirect

Maximal subgroups

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

subgroupindex
#213712
C4×#4282
#80982
#80712
#79482
C2×#12852
#79372