Finite Group Databasebeta

← #189980

Subgroups — #189980

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2#21851C212yesmaximal
3#8727C212yesmaximal, complemented
4#8727C212yesmaximal, complemented
5#8726C212yesmaximal, complemented
6#8726C212yesmaximal, complemented
7#8044C212yesmaximal, complemented
8#8040C211yesmaximal, characteristic, complemented
9C2×#1296C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#432C2212yes—
12C22×C4⋊C8C2212yes—
13#2355—22no—
14#2354—22no—
15#2352—22no—
16#2352—22no—
17#2314C414yes—
18#2311C412yes—
19#2285C2212yes—
20#2285C2212yes—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2274C412yes—
24#1985—22no—
25#1955—22no—
26#1869—22no—
27#1869—22no—
28#1797C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1296C2211yescharacteristic
33#1296C2211yescharacteristic
34#1296C2212yes—
35#1296C2212yes—
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42C22×#41C2×C411yescharacteristic
43C23×C8—22no—
44C4×#41—22no—
45C4×#41—24no—
46C4×#41—22no—
47C2×C4×Q8D412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117C2×C412yes—
54C2×#117C2×C412yes—
55C2×#117C2×C411yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2×C412yes—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60#432—24no—
61#432—24no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×#97—22no—
75#391—22no—
76#390—22no—
77#388—24no—
78#388—24no—
79#387—24no—
80#387—24no—
81#386—24no—
82#384—22no—
83#384C2×C412yes—
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#384C2312yes—
87#382—22no—
88#382—22no—
89#382C2×C412yes—
90#380—22no—
91#380—22no—
92#380—22no—
93#380C2×C414yes—
94#380C2×C414yes—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C414yes—
100C4×C4⋊C4C2×C414yes—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107C2×#41—22no—
108C2×#41—22no—
109C2×#41—42no—
110C2×#41—42no—
111C22×Q8C22⋊C411yescharacteristic
112C22×D4C22⋊C412yes—
113C22×D4C22⋊C411yescharacteristic
114C23×C4—22no—
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—44no—
125C22×C8—82no—
126C22×C8—82no—
127C4×Q8—24no—
128C4×Q8—24no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135#117—22no—
136#117—22no—
137#117—24no—
138#117—24no—
139#117—44no—
140#117—82no—
141#117—82no—
142#117—82no—
143#117—82no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C22×C414yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C22⋊C411yescharacteristic
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42#4112yes—
179C2×C42#4114yes—
180C2×C42#4112yes—
181C2×C42C2411yescharacteristic
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—44no—
191C22⋊C8—44no—
192C22⋊C8—82no—
193C22⋊C8—82no—
194C22⋊C8—82no—
195C22⋊C8—82no—
196#97—44no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202C24C2×C22⋊C412yes—
203C24C2×C22⋊C411yescharacteristic
204#41—48no—
205#41—48no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—84no—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4#11712yes—
248C22×C4#11712yes—
249C22×C4#12314yes—
250C22×C4#12314yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D412yes—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—88no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—84no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—44no—
287C22⋊C4—84no—
288C22⋊C4—84no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—44no—
296C42—84no—
297C42—84no—
298C42#11712yes—
299C42#11712yes—
300C42#12314yes—
301C42#12314yes—
302C42C2×#4111yescharacteristic
303C42C2×#4112yes—
304C42C2×#4111yescharacteristic
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23#38112yes—
315C23#41312yes—
316C23#41312yes—
317C23C22×C22⋊C411yescharacteristic
318Q8—44no—
319D4—44no—
320D4—48no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—164no—
360C2×C4#38114yes—
361C2×C4#38112yes—
362C2×C4C2×#11711yescharacteristic
363C2×C4C2×#11711yescharacteristic
364C2×C4C2×#12312yes—
365C2×C4C2×#12312yes—
366C2×C4#57612yes—
367C2×C4#57712yes—
368C2×C4#57712yes—
369C2×C4#57812yes—
370C8—82no—
371C8—82no—
372C8—82no—
373C8—82no—
374C22—22no—
375C22—24no—
376C22—22no—
377C22—24no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22#182214yes—
386C22C2×#38111yescharacteristic
387C22C2×#41311yescharacteristic
388C22C2×#41311yescharacteristic
389C22#289712yes—
390C22#289812yes—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—24no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—82no—
399C4—82no—
400C4—84no—
401C4#309612yes—
402C4#309712yes—
403C4#309812yes—
404C4#309912yes—
405C2—24no—
406C2—82no—
407C2#1777111yescharacteristic
408C2#1777512yes—
409C2#1778011yescharacteristic
410C2C2×#182211yescharacteristic
411C2#2070312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8727C2semidirect
4. #8727C2semidirect
5. #8726C2semidirect
6. #8726C2semidirect
7. #8044C2semidirect
8. #8040C2semidirect
9. C2×#1296C2semidirect

Maximal subgroups

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

subgroupindex
#341882
#218512
#87272
#87262
#80442
#80402
C2×#12962