Finite Group Databasebeta

← #120920

Subgroups — #120920

Order 512.

485 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21953C211yesmaximal, characteristic
2#10852C211yesmaximal, characteristic
3#8092C211yesmaximal, characteristic
4#8081C211yesmaximal, characteristic
5#8077C211yesmaximal, characteristic
6#7950C211yesmaximal, characteristic
7#7949C211yesmaximal, 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#1303—22no—
22#1301—21no—
23#1301—21no—
24#1301—21no—
25#1301—21no—
26#1285—22no—
27#1285C414yescomplemented
28C22×M16—21no—
29C2×#117—21no—
30C22×C4⋊C4—21no—
31C22×C42D412yes—
32C22×C42C2311yescharacteristic
33#432—21no—
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#432—21no—
39#432C2×C412yes—
40#432C2×C412yes—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431—21no—
46#431—21no—
47#431—21no—
48#431—21no—
49#423—21no—
50#423—21no—
51#423—21no—
52#423—21no—
53#423—21no—
54#423Q812yes—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8D411yescharacteristic
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8D411yescharacteristic
63#407—21no—
64#407—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C4×M16—21no—
68C4×M16—21no—
69C2×#97—21no—
70C2×C4×C8—21no—
71C4×C4⋊C4—21no—
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4C2×C412yes—
75C4×C4⋊C4D412yes—
76C4×C4⋊C4Q811yescharacteristic
77C4×C4⋊C4Q811yescharacteristic
78C4×C22⋊C4—21no—
79C4×C22⋊C4—21no—
80#376—21no—
81C2×#95—21no—
82C43—21no—
83#339—44no—
84#336—22no—
85#336—22no—
86C23×C4—22no—
87C23×C4C2×D411yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C2×M16—22no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—42no—
95C2×M16—42no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—41no—
101#117—22no—
102#117—22no—
103#117—41no—
104#117—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—44no—
110C2×C22⋊C4—41no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—41no—
114C2×C42—42no—
115C2×C42—42no—
116C2×C42—41no—
117C2×C42C4⋊C412yes—
118C2×C42C4⋊C412yes—
119C2×C42C22×C411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×Q811yescharacteristic
124C2×C42C2×Q811yescharacteristic
125C2×C42#4111yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128C2×C42#4111yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—82no—
148C4⋊C8—82no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—41no—
153C22⋊C8—41no—
154C22⋊C8—41no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157#97—22no—
158#97—22no—
159#97—41no—
160#97—41no—
161C4×C8—41no—
162C4×C8—41no—
163C4×C8—41no—
164C4×C8—41no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—41no—
170C24#12311yescharacteristic
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4—82no—
183C22×C4C2×C4⋊C411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4#12311yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12411yescharacteristic
194C22×C4#12411yescharacteristic
195C22×C4#12511yescharacteristic
196C22×C4#12511yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4#12611yescharacteristic
199C22×C4#12611yescharacteristic
200C22×C4#12611yescharacteristic
201M16—44no—
202M16—82no—
203M16—82no—
204M16—81no—
205M16—81no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—82no—
236C42—82no—
237C42#11711yescharacteristic
238C42#11711yescharacteristic
239C42C4×D411yescharacteristic
240C42C4×Q811yescharacteristic
241C42#12111yescharacteristic
242C42#12111yescharacteristic
243C42C22⋊Q811yescharacteristic
244C42C22⋊Q811yescharacteristic
245C42#12311yescharacteristic
246C42#12311yescharacteristic
247C42#12411yescharacteristic
248C42#12411yescharacteristic
249C42#12511yescharacteristic
250C42#12511yescharacteristic
251C42#12511yescharacteristic
252C42C4⋊Q811yescharacteristic
253C23—22no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23#38511yescharacteristic
258C23#39711yescharacteristic
259C23#40011yescharacteristic
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—88no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#38211yescharacteristic
293C2×C4#38711yescharacteristic
294C2×C4#38711yescharacteristic
295C2×C4#38811yescharacteristic
296C2×C4#38811yescharacteristic
297C2×C4#38911yescharacteristic
298C2×C4#39311yescharacteristic
299C2×C4#39411yescharacteristic
300C2×C4#39811yescharacteristic
301C2×C4#39811yescharacteristic
302C2×C4#40011yescharacteristic
303C2×C4#40011yescharacteristic
304C2×C4#45311yescharacteristic
305C2×C4#45411yescharacteristic
306C2×C4#45511yescharacteristic
307C2×C4#45611yescharacteristic
308C8—82no—
309C8—82no—
310C8—82no—
311C8—81no—
312C8—81no—
313C8—81no—
314C8—81no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22#144211yescharacteristic
325C22#185611yescharacteristic
326C22#185711yescharacteristic
327C22#208012yes—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—44no—
338C4—82no—
339C4—84no—
340C4#202711yescharacteristic
341C4#202811yescharacteristic
342C4#203911yescharacteristic
343C4#204011yescharacteristic
344C4#210212yes—
345C2—21no—
346C2—22no—
347C2—21no—
348C2—41no—
349C2#1126111yescharacteristic
350C2#1128111yescharacteristic
351C2#1129611yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1442
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
27. #1285C4semidirect

Maximal subgroups

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

subgroupindex
#219532
#108522
#80922
#80812
#80772
#79502
#79492