Finite Group Databasebeta

← #121509

Subgroups — #121509

Order 512.

537 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21953C211yesmaximal, characteristic
2#17822C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#10852C211yesmaximal, characteristic
5#8071C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1940—21no—
16#1940—21no—
17#1939—21no—
18#1939—21no—
19#1797—22no—
20C2×#339—22no—
21#1574—22no—
22#1573—22no—
23#1571—21no—
24#1571—21no—
25#1570—21no—
26#1570—21no—
27#1569—22no—
28#1565—22no—
29#1300—22no—
30#1285C412yescomplemented
31#1285C412yescomplemented
32#1276—22no—
33C22×M16—21no—
34C2×#117—21no—
35C22×C4⋊C4—21no—
36C22×C42D412yes—
37C22×C42C2311yescharacteristic
38#432—21no—
39#432—21no—
40#432—21no—
41#432—21no—
42#432—21no—
43#432C2×C412yes—
44#432C2×C412yes—
45#432Q812yes—
46#431—21no—
47#431—21no—
48#431—21no—
49#431—21no—
50#431—21no—
51#431—21no—
52#431—21no—
53#431D412yes—
54#423—21no—
55#423—21no—
56#423—21no—
57#423—21no—
58#423D412yes—
59#423Q812yes—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8D411yescharacteristic
65C2×C4⋊C8D411yescharacteristic
66C2×C4⋊C8D411yescharacteristic
67C2×C4⋊C8D411yescharacteristic
68#407—21no—
69#407—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16D412yes—
74C2×#97—21no—
75C2×C4×C8—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4C2×C411yescharacteristic
80C4×C4⋊C4C2×C411yescharacteristic
81C4×C4⋊C4Q812yes—
82C4×C4⋊C4Q812yes—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85#376—21no—
86C2×#95—21no—
87C43—21no—
88#339—44no—
89#336—22no—
90#336—22no—
91#335—24no—
92#335—24no—
93#334—24no—
94#334—24no—
95C23×C4—22no—
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—41no—
103C2×M16—42no—
104C2×M16—42no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—41no—
110#117—22no—
111#117—22no—
112#117—41no—
113#117—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—44no—
119C2×C22⋊C4—41no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—41no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42—41no—
126C2×C42C4⋊C412yes—
127C2×C42C4⋊C412yes—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×Q811yescharacteristic
135C2×C42C2×Q811yescharacteristic
136C2×C42C2×Q811yescharacteristic
137C2×C42C2×Q811yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42#4111yescharacteristic
140C4⋊C8—22no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—41no—
154C4⋊C8—41no—
155C4⋊C8—82no—
156C4⋊C8—82no—
157C4⋊C8C4⋊C414yes—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166#97—22no—
167#97—41no—
168#97—41no—
169#97C4⋊C414yes—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—41no—
173C4×C8—41no—
174#95—21no—
175#95—21no—
176#95—22no—
177#95—22no—
178#95—41no—
179C24#12311yescharacteristic
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—82no—
192C22×C4C2×C4⋊C411yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4C22⋊Q811yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12711yescharacteristic
209C22×C4C4⋊Q811yescharacteristic
210M16—44no—
211M16—44no—
212M16—82no—
213M16—81no—
214M16—81no—
215C2×C8—24no—
216C2×C8—24no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—82no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42—42no—
242C42—44no—
243C42—42no—
244C42—42no—
245C42—82no—
246C42C2×C4⋊C411yescharacteristic
247C42C2×C4⋊C411yescharacteristic
248C42C4×D411yescharacteristic
249C42C4×Q811yescharacteristic
250C42#12011yescharacteristic
251C42#12011yescharacteristic
252C42#12111yescharacteristic
253C42#12111yescharacteristic
254C42C22⋊Q811yescharacteristic
255C42C22⋊Q811yescharacteristic
256C42#12511yescharacteristic
257C42C4⋊Q811yescharacteristic
258C42C4⋊Q811yescharacteristic
259C42C4⋊Q811yescharacteristic
260C42C4⋊Q811yescharacteristic
261C42C4⋊Q811yescharacteristic
262C23—22no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23#38511yescharacteristic
267C23#39811yescharacteristic
268C23#39911yescharacteristic
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—88no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#38411yescharacteristic
302C2×C4#38511yescharacteristic
303C2×C4#38511yescharacteristic
304C2×C4#38911yescharacteristic
305C2×C4#38911yescharacteristic
306C2×C4#38911yescharacteristic
307C2×C4#39211yescharacteristic
308C2×C4#39511yescharacteristic
309C2×C4#39511yescharacteristic
310C2×C4#39511yescharacteristic
311C2×C4#39811yescharacteristic
312C2×C4#39811yescharacteristic
313C2×C4#45311yescharacteristic
314C2×C4#45411yescharacteristic
315C2×C4#45511yescharacteristic
316C2×C4#45611yescharacteristic
317C8—44no—
318C8—44no—
319C8—81no—
320C8—81no—
321C8—82no—
322C8—81no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—42no—
333C22#144411yescharacteristic
334C22#185611yescharacteristic
335C22#185711yescharacteristic
336C22#209812yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—82no—
348C4—84no—
349C4#186112yes—
350C4#200611yescharacteristic
351C4#200711yescharacteristic
352C4#206011yescharacteristic
353C4#206111yescharacteristic
354C2—21no—
355C2—22no—
356C2—21no—
357C2—41no—
358C2#1133611yescharacteristic
359C2#1134111yescharacteristic
360C2#1134211yescharacteristic
subgroupG/H
centerC2×C4#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1444
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1285C4semidirect
31. #1285C4semidirect

Maximal subgroups

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

subgroupindex
#219532
#178222
#178092
#108522
#80712
#79502
#78692