Finite Group Databasebeta

← #106807

Subgroups — #106807

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34048C211yesmaximal, characteristic
2#20627C211yesmaximal, characteristic
3C2×#1458C211yesmaximal, characteristic
4#8095C212yesmaximal, complemented
5#7957C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7C2×#431C2212yes—
8C2×#423C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C2×#407—21no—
11#2314—24no—
12#2313—21no—
13#2313C412yes—
14#2312—21no—
15#2312C412yes—
16#2298—24no—
17#2285C2212yes—
18#2280—21no—
19C4×#117—21no—
20#1798—22no—
21#1798C414yescomplemented
22#1728—21no—
23#1458—22no—
24#1458C414yescomplemented
25C2×#324—21no—
26C2×#323—21no—
27#1303—24no—
28#1286—24no—
29C24×C4D411yescharacteristic
30C22×M16—21no—
31C22×M16—21no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C2×#117D412yes—
40C2×#117Q812yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C22⋊C4C2×C411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42C2311yescharacteristic
47#431—22no—
48#431—22no—
49#431—22no—
50#431C2×C414yes—
51#423—22no—
52#423—22no—
53#423—22no—
54#423—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2×C412yes—
59C2×C4⋊C8C2×C412yes—
60#407—44no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×C4×C8—22no—
64#388—28no—
65#388—28no—
66#387—28no—
67#387—28no—
68#384—22no—
69#382D414yes—
70#381—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380C2×C414yes—
75#379—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4D414yes—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84#376—22no—
85C43—22no—
86#336—24no—
87#324—24no—
88#323—24no—
89C25C22⋊C411yescharacteristic
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C412yes—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C4⋊C412yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—41no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—42no—
117#117—24no—
118#117—24no—
119#117—24no—
120#117—24no—
121#117—24no—
122#117—24no—
123#117—24no—
124#117—24no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4C4212yes—
137C2×C4⋊C4C22⋊C412yes—
138C2×C4⋊C4C4⋊C412yes—
139C2×C4⋊C4C4⋊C412yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4C4212yes—
148C2×C22⋊C4C22⋊C412yes—
149C2×C22⋊C4C4⋊C412yes—
150C2×C22⋊C4C4⋊C412yes—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42C22⋊C412yes—
159C2×C42C4⋊C412yes—
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42#4112yes—
167C2×C42#4112yes—
168C4⋊C8—24no—
169C4⋊C8—24no—
170C4⋊C8—24no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173C4⋊C8—42no—
174C22⋊C8—24no—
175C22⋊C8—48no—
176C22⋊C8—42no—
177C4×C8—24no—
178C4×C8—42no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95C22⋊C418yes—
185C24—21no—
186C24—21no—
187C24—22no—
188C24—22no—
189C24—81no—
190C24—81no—
191C24#9512yes—
192C24C2×C22⋊C411yescharacteristic
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—28no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4#9512yes—
225C22×C4#9512yes—
226C22×C4#9512yes—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4C2×C4⋊C411yescharacteristic
234C22×C4#11712yes—
235C22×C4#12012yes—
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12312yes—
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#13512yes—
242C22×C4#13512yes—
243C22×C4AGL(1,ℤ/8)12yes—
244C22×C4#13712yes—
245M16—82no—
246M16—82no—
247M16—82no—
248M16—82no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—84no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—44no—
278C42—44no—
279C42—84no—
280C42—84no—
281C42C4×D412yes—
282C42C4×D412yes—
283C42C4×D412yes—
284C42C4×D412yes—
285C42#12112yes—
286C42#12112yes—
287C42C22⋊Q812yes—
288C42C22⋊Q812yes—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—81no—
301C23—81no—
302C23—84no—
303C23—84no—
304C23C2×#9511yescharacteristic
305C23#37911yescharacteristic
306C23#38111yescharacteristic
307C23#41612yes—
308C23#41712yes—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C4×C22⋊C412yes—
340C2×C4#38012yes—
341C2×C4#38512yes—
342C2×C4#38511yescharacteristic
343C2×C4#38511yescharacteristic
344C2×C4#38612yes—
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C2×C4#41611yescharacteristic
348C2×C4#41611yescharacteristic
349C2×C4#41811yescharacteristic
350C2×C4#41911yescharacteristic
351C2×C4#42712yes—
352C2×C4#42812yes—
353C2×C4#44912yes—
354C2×C4#44912yes—
355C2×C4#45212yes—
356C2×C4#45212yes—
357C2×C4#48212yes—
358C2×C4#48212yes—
359C2×C4#48512yes—
360C2×C4#48512yes—
361C8—84no—
362C8—82no—
363C8—82no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—42no—
371C22—42no—
372C22—44no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—82no—
377C22—84no—
378C22—84no—
379C22C22⋊#9511yescharacteristic
380C22#173511yescharacteristic
381C22#173611yescharacteristic
382C22#178812yes—
383C22#182512yes—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—88no—
392C4#185412yes—
393C4#185412yes—
394C4#187311yescharacteristic
395C4#187311yescharacteristic
396C4#187511yescharacteristic
397C4#187511yescharacteristic
398C2—22no—
399C2—21no—
400C2—22no—
401C2—21no—
402C2—82no—
403C2#1073111yescharacteristic
404C2#1074311yescharacteristic
405C2#1075611yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ 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
4. #8095C2semidirect
5. #7957C2semidirect
21. #1798C4semidirect
24. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#340482
#206272
C2×#14582
#80952
#79572